This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

2015 Sharygin Geometry Olympiad, 2
Tags: geometry , orthocenter , angle
A circle passing through $A, B$ and the orthocenter of triangle $ABC$ meets sides $AC, BC$ at their inner points. Prove that $60^o < \angle C < 90^o$ . (A. Blinkov)

1993 Italy TST, 4
Tags: domino , tiling , graph theory , combinatorics
An $m \times n$ chessboard with $m,n \ge 2$ is given. Some dominoes are placed on the chessboard so that the following conditions are satisfied: (i) Each domino occupies two adjacent squares of the chessboard, (ii) It is not possible to put another domino onto the chessboard without overlapping, (iii) It is not possible to slide a domino horizontally or vertically without overlapping. Prove that the number of squares that are not covered by a domino is less than $\frac15 mn$.

2001 Spain Mathematical Olympiad, Problem 3
Tags: triangle inequality , geometry
You have five segments of lengths $a_1, a_2, a_3, a_4,$ and $a_5$ such that it is possible to form a triangle with any three of them. Demonstrate that at least one of those triangles has angles that are all acute.

MathLinks Contest 4th, 7.3
Tags: algebra
Let $\{f_n\}_{n \ge 0}$ be the Fibonacci sequence, given by $f_0 = f_1 = 1$, and for all positive integers $n$ the recurrence $f_{n+1} = f_n + f_{n-1}$. Let $a_n = f_{n+1}f_n$ for any non-negative integer $n$, and let $$P_n(X) = X^n + a_{n-1}X^{n-1} + ... + a_1X + a_0.$$ Prove that for all positive integers $n \ge 3$ the polynomial $P_n(X)$ is irreducible in $Z[X]$.

2012 AMC 12/AHSME, 7
Tags:
Small lights are hung on a string 6 inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of 2 red lights followed by 3 green lights. How many feet separate the third red light and the 21st red light? [b]Note:[/b] 1 foot is equal to 12 inches. $\textbf{(A)}\ 18 \qquad\textbf{(B)}\ 18.5 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 20.5 \qquad\textbf{(E)}\ 22.5 $

2015 Romania Team Selection Tests, 1
Tags: equal angles , perimeter , geometry
Let $ABC$ and $ABD$ be coplanar triangles with equal perimeters. The lines of support of the internal bisectrices of the angles $CAD$ and $CBD$ meet at $P$. Show that the angles $APC$ and $BPD$ are congruent.

2018 Peru Cono Sur TST, 6
Tags: queen , combinatorics , chessboard , chess
Let $n$ be a positive integer. In an $n \times n$ board, two opposite sides have been joined, forming a cylinder. Determine whether it is possible to place $n$ queens on the board such that no two threaten each other when: $a)\:$ $n=14$. $b)\:$ $n=15$.

2021 JHMT HS, 9
Tags: algebra , logarithm
Let $a$ and $b$ be positive real numbers such that $\log_{43}{a} = \log_{47} (3a + 4b) = \log_{2021}b^2$. Then, the value of $\tfrac{b^2}{a^2}$ can be written as $m + \sqrt{n}$, where $m$ and $n$ are integers. Find $m + n$.

2025 CMIMC Team, 2
Tags: team
We are searching for the number $7$ in the following binary tree: [center] [img] https://cdn.artofproblemsolving.com/attachments/8/c/70ad159d239e9fd8dd9775e6391965e1016f03.png [/img] [/center] We use the following algorithm (which terminates with probability $1$): [list=1] [*] Write down the number currently at the root node. [*] If we wrote down $7,$ terminate. [*] Else, pick a random edge, and swap the two numbers at the endpoints of that edge [*] Go back to step $1.$ [/list] Let $p(a)$ be the probability that we ever write down the number $a$ after running the algorithm once. Find $$p(1)+p(2)+p(3)+p(5)+p(6).$$

Kvant 2023, M2743
Tags: geometry
Perimeter of triangle $ABC$ is $1$. Circle $\omega$ touches side $BC$, continuation of side $AB$ at $P$ and continuation of side $AC$ in $Q$. Line through midpoints $AB$ and $AC$ intersects circumcircle of $APQ$ at $X$ and $Y$. Find length of $XY$.

2008 National Olympiad First Round, 28
Tags: geometry
A unit square from one of the corners of a $8\times 8$ chessboard is cut and thrown. At least how many triangles are necessary to divide the new board into triangles with equal areas? $ \textbf{(A)}\ 17 \qquad\textbf{(B)}\ 19 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ \text{None of the above} $

2017 239 Open Mathematical Olympiad, 4
Tags: polynomial , algebra
A polynomial $f(x)$ with integer coefficients is given. We define $d(a,k)=|f^k(a)-a|.$ It is known that for each integer $a$ and natural number $k$, $d(a,k)$ is positive. Prove that for all such $a,k$, $$d(a,k) \geq \frac{k}{3}.$$ ($f^k(x)=f(f^{k-1}(x)), f^0(x)=x.$)

2012 Bundeswettbewerb Mathematik, 1
Tags: number theory , algebra
given a positive integer $n$. the set $\{ 1,2,..,2n \}$ is partitioned into $a_1<a_2<...<a_n $ and $b_1>b_2>...>b_n$. find the value of : $ \sum_{i=1}^{n}|a_i - b_i| $

2018 AMC 12/AHSME, 4
Tags: geometry
A circle has a chord of length $10$, and the distance from the center of the circle to the chord is $5$. What is the area of the circle? $\textbf{(A) }25\pi\qquad\textbf{(B) }50\pi\qquad\textbf{(C) }75\pi\qquad\textbf{(D) }100\pi\qquad\textbf{(E) }125\pi$

Kvant 2021, M2653
Tags: combinatorics , combinatorial number theory
Let $p{}$ and $q{}$ be two coprime positive integers. A frog hops along the integer line so that on every hop it moves either $p{}$ units to the right or $q{}$ units to the left. Eventually, the frog returns to the initial point. Prove that for every positive integer $d{}$ with $d < p + q$ there are two numbers visited by the frog which differ just by $d{}$. [i]Nikolay Belukhov[/i]

Russian TST 2018, P4
Tags: number theory
Let $k$ be a given even positive integer. Sarah first picks a positive integer $N$ greater than $1$ and proceeds to alter it as follows: every minute, she chooses a prime divisor $p$ of the current value of $N$, and multiplies the current $N$ by $p^k -p^{-1}$ to produce the next value of $N$. Prove that there are infinitely many even positive integers $k$ such that, no matter what choices Sarah makes, her number $N$ will at some point be divisible by $2018$.

2006 China Team Selection Test, 1
Tags: polynomial , induction , modular arithmetic , pigeonhole principle , algebra
Let $k$ be an odd number that is greater than or equal to $3$. Prove that there exists a $k^{th}$-degree integer-valued polynomial with non-integer-coefficients that has the following properties: (1) $f(0)=0$ and $f(1)=1$; and. (2) There exist infinitely many positive integers $n$ so that if the following equation: \[ n= f(x_1)+\cdots+f(x_s), \] has integer solutions $x_1, x_2, \dots, x_s$, then $s \geq 2^k-1$.

2008 AMC 10, 15
Tags: algebra , difference of squares , special factorizations
How many right triangles have integer leg lengths $ a$ and $ b$ and a hypotenuse of length $ b\plus{}1$, where $ b<100$? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 10$

2019 India Regional Mathematical Olympiad, 2
Tags: geometry , construction , constructive geometry
Given a circle $\tau$, let $P$ be a point in its interior, and let $l$ be a line through $P$. Construct with proof using ruler and compass, all circles which pass through $P$, are tangent to $\tau$ and whose center lies on line $l$.

2022 Math Prize for Girls Problems, 19
Tags:
Let $S_-$ be the semicircular arc defined by \[ (x + 1)^2 + (y - \frac{3}{2})^2 = \frac{1}{4} \text{ and } x \le -1. \] Let $S_+$ be the semicircular arc defined by \[ (x - 1)^2 + (y - \frac{3}{2})^2 = \frac{1}{4} \text{ and } x \ge 1. \] Let $R$ be the locus of points $P$ such that $P$ is the intersection of two lines, one of the form $Ax + By = 1$ where $(A, B) \in S_-$ and the other of the form $Cx + Dy = 1$ where $(C, D) \in S_+$. What is the area of $R$?

2011 Tournament of Towns, 3
Tags: geometry , convex , quadrilateral , angle
In a convex quadrilateral $ABCD, AB = 10, BC = 14, CD = 11$ and $DA = 5$. Determine the angle between its diagonals.

2011 IFYM, Sozopol, 2
Tags: geometry , inscribed circle , circumcircle
On side $AB$ of $\Delta ABC$ is chosen point $M$. A circle is tangent internally to the circumcircle of $\Delta ABC$ and segments $MB$ and $MC$ in points $P$ and $Q$ respectively. Prove that the center of the inscribed circle of $\Delta ABC$ lies on line $PQ$.

2019 Azerbaijan Senior NMO, 4
Tags: plane geometry , 3-dimensional geometry , dimension
Is it possible to construct a equilateral triangle such that: $\text{a)}$ Coordinates of this triangle are integers in two dimensional plane? $\text{b)}$ Coordinates of this triangle are integers in three dimensional plane?

2008 ITest, 46
Tags: trigonometry
Let $S$ be the sum of all $x$ in the interval $[0,2\pi)$ that satisfy \[\tan^2 x - 2\tan x\sin x=0.\] Compute $\lfloor10S\rfloor$.

2016 Putnam, A6
Tags: putnam analysis
Find the smallest constant $C$ such that for every real polynomial $P(x)$ of degree $3$ that has a root in the interval $[0,1],$ \[\int_0^1|P(x)|\,dx\le C\max_{x\in[0,1]}|P(x)|.\]