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Integers | AIME I, 1993 Problem | Question 4

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1993 based on Integers.

Integer – AIME I, 1993

Find the number of four topics of integers (a,b,c,d) with 0<a<b<c<d<500 satisfy a+d=b+c and bc-ad=93.

• is 107
• is 870
• is 840
• cannot be determined from the given information

Key Concepts

Integers

Digits

Algebra

AIME I, 1993, Question 4

Elementary Algebra by Hall and Knight

Try with Hints

Let k=a+d=b+c

or, d=k-a, b=k-c,

or, (k-c)c-a(k-a)=k(c-a)-(c-a)(c+a)

=(a-c)(a+c-k)

=(c-a)(d-c)=93

(c-a)(d-c)=(1,93),(3,31),(31,3),(93,1)

solving for c

(a,b,c,d)=(c-93,c-92,c,c+1),(c-31,c-28,c,c+3),(c-1,c+92,c,c+93),(c-3,c+28,c,c+31)

taking first two solutions a<b<c<d<500

or,$1 \leq c-93, c+1 \leq 499$

or, $94 \leq c \leq 498$ gives 405 solutions

and $1 \leq c-31, c+3 \leq 499$

or, $32 \leq c \leq 496$ gives 465 solutions

or, 405+465=870 solutions.

Categories

Trapezoid Problem | AIME I, 1992 | Question 9

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1992 based on trapezoid.

Trapezoid – AIME I, 1992

Trapezoid ABCD has sides AB=92, BC=50,CD=19,AD=70 with AB parallel to CD. A circle with centre P on AB is drawn tangent to BC and AD. Given that AP=$\frac{m}{n}$, where m and n are relatively prime positive integers, find m+n.

• is 107
• is 164
• is 840
• cannot be determined from the given information

Key Concepts

Integers

Trapezoid

Angle Bisectors

AIME I, 1992, Question 9

Coordinate Geometry by Loney

Try with Hints

Let AP=y or, PB=92-y

extending AD and BC to meet at Y

and YP bisects angle AYB

Let F be point on CD where it meets

Taking angle bisector theorem,

let YB=z(92-y), YA=zy for some z

YD=zy-70, YC=z(92-y)-50

$\frac{yz-79}{z(92-y)-50}=\frac{YD}{YC}=\frac{FD}{FC}=\frac{AP}{PB}=\frac{y}{42-y}$

solving we get 120y=(70)(92)

or, AP=y=$\frac{161}{3}$

or, 161+3=164.

Categories

Problem on Fibonacci sequence | AIME I, 1988 | Question 13

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1988 based on Fibonacci sequence.

Fibonacci sequence Problem – AIME I, 1988

Find a if a and b are integers such that $x^{2}-x-1$ is a factor of $ax^{17}+bx^{16}+1$.

• is 107
• is 987
• is 840
• cannot be determined from the given information

Key Concepts

Integers

Digits

Sets

AIME I, 1988, Question 13

Elementary Number Theory by David Burton

Try with Hints

Let F(x)=$ax^{17}+bx^{16}+1$

Let P(x) be polynomial such that

$P(x)(x^{2}-x-1)=F(x)$

constant term of P(x) =(-1)

now $(x^{2}-x-1)(c_1x^{15}+c_2x^{14}+….+c_{15}x-1)$ where $c_{i}$=coefficient

comparing the coefficients of x we get the terms

since F(x) has no x term, then $c_{15}$=1

getting $c_{14}$

$(x^{2}-x-1)(c_1x^{15}+c_2x^{14}+….+c_{15}x-1)$

=terms +$0x^{2}$ +terms

or, $c_{14}=-2$

proceeding in the same way $c_{13}=3$, $c_{12}=-5$, $c_{11}=8$ gives a pattern of Fibonacci sequence

or, coefficients of P(x) are Fibonacci sequence with alternating signs

or, a=$c_1=F_{16}$ where $F_{16}$ is 16th Fibonacci number

or, a=987.

Categories

Problem on Complex plane | AIME I, 1988| Question 11

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1988 based on Complex Plane.

Problem on Complex Plane – AIME I, 1988

Let w_1,w_2,….,w_n be complex numbers. A line L in the complex plane is called a mean line for the points w_1,w_2,….w_n if L contains points (complex numbers) z_1,z_2, …..z_n such that $\sum_{k=1}^{n}(z_{k}-w_{k})=0$ for the numbers $w_1=32+170i, w_2=-7+64i, w_3=-9+200i, w_4=1+27i$ and $w_5=-14+43i$, there is a unique mean line with y-intercept 3. Find the slope of this mean line.

• is 107
• is 163
• is 634
• cannot be determined from the given information

Key Concepts

Integers

Equations

Algebra

AIME I, 1988, Question 11

Elementary Algebra by Hall and Knight

Try with Hints

$\sum_{k=1}^{5}w_k=3+504i$

and $\sum_{k-1}^{5}z_k=3+504i$

taking the numbers in the form a+bi

$\sum_{k=1}^{5}a_k=3$ and $\sum_{k=1}^{5}b_k=504$

or, y=mx+3 where $b_k=ma_k+3$ adding all 5 equations given for each k

or, 504=3m+15

or, m=163.

Categories

Problem on Real Numbers | AIME I, 1990| Question 15

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Real Numbers.

Problem on Real Numbers – AIME I, 1990

Find $ax^{5}+by^{5}$ if real numbers a,b,x,y satisfy the equations

ax+by=3

$ax^{2}+by^{2}=7$

$ax^{3}+by^{3}=16$

$ax^{4}+by^{4}=42$

• is 107
• is 20
• is 634
• cannot be determined from the given information

Key Concepts

Integers

Equations

Algebra

AIME I, 1990, Question 15

Elementary Algebra by Hall and Knight

Try with Hints

Let S=x+y, P=xy

$(ax^{n}+by^{n})(x+y)$

$=(ax^{n+1}+by^{n+1})+(xy)(ax^{n-1}+by^{n-1})$

or,$(ax^{2}+by^{2})(x+y)=(ax^{3}+by^{3})+(xy)(ax+by)$ which is first equation

or,$(ax^{3}+by^{3})(x+y)=(ax^{4}+by^{4})+(xy)(ax^{2}+by^{2})$ which is second equation

or, 7S=16+3P

16S=42+7P

or, S=-14, P=-38

or, $(ax^{4}+by^{4})(x+y)=(ax^{5}+by^{5})+(xy)(ax^{3}+by^{3})$

or, $42S=(ax^{5}+by^{5})+P(16)$

or, $42(-14)=(ax^{5}+by^{5})+(-38)(16)$

or, $ax^{5}+by^{5}=20$.

Categories

Digits and Integers | AIME I, 1990 | Question 13

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Digits and Integers.

Digits and Integers – AIME I, 1990

Let T={$9^{k}$: k is an integer, $0 \leq k \leq 4000$} given that $9^{4000}$ has 3817 digits and that its first (leftmost) digit is 9, how many elements of T have 9 as their leftmost digit?

• is 107
• is 184
• is 840
• cannot be determined from the given information

Key Concepts

Integers

Digits

Sets

AIME I, 1990, Question 13

Elementary Number Theory by David Burton

Try with Hints

here $9^{4000}$ has 3816 digits more than 9,

or, 4000-3816=184

or, 184 numbers have 9 as their leftmost digits.

Categories

Complex numbers and Sets | AIME I, 1990 | Question 10

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on Complex Numbers and Sets.

Complex Numbers and Sets – AIME I, 1990

The sets A={z:$z^{18}=1$} and B={w:$w^{48}=1$} are both sets of complex roots with unity, the set C={zw: $z \in A and w \in B$} is also a set of complex roots of unity. How many distinct elements are in C?.

• is 107
• is 144
• is 840
• cannot be determined from the given information

Key Concepts

Integers

Complex Numbers

Sets

AIME I, 1990, Question 10

Complex Numbers from A to Z by Titu Andreescue

Try with Hints

18th and 48th roots of 1 found by de Moivre’s Theorem

=$cis(\frac{2k_1\pi}{18})$ and $cis(\frac{2k_2\pi}{48})$

where $k_1$, $K_2$ are integers from 0 to 17 and 0 to 47 and $cis \theta = cos \theta +i sin \theta$

zw= $cis(\frac{k_1\pi}{9}+\frac{k_2\pi}{24})=cis(\frac{8k_1\pi+3k_2\pi}{72})$

and since the trigonometric functions are periodic every period ${2\pi}$

or, at (72)(2)=144 distinct elements in C.

Categories

Consecutive positive Integers | AIME I, 1990| Question 11

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on consecutive positive integers.

Consecutive positive integer – AIME I, 1990

Someone observed that 6!=(8)(9)(10). Find the largest positive integer n for which n! can be expressed as the product of n-3 consecutive positive integers.

• is 107
• is 23
• is 634
• cannot be determined from the given information

Key Concepts

Integers

Inequality

Algebra

AIME I, 1990, Question 11

Elementary Number Theory by David Burton

Try with Hints

The product of (n-3) consecutive integers=$\frac{(n-3+a)!}{a!}$ for a is an integer

$n!=\frac{(n-3+a)!}{a!}$ for $a \geq 3$ $(n-3+a)! \geq n!$

or, $n!=\frac{n!(n+1)(n+2)….(n-3+a)}{a!}$

for a=4, n+1=4! or, n=23 which is greatest here

n=23.

Categories

Convex polyhedron Problem | AIME I, 1988 | Question 10

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1988 based on convex polyhedron.

Convex polyhedron Problem – AIME I, 1988

A convex polyhedron has for its faces 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex of the polyhedron one square, one hexagon, and one octagon meet. How many segments joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an edge or a face?

• is 107
• is 840
• is 634
• cannot be determined from the given information

Key Concepts

Integers

Edges

Algebra

AIME I, 1988, Question 10

Geometry Revisited by Coxeter

Try with Hints

${48 \choose 2}$=1128

Every vertex lies on exactly one vertex of a square/hexagon/octagon

V=(12)(4)=(8)(6)=(6)(8)=48

each vertex is formed by the trisection of three edges and every edge is counted twice, once at each of its endpoints, the number of edges E=$\frac{3V}{2}$=72

each of the segment on face of polyhedron is diagonal of that face, so each square gives $\frac{n(n-3)}{2}=2$ diagonals, each hexagon=9,each octagon=20. The number of diagonals is $(2)(12)+(9)(8)+(20)(6)$=216

or, number of space diagonals =1128-72-216=840.

Categories

Fair coin Problem | AIME I, 1990 | Question 9

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1990 based on fair coin.

Fair Coin Problem – AIME I, 1990

A fair coin is to be tossed 10 times. Let i|j, in lowest terms, be the probability that heads never occur on consecutive tosses, find i+j.

• is 107
• is 73
• is 840
• cannot be determined from the given information

Key Concepts

Integers

Combinatorics

Algebra

AIME I, 1990, Question 9

Elementary Algebra by Hall and Knight

Try with Hints

5 tails flipped, any less,

by Pigeonhole principle there will be heads that appear on consecutive tosses

(H)T(H)T(H)T(H)T(H)T(H) 5 tails occur there are 6 slots for the heads to be placed but only 5H remaining, ${6 \choose 5}$ possible combination of 6 heads there are

$\sum_{i=6}^{11}{i \choose 11-i}$=${6 \choose 5} +{7 \choose 4}+{8 \choose 3}+{9 \choose 2} +{10 \choose 1} +{11 \choose 0}$=144

there are $2^{10}$ possible flips of 10 coins

or, probability=$\frac{144}{1024}=\frac{9}{64}$ or, 9+64=73.