Categories
AMC 8 USA Math Olympiad

Powers of Numbers AMC 8 ,2013 problem 15

[et_pb_section fb_built=”1″ _builder_version=”4.0″][et_pb_row _builder_version=”3.25″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

What are we learning ?

[/et_pb_text][et_pb_text _builder_version=”4.1″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” box_shadow_style=”preset2″]Competency in Focus: Powers of Numbers This problem from American Mathematics contest (AMC 8, 2013) is based on basic  algebra and Powers of Numbers.[/et_pb_text][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

First look at the knowledge graph.

[/et_pb_text][et_pb_image src=”https://www.cheenta.com/wp-content/uploads/2020/01/AMC-8-2013-problem-15-1.png” align=”center” force_fullwidth=”on” _builder_version=”4.1″ min_height=”298px” height=”189px” max_height=”207px”][/et_pb_image][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Next understand the problem

[/et_pb_text][et_pb_text _builder_version=”4.1″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” box_shadow_style=”preset2″]If $3^p + 3^4 = 90$$2^r + 44 = 76$, and $5^3 + 6^s = 1421$, what is the product of $p$$r$, and $s$?[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version=”4.0″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_accordion open_toggle_text_color=”#0c71c3″ _builder_version=”4.1″ toggle_font=”||||||||” body_font=”Raleway||||||||” text_orientation=”center” custom_margin=”10px||10px”][et_pb_accordion_item title=”Source of the problem” open=”on” _builder_version=”4.1″]American Mathematical Contest 2013, AMC 8 Problem 15[/et_pb_accordion_item][et_pb_accordion_item title=”Key Competency” _builder_version=”4.1″ open=”off”]Basic algebra and Powers of Numbers[/et_pb_accordion_item][et_pb_accordion_item title=”Difficulty Level” _builder_version=”4.1″ open=”off”]4/10[/et_pb_accordion_item][et_pb_accordion_item title=”Suggested Book” _builder_version=”4.0.9″ open=”off”]Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics 

[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version=”4.0.9″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_margin=”48px||48px” custom_padding=”20px|20px|0px|20px||” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Start with hints 

[/et_pb_text][et_pb_tabs _builder_version=”4.1″][et_pb_tab title=”HINT 0″ _builder_version=”4.0.9″]Do you really need a hint? Try it first![/et_pb_tab][et_pb_tab title=”HINT 1″ _builder_version=”4.1″]First, we’re going to solve for $p$Start with $3^p+3^4=90$. Then, change $3^4$ to $81$. Subtract $81$ from both sides to get $3^p=9$ .Now we can write 9 as \(3^2\) .So, from here we can say that p=2.[/et_pb_tab][et_pb_tab title=”HINT 2″ _builder_version=”4.1″]Now, solve for $r$. Since $2^r+44=76$$2^r$ must equal $32$,  and 32 can be written as \( 2^5 \) .So from here we have r=5.[/et_pb_tab][et_pb_tab title=”HINT 3″ _builder_version=”4.1″]Similarly now, solve for $s$$5^3+6^s=1421$ can be simplified to $125+6^s=1421$ which simplifies further to $6^s=1296$=\(6^4\) , which gives s=4.[/et_pb_tab][et_pb_tab title=”HINT 4″ _builder_version=”4.1″]Lastly, $prs$ equals $2*5*4$ which equals $40$. So, the answer is 40.[/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ min_height=”12px” custom_margin=”50px||50px” custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Connected Program at Cheenta

[/et_pb_text][et_pb_blurb title=”Amc 8 Master class” url=”https://www.cheenta.com/matholympiad/” url_new_window=”on” image=”https://www.cheenta.com/wp-content/uploads/2018/03/matholympiad.png” _builder_version=”4.0.9″ header_font=”||||||||” header_text_color=”#0c71c3″ header_font_size=”48px” body_font_size=”20px” body_letter_spacing=”1px” body_line_height=”1.5em” link_option_url=”https://www.cheenta.com/matholympiad/” link_option_url_new_window=”on”]

Cheenta AMC Training Camp consists of live group and one on one classes, 24/7 doubt clearing and continuous problem solving streams.[/et_pb_blurb][et_pb_button button_url=”https://www.cheenta.com/amc-8-american-mathematics-competition/” url_new_window=”on” button_text=”Learn More” button_alignment=”center” _builder_version=”4.0.9″ custom_button=”on” button_bg_color=”#0c71c3″ button_border_color=”#0c71c3″ button_border_radius=”0px” button_font=”Raleway||||||||” button_icon=”%%3%%” background_layout=”dark” button_text_shadow_style=”preset1″ box_shadow_style=”preset1″ box_shadow_color=”#0c71c3″][/et_pb_button][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_margin=”50px||50px” custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Similar Problems

[/et_pb_text][et_pb_post_nav in_same_term=”off” _builder_version=”4.0.9″][/et_pb_post_nav][et_pb_divider _builder_version=”3.22.4″ background_color=”#0c71c3″][/et_pb_divider][/et_pb_column][/et_pb_row][/et_pb_section]
Categories
AMC 8 USA Math Olympiad

Representation of numbers in base 10 AMC 8 2013 , problem number 13

[et_pb_section fb_built=”1″ _builder_version=”4.0″][et_pb_row _builder_version=”3.25″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

What are we learning ?

[/et_pb_text][et_pb_text _builder_version=”4.1″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” hover_enabled=”0″ box_shadow_style=”preset2″]Competency in Focus: Representation of numbers in base 10 This problem from American Mathematics contest (AMC 8, 2013) is a digit problem . [/et_pb_text][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

First look at the knowledge graph.

[/et_pb_text][et_pb_image src=”https://www.cheenta.com/wp-content/uploads/2020/01/AMC-8-2013-problem-13-1.png” align=”center” _builder_version=”4.1″ hover_enabled=”0″][/et_pb_image][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Next understand the problem

[/et_pb_text][et_pb_text _builder_version=”4.1″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” hover_enabled=”0″ box_shadow_style=”preset2″]When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one? [/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version=”4.0″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_accordion open_toggle_text_color=”#0c71c3″ _builder_version=”4.1″ toggle_font=”||||||||” body_font=”Raleway||||||||” text_orientation=”center” custom_margin=”10px||10px” hover_enabled=”0″][et_pb_accordion_item title=”Source of the problem” open=”off” _builder_version=”4.1″ hover_enabled=”0″]American Mathematical Contest 2013, AMC 8 Problem 13 [/et_pb_accordion_item][et_pb_accordion_item title=”Key Competency” _builder_version=”4.1″ hover_enabled=”0″ open=”on”]Representation of numbers in base 10 [/et_pb_accordion_item][et_pb_accordion_item title=”Difficulty Level” _builder_version=”4.1″ hover_enabled=”0″ open=”off”]4/10[/et_pb_accordion_item][et_pb_accordion_item title=”Suggested Book” _builder_version=”4.0.9″ open=”off”]Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics 

[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version=”4.0.9″ header_font=”Raleway|300|||||||” background_color=”#0c71c3″ min_height=”12px” custom_margin=”30px||30px||false|false” custom_padding=”20px|20px|20px|20px|false|false” border_radii=”on|5px|5px|5px|5px”]

Start with hints

[/et_pb_text][et_pb_tabs _builder_version=”4.1″ hover_enabled=”0″][et_pb_tab title=”Hint 0″ _builder_version=”4.0.9″]Do you really need a hint ? Try it first![/et_pb_tab][et_pb_tab title=”Hint 1″ _builder_version=”4.1″ hover_enabled=”0″]Let the two digits be $a$ and $b$ that is total score of Clara is ab . [/et_pb_tab][et_pb_tab title=”Hint 2″ _builder_version=”4.1″ hover_enabled=”0″]Now we know that any number in base 10 can be represented as ab=10 a+ b.  Given, when Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. So, Clara misinterpreted ab as ba . Again , ba= 10b+a . [/et_pb_tab][et_pb_tab title=”Hint 3″ _builder_version=”4.1″ hover_enabled=”0″]The difference between the two is $|9a-9b|$ which factors into $|9(a-b)|$. So, what can we say from here? [/et_pb_tab][et_pb_tab title=”Hint 4″ _builder_version=”4.1″ hover_enabled=”0″]Therefore, since the difference is a multiple of 9, the only answer choice that is a multiple of 9 that is 45. [/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ min_height=”12px” custom_margin=”50px||50px” custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Connected Program at Cheenta

[/et_pb_text][et_pb_blurb title=”Amc 8 Master class” url=”https://www.cheenta.com/matholympiad/” url_new_window=”on” image=”https://www.cheenta.com/wp-content/uploads/2018/03/matholympiad.png” _builder_version=”4.0.9″ header_font=”||||||||” header_text_color=”#0c71c3″ header_font_size=”48px” body_font_size=”20px” body_letter_spacing=”1px” body_line_height=”1.5em” link_option_url=”https://www.cheenta.com/matholympiad/” link_option_url_new_window=”on”]

Cheenta AMC Training Camp consists of live group and one on one classes, 24/7 doubt clearing and continuous problem solving streams.[/et_pb_blurb][et_pb_button button_url=”https://www.cheenta.com/amc-8-american-mathematics-competition/” url_new_window=”on” button_text=”Learn More” button_alignment=”center” _builder_version=”4.0.9″ custom_button=”on” button_bg_color=”#0c71c3″ button_border_color=”#0c71c3″ button_border_radius=”0px” button_font=”Raleway||||||||” button_icon=”%%3%%” background_layout=”dark” button_text_shadow_style=”preset1″ box_shadow_style=”preset1″ box_shadow_color=”#0c71c3″][/et_pb_button][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_margin=”50px||50px” custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Similar Problems

[/et_pb_text][et_pb_post_slider include_categories=”869″ show_meta=”off” image_placement=”left” _builder_version=”4.0.9″][/et_pb_post_slider][et_pb_post_nav in_same_term=”off” _builder_version=”4.0.9″][/et_pb_post_nav][et_pb_divider _builder_version=”3.22.4″ background_color=”#0c71c3″][/et_pb_divider][/et_pb_column][/et_pb_row][/et_pb_section]
Categories
AMC 8 USA Math Olympiad

Mean and Median calculation AMC 8, 2013 Problem 5

[et_pb_section fb_built=”1″ _builder_version=”4.0″][et_pb_row _builder_version=”4.2.2″ hover_enabled=”0″ width=”100%”][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_text _builder_version=”4.2.2″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”10px|10px|10px|10px|false|false” hover_enabled=”0″ border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″ inline_fonts=”Aclonica”]

What are we learning ?

[/et_pb_text][et_pb_text _builder_version=”4.2.2″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||20px||false|false” custom_padding=”10px|20px|10px|20px” hover_enabled=”0″ box_shadow_style=”preset2″]

Competency in Focus: Mean and Median calculation

This problem from American Mathematics Contest 8 (AMC 8, 2013) is based on calculation of mean and median. It is Question no. 5 of the AMC 8 2013 Problem series.

[/et_pb_text][et_pb_text _builder_version=”4.2.2″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”10px|10px|10px|10px|false|false” hover_enabled=”0″ border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″ inline_fonts=”Aclonica” custom_margin=”10px||10px||false|false”]

First look at the knowledge graph:-

[/et_pb_text][et_pb_image src=”https://www.cheenta.com/wp-content/uploads/2020/01/AMC-8-2013-Problem-5-1.png” alt=”calculation of mean and median- AMC 8 2013 Problem” title_text=” mean and median- AMC 8 2013 Problem” align=”center” force_fullwidth=”on” _builder_version=”4.2.2″ min_height=”429px” height=”189px” max_height=”198px” hover_enabled=”0″ custom_padding=”10px|10px|10px|10px|false|false”][/et_pb_image][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″ inline_fonts=”Aclonica”]

Next understand the problem

[/et_pb_text][et_pb_text _builder_version=”4.1″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” box_shadow_style=”preset2″]Hammie is in the $6^\text{th}$ grade and weighs 106 pounds. His quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?  [/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version=”4.0″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_accordion open_toggle_text_color=”#0c71c3″ _builder_version=”4.1″ toggle_font=”||||||||” body_font=”Raleway||||||||” text_orientation=”center” custom_margin=”10px||10px”][et_pb_accordion_item title=”Source of the problem” _builder_version=”4.1″ open=”on”]American Mathematical Contest 2013, AMC 8 Problem 5

[/et_pb_accordion_item][et_pb_accordion_item title=”Key Competency” _builder_version=”4.1″ inline_fonts=”Abhaya Libre” open=”off”]

Basic Statistics and Data Representation mainly calculation of mean and median.

[/et_pb_accordion_item][et_pb_accordion_item title=”Difficulty Level” _builder_version=”4.1″ open=”off”]4/10[/et_pb_accordion_item][et_pb_accordion_item title=”Suggested Book” open=”off” _builder_version=”4.0.9″]Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics 

[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version=”4.0.9″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_margin=”48px||48px” custom_padding=”20px|20px|0px|20px||” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″ inline_fonts=”Aclonica”]

Start with hints 

[/et_pb_text][et_pb_tabs _builder_version=”4.1″][et_pb_tab title=”HINT 0″ _builder_version=”4.0.9″]Do you really need a hint? Try it first![/et_pb_tab][et_pb_tab title=”HINT 1″ _builder_version=”4.1″]

Let us first find the median of the weight of the five children. For this, we first have to arrange the weights of the five children in increasing order. As we know, the median is the middle value, if there is an odd number of observations, and if there is an even number of observations, it is the average of the two middle values. Thus, lining up the numbers (5, 5, 6, 8, 106), we see that it  is 6 pounds.

[/et_pb_tab][et_pb_tab title=”HINT 2″ _builder_version=”4.1″]Now what we have to find is the mean of the weights of five children .The average weight of the five kids is $\dfrac{5+5+6+8+106}{5} = \dfrac{130}{5} = 26$.[/et_pb_tab][et_pb_tab title=”HINT 3″ _builder_version=”4.1″]The median here is obviously less than the mean.[/et_pb_tab][et_pb_tab title=”HINT 4″ _builder_version=”4.1″]Therefore, the average weight is bigger than median weight , by $26-6 = 20$ pounds, making the answer , average by 20.[/et_pb_tab][/et_pb_tabs][/et_pb_column][/et_pb_row][/et_pb_section][et_pb_section fb_built=”1″ fullwidth=”on” _builder_version=”4.2.2″ global_module=”50833″][et_pb_fullwidth_header title=”AMC – AIME Program” button_one_text=”Learn More” button_one_url=”https://www.cheenta.com/amc-aime-usamo-math-olympiad-program/” header_image_url=”https://www.cheenta.com/wp-content/uploads/2018/03/matholympiad.png” _builder_version=”4.2.2″ background_color=”#00457a” custom_button_one=”on” button_one_text_color=”#44580e” button_one_bg_color=”#ffffff” button_one_border_color=”#ffffff” button_one_border_radius=”5px”]

AMC – AIME – USAMO Boot Camp for brilliant students. Use our exclusive one-on-one plus group class system to prepare for Math Olympiad

[/et_pb_fullwidth_header][/et_pb_section][et_pb_section fb_built=”1″ fullwidth=”on” _builder_version=”4.2.2″ global_module=”50840″ saved_tabs=”all”][et_pb_fullwidth_post_slider _builder_version=”4.2.2″ include_categories=”879,878,869″ show_arrows=”off” show_pagination=”off” show_meta=”off” image_placement=”left” custom_margin=”20px||20px||false|false” custom_padding=”20px||20px||false|false” custom_button=”on” button_bg_color=”#ffffff” button_bg_enable_color=”on” button_text_color=”#0c71c3″ hover_enabled=”0″][/et_pb_fullwidth_post_slider][/et_pb_section]
Categories
AMC 8 USA Math Olympiad

Geometry of circles and rectangles AMC 8 2014 problem 20

[et_pb_section fb_built=”1″ _builder_version=”4.0″][et_pb_row _builder_version=”3.25″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

What are we learning ?

[/et_pb_text][et_pb_text _builder_version=”4.0.9″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” box_shadow_style=”preset2″]Competency in Focus: Geometry of circles and rectangles This problem from American Mathematics contest (AMC 8, 2014) will help us to learn more about geometry of circles and rectangles.

[/et_pb_text][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

First look at the knowledge graph.

[/et_pb_text][et_pb_image src=”https://www.cheenta.com/wp-content/uploads/2020/01/Untitled-Diagram-2.png” align=”center” force_fullwidth=”on” _builder_version=”4.1″ min_height=”166px” height=”339px” max_height=”452px”][/et_pb_image][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Next understand the problem

[/et_pb_text][et_pb_text _builder_version=”4.1″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” box_shadow_style=”preset2″]

Rectangle $ABCD$ has sides $CD=3$ and $DA=5$. A circle of radius $1$ is centered at $A$, a circle of radius $2$ is centered at $B$, and a circle of radius $3$ is centered at $C$. Which of the following is closest to the area of the region inside the rectangle but outside all three circles?[asy] draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0)); draw(Circle((0,0),1)); draw(Circle((0,3),2)); draw(Circle((5,3),3)); label("A",(0.2,0),W); label("B",(0.2,2.8),NW); label("C",(4.8,2.8),NE); label("D",(5,0),SE); label("5",(2.5,0),N); label("3",(5,1.5),E); [/asy][/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version=”4.0″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_accordion open_toggle_text_color=”#0c71c3″ _builder_version=”4.1″ toggle_font=”||||||||” body_font=”Raleway||||||||” text_orientation=”center” custom_margin=”10px||10px”][et_pb_accordion_item title=”Source of the problem” _builder_version=”4.0.9″ open=”on”]American Mathematical Contest 2014, AMC 8 Problem 20

[/et_pb_accordion_item][et_pb_accordion_item title=”Key Competency” open=”off” _builder_version=”4.1″]

Geometry of circles and rectangles [/et_pb_accordion_item][et_pb_accordion_item title=”Difficulty Level” _builder_version=”4.1″ open=”off”]6/10[/et_pb_accordion_item][et_pb_accordion_item title=”Suggested Book” _builder_version=”4.0.9″ open=”off”]Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics 

[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version=”4.1″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_margin=”48px||48px” custom_padding=”20px|20px|0px|20px||” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Start with hints 

[/et_pb_text][et_pb_tabs _builder_version=”4.1″][et_pb_tab title=”Hint 0″ _builder_version=”4.1″]Do you really need a hint? Try it first![/et_pb_tab][et_pb_tab title=”Hint 1″ _builder_version=”4.1″]The area in the rectangle but outside the circles is the area of the rectangle minus the area of all three of the quarter circles in the rectangle.[/et_pb_tab][et_pb_tab title=”Hint 2″ _builder_version=”4.1″]Here the area of the rectangle is 3.5=15. Area of quater circles is (Area of the circle )/4 = \( \frac{\pi  . r^2}{4} \) , where r= radius of the circle . so, The area of all 3 quarter circles is $\frac{\pi}{4}+\frac{\pi(2)^2}{4}+\frac{\pi(3)^2}{4} = \frac{14\pi}{4} = \frac{7\pi}{2}$, where area of the quater for circle A is \( \frac{\pi}{4} \) ,for circle B is \( \frac {\pi .2^2}{4} \) , for circle C is \( \frac{\pi.3^2}{4} \).Therefore the area in the rectangle but outside the circles is $15-\frac{7\pi}{2}$.[/et_pb_tab][et_pb_tab title=”Hint 3″ _builder_version=”4.1″]Now what can we do with  $15-\frac{7\pi}{2}$ to get an approximate value ?[/et_pb_tab][et_pb_tab title=”Hint 4 ” _builder_version=”4.1″]As we know that we can approximate \( \pi \) by \( \frac{22}{7} \) .  and substituting that in will give 15-11=4.[/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ min_height=”12px” custom_margin=”50px||50px” custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Connected Program at Cheenta

[/et_pb_text][et_pb_blurb title=”Amc 8 Master class” url=”https://www.cheenta.com/matholympiad/” url_new_window=”on” image=”https://www.cheenta.com/wp-content/uploads/2018/03/matholympiad.png” _builder_version=”4.0.9″ header_font=”||||||||” header_text_color=”#0c71c3″ header_font_size=”48px” body_font_size=”20px” body_letter_spacing=”1px” body_line_height=”1.5em” link_option_url=”https://www.cheenta.com/matholympiad/” link_option_url_new_window=”on”]

Cheenta AMC Training Camp consists of live group and one on one classes, 24/7 doubt clearing and continuous problem solving streams.[/et_pb_blurb][et_pb_button button_url=”https://www.cheenta.com/amc-8-american-mathematics-competition/” url_new_window=”on” button_text=”Learn More” button_alignment=”center” _builder_version=”4.0.9″ custom_button=”on” button_bg_color=”#0c71c3″ button_border_color=”#0c71c3″ button_border_radius=”0px” button_font=”Raleway||||||||” button_icon=”%%3%%” background_layout=”dark” button_text_shadow_style=”preset1″ box_shadow_style=”preset1″ box_shadow_color=”#0c71c3″][/et_pb_button][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_margin=”50px||50px” custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Similar Problems

[/et_pb_text][et_pb_post_nav in_same_term=”off” _builder_version=”4.0.9″][/et_pb_post_nav][et_pb_divider _builder_version=”3.22.4″ background_color=”#0c71c3″][/et_pb_divider][/et_pb_column][/et_pb_row][/et_pb_section]
Categories
AMC 8 USA Math Olympiad

Number theory AMC 8 2014 Problem Number 23

[et_pb_section fb_built=”1″ _builder_version=”4.0″][et_pb_row _builder_version=”3.25″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

What are we learning ?

[/et_pb_text][et_pb_text _builder_version=”4.0.9″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” box_shadow_style=”preset2″]Competency in Focus: Number theory This problem from American Mathematics contest (AMC 8, 2014) is based on basic knowledge about prime numbers and simple logical reasoning and number theory .[/et_pb_text][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

First look at the knowledge graph.

[/et_pb_text][et_pb_image src=”https://www.cheenta.com/wp-content/uploads/2020/01/AMC-8-2014-problem-number-23-2.png” align=”center” force_fullwidth=”on” _builder_version=”4.0.9″ hover_enabled=”0″ min_height=”516px” height=”198px” max_height=”207px”][/et_pb_image][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Next understand the problem

[/et_pb_text][et_pb_text _builder_version=”4.0.9″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” box_shadow_style=”preset2″]Three members of the Euclid Middle School girls’ softball team had the following conversation. Ashley: I just realized that our uniform numbers are all $2$-digit primes. Brittany : And the sum of your two uniform numbers is the date of my birthday earlier this month. Caitlin: That’s funny. The sum of your two uniform numbers is the date of my birthday later this month. Ashley: And the sum of your two uniform numbers is today’s date. What number does Caitlin wear?[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version=”4.0″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_accordion open_toggle_text_color=”#0c71c3″ _builder_version=”4.0.9″ toggle_font=”||||||||” body_font=”Raleway||||||||” text_orientation=”center” custom_margin=”10px||10px”][et_pb_accordion_item title=”Source of the problem” _builder_version=”4.0.9″ open=”off”]American Mathematical Contest 2014, AMC 8 Problem 23[/et_pb_accordion_item][et_pb_accordion_item title=”Key Competency” open=”on” _builder_version=”4.0.9″]Basic knowledge about prime numbers , simple logical reasoning and number theory.[/et_pb_accordion_item][et_pb_accordion_item title=”Difficulty Level” _builder_version=”4.0.9″ open=”off”]6/10[/et_pb_accordion_item][et_pb_accordion_item title=”Suggested Book” _builder_version=”4.0.9″ open=”off”]Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics 

[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version=”4.0.9″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_margin=”48px||48px” custom_padding=”20px|20px|0px|20px||” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Start with hints 

[/et_pb_text][et_pb_tabs _builder_version=”4.0.9″][et_pb_tab title=”HINT 0″ _builder_version=”4.0.9″]Do you really need a hint? Try it first![/et_pb_tab][et_pb_tab title=”HINT 1″ _builder_version=”4.0.9″]The maximum amount of days any given month can have is $31$, and the smallest two digit primes are 11,13,17. Now see that there are a few different sums that can be deduced from the following numbers, which are $24, 30,$ and $28$, all of which represent the three days. So,the uniform numbers must be 11,13,17 otherwise there different sums will be more than 31 which gives a contradiction as the maximum amount of days any given moth can have is 31.[/et_pb_tab][et_pb_tab title=”HINT 2″ _builder_version=”4.0.9″]Now we have to find which member has what uniform numbers from the given  conversation .[/et_pb_tab][et_pb_tab title=”HINT 3″ _builder_version=”4.0.9″] since Brittany says that the other two people’s uniform numbers are earlier, so that means Caitlin and Ashley’s numbers must add up to $24$. Similarly, Caitlin says that the other two people’s uniform numbers is later, so the sum must add up to $30$. This leaves $28$ as today’s date.[/et_pb_tab][et_pb_tab title=”HINT 4″ _builder_version=”4.0.9″]From this, Caitlin was referring to the uniform wearers $13$ and $17$, telling us that her number is $11$, giving our solution as 11.[/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ min_height=”12px” custom_margin=”50px||50px” custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Connected Program at Cheenta

[/et_pb_text][et_pb_blurb title=”Amc 8 Master class” url=”https://www.cheenta.com/matholympiad/” url_new_window=”on” image=”https://www.cheenta.com/wp-content/uploads/2018/03/matholympiad.png” _builder_version=”4.0.9″ header_font=”||||||||” header_text_color=”#0c71c3″ header_font_size=”48px” body_font_size=”20px” body_letter_spacing=”1px” body_line_height=”1.5em” link_option_url=”https://www.cheenta.com/matholympiad/” link_option_url_new_window=”on”]

Cheenta AMC Training Camp consists of live group and one on one classes, 24/7 doubt clearing and continuous problem solving streams.[/et_pb_blurb][et_pb_button button_url=”https://www.cheenta.com/amc-8-american-mathematics-competition/” url_new_window=”on” button_text=”Learn More” button_alignment=”center” _builder_version=”4.0.9″ custom_button=”on” button_bg_color=”#0c71c3″ button_border_color=”#0c71c3″ button_border_radius=”0px” button_font=”Raleway||||||||” button_icon=”%%3%%” background_layout=”dark” button_text_shadow_style=”preset1″ box_shadow_style=”preset1″ box_shadow_color=”#0c71c3″][/et_pb_button][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_margin=”50px||50px” custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Similar Problems

[/et_pb_text][et_pb_post_nav in_same_term=”off” _builder_version=”4.0.9″][/et_pb_post_nav][et_pb_divider _builder_version=”3.22.4″ background_color=”#0c71c3″][/et_pb_divider][/et_pb_column][/et_pb_row][/et_pb_section]
Categories
AMC 8 USA Math Olympiad

Calculating the median of observations AMC 8 2014 Problem 24

[et_pb_section fb_built=”1″ _builder_version=”4.0″][et_pb_row _builder_version=”3.25″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

What are we learning ?

[/et_pb_text][et_pb_text _builder_version=”4.1″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” hover_enabled=”0″ box_shadow_style=”preset2″]Competency in Focus: Calculating the median of even number of observations  This problem from American Mathematics contest (AMC 8, 2014) is based on maximising median of even number of observations. [/et_pb_text][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

First look at the knowledge graph.

[/et_pb_text][et_pb_image src=”https://www.cheenta.com/wp-content/uploads/2020/01/amc-8-2014-problem-number-24-1.png” align=”center” force_fullwidth=”on” _builder_version=”4.0.9″ min_height=”252px” height=”316px” max_height=”411px”][/et_pb_image][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Next understand the problem

[/et_pb_text][et_pb_text _builder_version=”4.0.9″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” box_shadow_style=”preset2″]One day the Beverage Barn sold $252$ cans of soda to $100$ customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day?[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version=”4.0″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_accordion open_toggle_text_color=”#0c71c3″ _builder_version=”4.0.9″ toggle_font=”||||||||” body_font=”Raleway||||||||” text_orientation=”center” custom_margin=”10px||10px”][et_pb_accordion_item title=”Source of the problem” open=”on” _builder_version=”4.0.9″]American Mathematical Contest 2014, AMC 8 Problem 24[/et_pb_accordion_item][et_pb_accordion_item title=”Key Competency” _builder_version=”4.0.9″ open=”off”]Calculation of median for even number of observations .[/et_pb_accordion_item][et_pb_accordion_item title=”Difficulty Level” _builder_version=”4.0.9″ open=”off”]

5/10[/et_pb_accordion_item][et_pb_accordion_item title=”Suggested Book” _builder_version=”4.0.9″ open=”off”]Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics 

[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version=”4.0.9″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_margin=”48px||48px” custom_padding=”20px|20px|0px|20px||” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Start with hints

[/et_pb_text][et_pb_tabs _builder_version=”4.0.9″][et_pb_tab title=”HINT 0″ _builder_version=”4.0.9″]Do you really need a hint? Try it first![/et_pb_tab][et_pb_tab title=”HINT 1″ _builder_version=”4.0.9″]Suppose the numbers of cans purchased by the 100 customers are listed in increasing order.Now median is the average of the 50th and 51th numbers in the ordered list.  [/et_pb_tab][et_pb_tab title=”HINT 2″ _builder_version=”4.0.9″]Now look -How can you maximise the median ?  [/et_pb_tab][et_pb_tab title=”HINT 3″ _builder_version=”4.0.9″]In order to maximize the median, we need to make the first half of the numbers as small as possible. To minimise the median ,minimise the first 49 numbers by taking them all to be 1.As given  every customer bought at least one can of soda.[/et_pb_tab][et_pb_tab title=”HINT 4″ _builder_version=”4.0.9″] To minimize the first 49, they would each have one can. Subtracting these $49$ cans from the $252$ cans gives us $203$ cans left to divide among $51$ people.If the 50th number is 4,then the sum of all 100 numbers would  at least 49+51.4=253, which is too large .If instead the 50th number is 3 and the following numbers all equal 4 ,then the sum of the 100 numbers is 49+3+50.4=252.Now it’s fine right![/et_pb_tab][et_pb_tab title=”HINT 5″ _builder_version=”4.0.9″]Thus the median is  the average of $3$ and $4$ is $3.5$. Thus our answer is 3.5.[/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ min_height=”12px” custom_margin=”50px||50px” custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Connected Program at Cheenta

[/et_pb_text][et_pb_blurb title=”Amc 8 Master class” url=”https://www.cheenta.com/matholympiad/” url_new_window=”on” image=”https://www.cheenta.com/wp-content/uploads/2018/03/matholympiad.png” _builder_version=”4.0.9″ header_font=”||||||||” header_text_color=”#0c71c3″ header_font_size=”48px” body_font_size=”20px” body_letter_spacing=”1px” body_line_height=”1.5em” link_option_url=”https://www.cheenta.com/matholympiad/” link_option_url_new_window=”on”]

Cheenta AMC Training Camp consists of live group and one on one classes, 24/7 doubt clearing and continuous problem solving streams.[/et_pb_blurb][et_pb_button button_url=”https://www.cheenta.com/amc-8-american-mathematics-competition/” url_new_window=”on” button_text=”Learn More” button_alignment=”center” _builder_version=”4.0.9″ custom_button=”on” button_bg_color=”#0c71c3″ button_border_color=”#0c71c3″ button_border_radius=”0px” button_font=”Raleway||||||||” button_icon=”%%3%%” background_layout=”dark” button_text_shadow_style=”preset1″ box_shadow_style=”preset1″ box_shadow_color=”#0c71c3″][/et_pb_button][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_margin=”50px||50px” custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Similar Problems

[/et_pb_text][et_pb_post_nav in_same_term=”off” _builder_version=”4.0.9″][/et_pb_post_nav][et_pb_divider _builder_version=”3.22.4″ background_color=”#0c71c3″][/et_pb_divider][/et_pb_column][/et_pb_row][/et_pb_section]
Categories
AMC 8 USA Math Olympiad

Geometry of circles in AMC 8 2014 problem 25

[et_pb_section fb_built=”1″ _builder_version=”4.0″][et_pb_row _builder_version=”3.25″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

What are we learning ?

[/et_pb_text][et_pb_text _builder_version=”4.0.9″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” box_shadow_style=”preset2″]Competency in Focus:Geometry of circles. This problem from American Mathematics contest (AMC 8, 2014) is based on simple counting of semicircles.[/et_pb_text][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

First look at the knowledge graph.

[/et_pb_text][et_pb_image src=”https://www.cheenta.com/wp-content/uploads/2020/01/AMC8-2014-problem-25.png” align=”center” _builder_version=”4.0.9″][/et_pb_image][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Next understand the problem

[/et_pb_text][et_pb_text _builder_version=”4.0.9″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” box_shadow_style=”preset2″]A straight one-mile stretch of highway, 40 feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at 5 miles per hour, how many hours will it take to cover the one-mile stretch?
Note: 1 mile = 5280feet amc 8 2014 problem no 25[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version=”4.0″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_accordion open_toggle_text_color=”#0c71c3″ _builder_version=”4.0.9″ toggle_font=”||||||||” body_font=”Raleway||||||||” text_orientation=”center” custom_margin=”10px||10px”][et_pb_accordion_item title=”Source of the problem” open=”on” _builder_version=”4.0.9″]American Mathematical Contest 2014, AMC 8 Problem 25[/et_pb_accordion_item][et_pb_accordion_item title=”Key Competency” _builder_version=”4.0.9″ open=”off”]Geometry of circles[/et_pb_accordion_item][et_pb_accordion_item title=”Difficulty Level” _builder_version=”4.0.9″ open=”off”]4/10[/et_pb_accordion_item][et_pb_accordion_item title=”Suggested Book” _builder_version=”4.0.9″ open=”off”]Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics 

[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version=”4.0.9″ header_font=”Raleway|300|||||||” background_color=”#0c71c3″ min_height=”12px” custom_margin=”30px||30px||false|false” custom_padding=”20px|20px|20px|20px|false|false” border_radii=”on|5px|5px|5px|5px”]

Start with hints

[/et_pb_text][et_pb_tabs _builder_version=”4.0.9″ hover_enabled=”0″][et_pb_tab title=”Hint 0″ _builder_version=”4.0.9″]Do you really need a hint ? Try it first![/et_pb_tab][et_pb_tab title=”Hint 1″ _builder_version=”4.0.9″]How many lanes the highway consists of ? 2 right! given highway is 40 feet wide .Then width of each lane will be 40/2=20 feet wide .[/et_pb_tab][et_pb_tab title=”Hint 2″ _builder_version=”4.0.9″]Look at the diagram .See that the radius of each semicircle will be 20 feet on which Robert must be riding his bike .Again see that each semicrcle covers 40 feet of highway i.e. the diameter of the semicircle .[/et_pb_tab][et_pb_tab title=”Hint 3″ _builder_version=”4.0.9″]Calculate the number of semicircles over the whole mile . Number of semicircles=(length of the highway covered in total by Robert)/(length of highway covered by each semicircle)=5280/40 [since 1 mile=5280 feet] =132.[/et_pb_tab][et_pb_tab title=”Hint 4″ _builder_version=”4.0.9″ hover_enabled=”0″]Where the semicircles full circles ,their circumference would be 2.\( \pi \) r =2.\( \pi \).20=40 \( \pi \) feet (since r=radius=20 feet). Therefore the circumference of semicircles is half that, or 20.\( \pi \) feet. [/et_pb_tab][et_pb_tab title=”Hint 5″ _builder_version=”4.0.9″ hover_enabled=”0″]Therefore over the stretch of hghway, Robert rides a total of 132.20.\( \pi \)=2640. \( \pi \) feet equivalent to \(\frac{ \pi}{2} \)  mile( since 1 mile=5280 feet) Given Robert rides at 5 miles per hour.So, time required by Robert =distance travelled/rate=(\( \frac{\pi}{2} \) miles)/(5 miles per hour)= \( \frac{\pi}{10} \)hours.   [/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ min_height=”12px” custom_margin=”50px||50px” custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Connected Program at Cheenta

[/et_pb_text][et_pb_blurb title=”Amc 8 Master class” url=”https://www.cheenta.com/matholympiad/” url_new_window=”on” image=”https://www.cheenta.com/wp-content/uploads/2018/03/matholympiad.png” _builder_version=”4.0.9″ header_font=”||||||||” header_text_color=”#0c71c3″ header_font_size=”48px” body_font_size=”20px” body_letter_spacing=”1px” body_line_height=”1.5em” link_option_url=”https://www.cheenta.com/matholympiad/” link_option_url_new_window=”on”]

Cheenta AMC Training Camp consists of live group and one on one classes, 24/7 doubt clearing and continuous problem solving streams.[/et_pb_blurb][et_pb_button button_url=”https://www.cheenta.com/amc-8-american-mathematics-competition/” url_new_window=”on” button_text=”Learn More” button_alignment=”center” _builder_version=”4.0.9″ custom_button=”on” button_bg_color=”#0c71c3″ button_border_color=”#0c71c3″ button_border_radius=”0px” button_font=”Raleway||||||||” button_icon=”%%3%%” background_layout=”dark” button_text_shadow_style=”preset1″ box_shadow_style=”preset1″ box_shadow_color=”#0c71c3″][/et_pb_button][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_margin=”50px||50px” custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Similar Problems

[/et_pb_text][et_pb_post_slider include_categories=”869″ show_meta=”off” image_placement=”left” _builder_version=”4.0.9″][/et_pb_post_slider][et_pb_post_nav in_same_term=”off” _builder_version=”4.0.9″][/et_pb_post_nav][et_pb_divider _builder_version=”3.22.4″ background_color=”#0c71c3″][/et_pb_divider][/et_pb_column][/et_pb_row][/et_pb_section]
Categories
AMC 8 USA Math Olympiad

Menalaus Theorem in AMC 8 2019

[et_pb_section fb_built=”1″ _builder_version=”4.0″][et_pb_row _builder_version=”3.25″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

What are we learning ?

[/et_pb_text][et_pb_text _builder_version=”4.0.9″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” box_shadow_style=”preset2″]Competency in Focus: Menalaus’s Theorem This problem from American Mathematics contest (AMC 8, 2019) will help us to learn more about Menalaus’s Theorem. 

[/et_pb_text][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

First look at the knowledge graph.

[/et_pb_text][et_pb_image src=”https://www.cheenta.com/wp-content/uploads/2019/12/AMC-8-2019-Problem-24.png” align=”center” force_fullwidth=”on” _builder_version=”4.0.9″][/et_pb_image][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Next understand the problem

[/et_pb_text][et_pb_text _builder_version=”4.0.9″ text_font=”Raleway||||||||” text_font_size=”20px” text_letter_spacing=”1px” text_line_height=”1.5em” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” box_shadow_style=”preset2″]In triangle ???, point ? divides side AC so that ?? ∶ ?? = 1 ∶ 2. Let ? be the midpoint of BD and ? be the point of intersection of line BC and line AE. Given that the area of ∆??? is 360, what is the area of ∆????[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version=”4.0″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_accordion open_toggle_text_color=”#0c71c3″ _builder_version=”4.0.9″ toggle_font=”||||||||” body_font=”Raleway||||||||” text_orientation=”center” custom_margin=”10px||10px”][et_pb_accordion_item title=”Source of the problem” _builder_version=”4.0.9″ open=”off”]American Mathematical Contest 2019, AMC 8 Problem 25

[/et_pb_accordion_item][et_pb_accordion_item title=”Key Competency” open=”on” _builder_version=”4.0.9″]Menalaus’s Theorem:   Given a triangle ABC, and a transversal line that crosses BC, AC, and AB at points D, E, and F respectively, with D, E, and F distinct from A, B, and C, then

$$ \displaystyle {\frac {AF}{FB}\times \frac {BD}{DC}\times \frac {CE}{EA}=-1.}$$

[/et_pb_accordion_item][et_pb_accordion_item title=”Difficulty Level” _builder_version=”4.0.9″ open=”off”]

7/10[/et_pb_accordion_item][et_pb_accordion_item title=”Suggested Book” _builder_version=”4.0.9″ open=”off”]Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics 

[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_margin=”48px||48px” custom_padding=”20px|20px|0px|20px||” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Watch video

[/et_pb_text][et_pb_video src=”https://youtu.be/1O-BxsyPw6E” _builder_version=”4.0.9″ hover_enabled=”0″][/et_pb_video][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ min_height=”12px” custom_margin=”50px||50px” custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Connected Program at Cheenta

[/et_pb_text][et_pb_blurb title=”Amc 8 Master class” url=”https://www.cheenta.com/matholympiad/” url_new_window=”on” image=”https://www.cheenta.com/wp-content/uploads/2018/03/matholympiad.png” _builder_version=”4.0.9″ header_font=”||||||||” header_text_color=”#0c71c3″ header_font_size=”48px” body_font_size=”20px” body_letter_spacing=”1px” body_line_height=”1.5em” link_option_url=”https://www.cheenta.com/matholympiad/” link_option_url_new_window=”on”]

Cheenta AMC Training Camp consists of live group and one on one classes, 24/7 doubt clearing and continuous problem solving streams.[/et_pb_blurb][et_pb_button button_url=”https://www.cheenta.com/amc-8-american-mathematics-competition/” url_new_window=”on” button_text=”Learn More” button_alignment=”center” _builder_version=”4.0.9″ custom_button=”on” button_bg_color=”#0c71c3″ button_border_color=”#0c71c3″ button_border_radius=”0px” button_font=”Raleway||||||||” button_icon=”%%3%%” background_layout=”dark” button_text_shadow_style=”preset1″ box_shadow_style=”preset1″ box_shadow_color=”#0c71c3″][/et_pb_button][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_margin=”50px||50px” custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Similar Problems

[/et_pb_text][et_pb_post_nav in_same_term=”off” _builder_version=”4.0.9″][/et_pb_post_nav][et_pb_divider _builder_version=”3.22.4″ background_color=”#0c71c3″][/et_pb_divider][/et_pb_column][/et_pb_row][/et_pb_section]
Categories
AMC 8 Math Olympiad Videos USA Math Olympiad Videos

AMC 8 2019 – Stick and Dot Method

[et_pb_section fb_built=”1″ _builder_version=”4.0″][et_pb_row _builder_version=”3.25″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Understand the problem

[/et_pb_text][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway||||||||” background_color=”#f4f4f4″ custom_margin=”10px||10px” custom_padding=”10px|20px|10px|20px” box_shadow_style=”preset2″]Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?

[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version=”4.0″][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” custom_padding__hover=”|||”][et_pb_accordion open_toggle_text_color=”#0c71c3″ _builder_version=”4.0″ toggle_font=”||||||||” body_font=”Raleway||||||||” text_orientation=”center” custom_margin=”10px||10px”][et_pb_accordion_item title=”Source of the problem” open=”on” _builder_version=”4.0.9″]American Mathematical Contest 2019, AMC 8 Problem 25

[/et_pb_accordion_item][et_pb_accordion_item title=”Topic” _builder_version=”4.0.9″ open=”off”]Combinatorics 

[/et_pb_accordion_item][et_pb_accordion_item title=”Difficulty Level” _builder_version=”4.0.9″ open=”off”]

7/10[/et_pb_accordion_item][et_pb_accordion_item title=”Suggested Book” _builder_version=”4.0.9″ open=”off”]Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics 

[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version=”4.0.9″ text_font_size=”30px” text_font=”Raleway||||||||” text_letter_spacing=”2px” text_text_shadow_style=”preset5″ hover_enabled=”0″]

Competency map leading to the problem [/et_pb_text][et_pb_image src=”https://www.cheenta.com/wp-content/uploads/2019/12/Concept-map-for-AMC-.png” _builder_version=”4.0.9″ hover_enabled=”0″ align=”center” custom_margin=”20px||20px||false|false” background_color=”#474747″][/et_pb_image][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_margin=”48px||48px” custom_padding=”20px|20px|0px|20px||” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Watch video

[/et_pb_text][et_pb_video src=”https://youtu.be/M9_p8E_7Vt8″ _builder_version=”4.0.9″ hover_enabled=”0″][/et_pb_video][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ min_height=”12px” custom_margin=”50px||50px” custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Connected Program at Cheenta

[/et_pb_text][et_pb_blurb title=”Amc 8 Master class” url=”https://www.cheenta.com/matholympiad/” url_new_window=”on” image=”https://www.cheenta.com/wp-content/uploads/2018/03/matholympiad.png” _builder_version=”4.0.9″ header_font=”||||||||” header_text_color=”#0c71c3″ header_font_size=”48px” body_font_size=”20px” body_letter_spacing=”1px” body_line_height=”1.5em” link_option_url=”https://www.cheenta.com/matholympiad/” link_option_url_new_window=”on”]

Cheenta AMC Training Camp consists of live group and one on one classes, 24/7 doubt clearing and continuous problem solving streams.[/et_pb_blurb][et_pb_button button_url=”https://www.cheenta.com/amc-8-american-mathematics-competition/” url_new_window=”on” button_text=”Learn More” button_alignment=”center” _builder_version=”4.0.9″ custom_button=”on” button_bg_color=”#0c71c3″ button_border_color=”#0c71c3″ button_border_radius=”0px” button_font=”Raleway||||||||” button_icon=”%%3%%” background_layout=”dark” button_text_shadow_style=”preset1″ box_shadow_style=”preset1″ box_shadow_color=”#0c71c3″][/et_pb_button][et_pb_text _builder_version=”3.27.4″ text_font=”Raleway|300|||||||” text_text_color=”#ffffff” header_font=”Raleway|300|||||||” header_text_color=”#e2e2e2″ background_color=”#0c71c3″ custom_margin=”50px||50px” custom_padding=”20px|20px|20px|20px” border_radii=”on|5px|5px|5px|5px” box_shadow_style=”preset3″]

Similar Problems

[/et_pb_text][et_pb_post_nav _builder_version=”4.0.9″ in_same_term=”off” hover_enabled=”0″][/et_pb_post_nav][et_pb_divider _builder_version=”3.22.4″ background_color=”#0c71c3″][/et_pb_divider][/et_pb_column][/et_pb_row][/et_pb_section]