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## Median of numbers | AMC-10A, 2020 | Problem 11

Try this beautiful problem from Geometry based on Median of numbers from AMC 10A, 2020.

## Median of numbers – AMC-10A, 2020- Problem 11

What is the median of the following list of $4040$ numbers$?$

$1,2,3,…….2020,1^2,2^2,3^2………..{2020}^2$

• $1989.5$
• $1976.5$
• $1972.5$

### Key Concepts

Median

Algebra

square numbers

Answer: $1976.5$

AMC-10A (2020) Problem 11

Pre College Mathematics

## Try with Hints

To find the median we need to know how many terms are there and the position of the numbers .here two types of numbers, first nonsquare i.e (1,2,3……2020) and squares numbers i.e $(1^2,2^2,3^2……2020^2)$.so We want to know the $2020$th term and the $2021$st term to get the median.

Can you now finish the problem ……….

Now less than 2020 the square number is ${44}^2$=1936 and if we take ${45}^2$=2025 which is greater than 2020.therefore we take the term that $1,2,3…2020$ trms + 44 terms=$2064$ terms.

can you finish the problem……..

since $44^{2}$ is $44+45=89$ less than $45^{2}=2025$ and 84 less than 2020 we will only need to consider the perfect square terms going down from the 2064 th term, 2020, after going down $84$ terms. Since the $2020$th and $2021$st terms are only $44$ and $43$ terms away from the $2064$th term, we can simply subtract $44$ from $2020$ and $43$ from $2020$ to get the two terms, which are $1976$ and $1977$. Averaging the two,=$1976.5$

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## Area of Triangle Problem | AMC-8, 2019 | Problem 21

Try this beautiful problem from Geometry based on the area of the triangle.

## Area of Triangle – AMC-8, 2019- Problem 21

What is the area of the triangle formed by the lines $y=5$, $y=1+x$, and $y=1-x$

• $8$
• $16$
• $15$

### Key Concepts

Geometry

Triangle

Linear equation

Answer: $16$

AMC-8 (2019) Problem 21

Pre College Mathematics

## Try with Hints

Find the three vertex of the triangle

Can you now finish the problem ……….

The area of the Triangle =$\frac{1}{2} \times \{x_1(y_2 – y_3)+x_2(y+3 -y_1)+x_3(y_1 -y_2)\}$

can you finish the problem……..

Solving two The lines y=5 and y=1+x are intersect at (4,5)=$(x_1,y_1)$(say)

Solving two The lines y=5 and y=1-x are intersect at (-4,5)=$(x_2,y_2)$(say)

Solving two The lines y=1-x and y=1+x are intersect at (0,1)=$(x_1,y_1)$(say)

Then the area of Triangle =$\frac{1}{2} \times\{ x_1(y_2 – y_3)+x_2(y+3 -y_1)+x_3(y_1 -y_2)\}$

= $\frac{1}{2} \times \{4(5-1)+(-4)(1-5)+0(5-5)=\frac{1}{2} (16+16)=16\}$

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# 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: Linear Equation This problem from American Mathematics contest (AMC 8, 2010 problem 21) is based on Linear Equation[/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″]

# 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″]Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the first day, Hui read $\frac{1}{5}$ of the pages plus 12 more, and on the second day she read 1/4 of the remaining pages plus 15 pages. On the third day she read 1/3 of the remaining pages plus 18 pages. She then realized that there were only 62 pages left to read, which she read the next day. How many pages are in this book? (A) 120  (B) 180  (C) 240  (D) 300  (E) 240[/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 2010, AMC 8 Problem 21 [/et_pb_accordion_item][et_pb_accordion_item title=”Key Competency” _builder_version=”4.1″ open=”off”]This number theory problem is based on the concept of linear equation[/et_pb_accordion_item][et_pb_accordion_item title=”Difficulty Level” _builder_version=”4.1″ open=”off”]5/10[/et_pb_accordion_item][et_pb_accordion_item title=”Suggested Book” _builder_version=”4.1″ open=”off”]Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics