Olympiad problems

2012 AIME Problems, 12

Tags: trigonometry , ratio , LaTeX , AMC , AIME , number theory , relatively prime

Let $\triangle ABC$ be a right triangle with right angle at $C$. Let $D$ and $E$ be points on $\overline{AB}$ with $D$ between $A$ and $E$ such that $\overline{CD}$ and $\overline{CE}$ trisect $\angle C$. If $\frac{DE}{BE} = \frac{8}{15}$, then $\tan B$ can be written as $\frac{m\sqrt{p}}{n}$, where $m$ and $n$ are relatively prime positive integers, and $p$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.

1999 AIME Problems, 2

Tags: geometry , parallelogram , analytic geometry , graphing lines , slope , AMC , AIME

Consider the parallelogram with vertices $(10,45),$ $(10,114),$ $(28,153),$ and $(28,84).$ A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

2003 AIME Problems, 8

Tags: ratio , modular arithmetic , AMC , AIME , arithmetic sequence , geometric sequence

In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by 30. Find the sum of the four terms.

2013 AIME Problems, 12

Tags: algebra , polynomial , trigonometry , AMC , AIME , Vieta , complex numbers

Let $S$ be the set of all polynomials of the form $z^3+az^2+bz+c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $\left\lvert z \right\rvert = 20$ or $\left\lvert z \right\rvert = 13$.

1991 AIME Problems, 11

Tags: geometry , trigonometry , AMC , AIME

Twelve congruent disks are placed on a circle $C$ of radius 1 in such a way that the twelve disks cover $C$, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of the twelve disks can be written in the from $\pi(a-b\sqrt{c})$, where $a,b,c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a+b+c$. [asy] real r=2-sqrt(3); draw(Circle(origin, 1)); int i; for(i=0; i<12; i=i+1) { draw(Circle(dir(30*i), r)); dot(dir(30*i)); } draw(origin--(1,0)--dir(30)--cycle); label("1", (0.5,0), S);[/asy]

2006 AIME Problems, 3

Tags: integration , AMC , AIME , AIME II , AIME I , number theory

Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is $\frac{1}{29}$ of the original integer.

2023 AIME, 7

Tags: AMC , AIME , AIME II

Each vertex of a regular dodecagon (12-gon) is to be colored either red or blue, and thus there are $2^{12}$ possible colorings. Find the number of these colorings with the property that no four vertices colored the same color are the four vertices of a rectangle.

1990 AIME Problems, 6

Tags: ratio , AMC , AIME , percent

A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on May 1, she assumes that $25\%$ of these fish are no longer in the lake on September 1 (because of death and emigrations), that $40\%$ of the fish were not in the lake May 1 (because of births and immigrations), and that the number of untagged fish and tagged fish in the September 1 sample are representative of the total population. What does the biologist calculate for the number of fish in the lake on May 1?

2012 AIME Problems, 11

Tags: function , AMC , AIME , number theory , relatively prime , AIME II , nice words

Let $f_1(x) = \frac{2}{3}-\frac{3}{3x+1}$, and for $n \ge 2$, define $f_n(x) = f_1(f_{n-1} (x))$. The value of x that satisfies $f_{1001}(x) = x - 3$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2000 AIME Problems, 13

Tags: geometry , calculus , AMC , AIME , number theory , relatively prime , AIME I

In the middle of a vast prairie, a firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at $50$ miles per hour along the highways and at $14$ miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within six minutes. The area of this region is $m/n$ square miles, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

2020 CHMMC Winter (2020-21), 7

Tags: combinatorics , AIME

Given $10$ points on a plane such that no three are collinear, we connect each pair of points with a segment and color each segment either red or blue. Assume that there exists some point $A$ among the $10$ points such that: 1. There is an odd number of red segments connected to $A$} 2. The number of red segments connected to each of the other points are all different Find the number of red triangles (i.e, a triangle whose three sides are all red segments) on the plane.

2016 AIME Problems, 13

Tags: 2016 AIME I , AIME

Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line $y = 24$. A fence is located at the horizontal line $y = 0$. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point where $y=0$, with equal likelihoods he chooses one of three directions where he either jumps parallel to the fence or jumps away from the fence, but he never chooses the direction that would have him cross over the fence to where $y < 0$. Freddy starts his search at the point $(0, 21)$ and will stop once he reaches a point on the river. Find the expected number of jumps it will take Freddy to reach the river.

2009 AIME Problems, 9

Tags: AMC , AIME

Let $ m$ be the number of solutions in positive integers to the equation $ 4x\plus{}3y\plus{}2z\equal{}2009$, and let $ n$ be the number of solutions in positive integers to the equation $ 4x\plus{}3y\plus{}2z\equal{}2000$. Find the remainder when $ m\minus{}n$ is divided by $ 1000$.

2019 AIME Problems, 12

Tags: geometry , AMC , AIME , AIME I , complex numbers , 2019 AIME I , 2019 AMC

Given $f(z) = z^2-19z$, there are complex numbers $z$ with the property that $z$, $f(z)$, and $f(f(z))$ are the vertices of a right triangle in the complex plane with a right angle at $f(z)$. There are positive integers $m$ and $n$ such that one such value of $z$ is $m+\sqrt{n}+11i$. Find $m+n$.

2024 AIME, 14

Tags: AMC , AIME , AIME 1 , 3B1B , puzzle

Let $ABCD$ be a tetrahedron such that $AB = CD = \sqrt{41}$, $AC = BD = \sqrt{80}$, and $BC = AD = \sqrt{89}$. There exists a point $I$ inside the tetrahedron such that the distances from $I$ to each of the faces of the tetrahedron are all equal. This distance can be written in the form $\frac{m \sqrt{n}}{p}$, when $m$, $n$, and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+p$.

2018 AIME Problems, 2

Tags: abstract algebra , AMC , AIME , AIME II

Let $a_0 = 2$, $a_1 = 5$, and $a_2 = 8$, and for $n>2$ define $a_n$ recursively to be the remainder when $4(a_{n-1} + a_{n-2} + a_{n-3})$ is divided by $11$. Find $a_{2018}\cdot a_{2020}\cdot a_{2022}$.

2019 AIME Problems, 11

Tags: 2019 AIME II , AIME , AIME II

Triangle $ABC$ has side lengths $AB=7, BC=8, $ and $CA=9.$ Circle $\omega_1$ passes through $B$ and is tangent to line $AC$ at $A.$ Circle $\omega_2$ passes through $C$ and is tangent to line $AB$ at $A.$ Let $K$ be the intersection of circles $\omega_1$ and $\omega_2$ not equal to $A.$ Then $AK=\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

2013 AIME Problems, 2

Tags: logarithms , AMC , algebra , AIME

Positive integers $a$ and $b$ satisfy the condition \[\log_2(\log_{2^a}(\log_{2^b}(2^{1000})))=0.\] Find the sum of all possible values of $a+b$.

2014 AIME Problems, 15

Tags: function , induction , number theory , prime factorization , AMC , 2014 AIME II , AIME

For any integer $k\ge1$, let $p(k)$ be the smallest prime which does not divide $k$. Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k)>2$, and $X(k)=1$ if $p(k)=2$. Let $\{x_n\}$ be the sequence defined by $x_0=1$, and $x_{n+1}X(x_n)=x_np(x_n)$ for $n\ge0$. Find the smallest positive integer, $t$ such that $x_t=2090$.

2014 AIME Problems, 7

Tags: trigonometry , logarithms , AMC , sums , AIME , AIME II , 2014 AIME II

Let $f(x) = (x^2+3x+2)^{\cos(\pi x)}$. Find the sum of all positive integers $n$ for which \[\left| \sum_{k=1}^n \log_{10} f(k) \right| = 1.\]

2017 AIME Problems, 2

Tags: AMC , AIME , 2017 AIME I

When each of 702, 787, and 855 is divided by the positive integer $m$, the remainder is always the positive integer $r$. When each of 412, 722, and 815 is divided by the positive integer $n$, the remainder is always the positive integer $s \neq r$. Fine $m+n+r+s$.

1983 AIME Problems, 9

Tags: trigonometry , inequalities , calculus , AIME

Find the minimum value of \[\frac{9x^2 \sin^2 x + 4}{x \sin x}\] for $0 < x < \pi$.

1987 AIME Problems, 3

Tags: AMC , AIME , calculus , integration , geometry , 3D geometry , tetrahedron

By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called "nice" if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers?

2016 AIME Problems, 4

Tags: 2016 AIME I , AIME , 3D geometry , geometry , prism , pyramid

A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain $A$ measures $60^\circ$. Find $h^2$.

2025 AIME, 14

Tags: AIME , 2025 AIME II , geometry , quadrilateral , 2025 AMC , AMC

Let ${\triangle ABC}$ be a right triangle with $\angle A = 90^\circ$ and $BC = 38.$ There exist points $K$ and $L$ inside the triangle such \[AK = AL = BK = CL = KL = 14.\] The area of the quadrilateral $BKLC$ can be expressed as $n\sqrt3$ for some positive integer $n.$ Find $n.$