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

2023 AMC 10, 21
Tags: probability
Each of $2023$ balls is placed in on of $3$ bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls? $\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } \frac{3}{10} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{1}{3} \qquad \textbf{(E) } \frac{1}{4}$

MOAA Team Rounds, 2021.2
Tags: team
Four students Alice, Bob, Charlie, and Diana want to arrange themselves in a line such that Alice is at either end of the line, i.e., she is not in between two students. In how many ways can the students do this? [i]Proposed by Nathan Xiong[/i]

2015 Peru Cono Sur TST, P2
Tags:
Let $a, b, c$ and $d$ be elements of the set $\{ 1, 2, 3,\ldots , 2014, 2015 \}$ such that $a < b < c < d$, $a + b$ is a divisor of $c + d$, and $a + c$ is a divisor of $b + d$. Determine the largest value that $a$ can take.

LMT Team Rounds 2021+, 1
Tags: number theory , algebra
Given the following system of equations: $$\begin{cases} R I +G +SP = 50 \\ R I +T + M = 63 \\ G +T +SP = 25 \\ SP + M = 13 \\ M +R I = 48 \\ N = 1 \end{cases}$$ Find the value of L that makes $LMT +SPR I NG = 2023$ true.

2009 Bundeswettbewerb Mathematik, 3
Tags: geometry , concurrency , symmetric , midpoint
Let $P$ be a point inside the triangle $ABC$ and $P_a, P_b ,P_c$ be the symmetric points wrt the midpoints of the sides $BC, CA,AB$ respectively. Prove that that the lines $AP_a, BP_b$ and $CP_c$ are concurrent.

2024 Girls in Mathematics Tournament, 3
Tags: geometry , sharky devil point
In a triangle scalene $ABC$, let $I$ be its incenter and $D$ the intersection of $AI$ and $BC$. Let $M$ and $N$ points where the incircle touches $AB$ and $AC$, respectively. Let $F$ be the second intersection of the circumcircle $(AMN)$ with the circumcircle $(ABC)$. Let $T$ the intersection of $AF$ and $BC$. Let $J$ be the intersection of $TI$ with the line parallel of $FI$ that passes through $D$. Prove that the line $AJ$ is perpendicular to $BC$.

2012 Argentina National Olympiad Level 2, 4
Tags: combinatorics
Given $2012$ stones divided into several groups, a [i]legal move[/i] is to merge two of the groups into one, as long as the size of the new group is less than or equal to $51$. Two players, $A$ and $B$, take turns making legal moves, starting with $A$. Initially, each stone is in a separate group. The player who cannot make a legal move on their turn loses. Determine which of the two players has a winning strategy and provide that strategy.

1999 National Olympiad First Round, 24
Tags: algebra , polynomial
Polynomial $ f\left(x\right)$ satisfies $ \left(x \minus{} 1\right)f\left(x \plus{} 1\right) \minus{} \left(x \plus{} 2\right)f\left(x\right) \equal{} 0$ for every $ x\in \Re$. If $ f\left(2\right) \equal{} 6$, $ f\left({\tfrac{3}{2}} \right) \equal{} ?$ $\textbf{(A)}\ -6 \qquad\textbf{(B)}\ 0 \qquad\textbf{(C)}\ \frac {3}{2} \qquad\textbf{(D)}\ \frac {15}{8} \qquad\textbf{(E)}\ \text{None}$

2008 Purple Comet Problems, 3
Tags:
Find the least integer $n$ greater than $345$ such that $\frac{3n+4}{5}, \frac{4n+5}{3},$ and $\frac{5n+3}{4}$ are all integers.

2011 Mathcenter Contest + Longlist, 6 sl8
Tags: inequalities , geometric inequality
Let $x,y,z$ represent the side lengths of any triangle, and $s=\dfrac{x+y+z}{2}$ and the area of this triangle be $\sqrt{s}$ square units. Prove that $$s\Big(\frac{1}{x(s-x)^2}+\frac{1}{y(s-y)^2}+\frac{1}{z(s-z)^ 2} \Big)\ge \frac{1}{2} \Big(\frac{1}{s-x}+\frac{1}{s-y}+\frac{1}{s-z}\Big)$$ [i](Zhuge Liang)[/i]

2024 Portugal MO, 1
Tags: number theory
A number is called cool if the sum of its digits is multiple of $17$ and the sum of digits of its successor is multiple of $17$. What is the smallest cool number?

2024 HMNT, 25
Tags: guts
Let $ABC$ be an equilateral triangle. A regular hexagon $PXQYRZ$ of side length $2$ is placed so that $P, Q,$ and $R$ lie on segments $\overline{BC}, \overline{CA},$ and $\overline{AB}$, respectively. If points $A, X,$ and $Y$ are collinear, compute $BC.$

2022 ELMO Revenge, 1
Tags: algebra
In terms of $p$ and $k$, compute the number of solutions in positive integers to the equation $ab+bc+ca=p^{2k}$ satisfying $a\leq b\leq c$ where $p$ is a fixed prime and $k$ is a fixed positive integer. [i]Proposed by Alexander Wang[/i]

2023 LMT Spring, 6
Tags: combinatorics
Aidan, Boyan, Calvin, Derek, Ephram, and Fanalex are all chamber musicians at a concert. In each performance, any combination of musicians of them can perform for all the others to watch. What is the minimum number of performances necessary to ensure that each musician watches every other musician play?

2021 Estonia Team Selection Test, 2
Tags: polynomial , algebra
Find all polynomials $P(x, y)$ with real coefficients which for all real numbers $x$ and $y$ satisfy $P(x + y, x - y) = 2P(x, y)$.

1971 IMO Longlists, 39
Tags: geometry , midpoint , congruent triangles , collinear
Two congruent equilateral triangles $ABC$ and $A'B'C'$ in the plane are given. Show that the midpoints of the segments $AA',BB', CC'$ either are collinear or form an equilateral triangle.

2015 HMIC, 3
Tags:
Let $M$ be a $2014\times 2014$ invertible matrix, and let $\mathcal{F}(M)$ denote the set of matrices whose rows are a permutation of the rows of $M$. Find the number of matrices $F\in\mathcal{F}(M)$ such that $\det(M + F) \ne 0$.

2005 Romania National Olympiad, 1
Tags: matrix , linear algebra
Let $n\geq 2$ a fixed integer. We shall call a $n\times n$ matrix $A$ with rational elements a [i]radical[/i] matrix if there exist an infinity of positive integers $k$, such that the equation $X^k=A$ has solutions in the set of $n\times n$ matrices with rational elements. a) Prove that if $A$ is a radical matrix then $\det A \in \{-1,0,1\}$ and there exists an infinity of radical matrices with determinant 1; b) Prove that there exist an infinity of matrices that are not radical and have determinant 0, and also an infinity of matrices that are not radical and have determinant 1. [i]After an idea of Harazi[/i]

2015 Polish MO Finals, 1
Tags: geometry
In triangle $ABC$ the angle $\angle A$ is the smallest. Points $D, E$ lie on sides $AB, AC$ so that $\angle CBE=\angle DCB=\angle BAC$. Prove that the midpoints of $AB, AC, BE, CD$ lie on one circle.

2007 Rioplatense Mathematical Olympiad, Level 3, 2
Tags: geometry , circumcircle , incenter
Let $ABC$ be a triangle with incenter $I$ . The circle of center $I$ which passes through $B$ intersects $AC$ at points $E$ and $F$, with $E$ and $F$ between $A $ and $C$ and different from each other. The circle circumscribed to triangle $IEF$ intersects segments $EB$ and $FB$ at $Q$ and $R$, respectively. Line $QR$ intersects the sides $A B$ and $B C$ at $P$ and $S$, respectively. If $a , b$ and $c$ are the measures of the sides $B C, CA$ and $A B$, respectively, calculate the measurements of $B P$ and $B S$.

2018 Thailand TSTST, 2
Tags: line , construction , combinatorics
$9$ horizontal and $9$ vertical lines are drawn through a square. Prove that it is possible to select $20$ rectangles so that the sides of each rectangle is a segment of one of the given lines (including the sides of the square), and for any two of the $20$ rectangles, it is possible to cover one of them with the other (rotations are allowed).

2010 Today's Calculation Of Integral, 606
Tags: geometry , calculus , integration , calculus computations
Find the area of the part bounded by two curves $y=\sqrt{x},\ \sqrt{x}+\sqrt{y}=1$ and the $x$-axis. 1956 Tokyo Institute of Technology entrance exam

2016 Harvard-MIT Mathematics Tournament, 35
Tags:
$\textbf{(Maximal Determinant)}$ In a $17 \times 17$ matrix $M$, all entries are $\pm 1$. The maximum possible value of $\left| \det M \right|$ is $N$. Estimate $N$. An estimate of $E > 0$ earns $\left\lfloor 20\min(N/E, E/N)^2 \right\rfloor$ points.

1974 Poland - Second Round, 3
Tags: angle bisector , geometry , tetrahedron , 3d geometry
Prove that the orthogonal projections of the vertex $ D $ of the tetrahedron $ ABCD $ onto the bisectors of the internal and external dihedral angles at the edges $ \overline{AB} $, $ \overline{BC} $ and $ \overline{CA} $ belong to one plane .

1983 Dutch Mathematical Olympiad, 4
Tags: combinatorics
Within an equilateral triangle of side $ 15$ are $ 111$ points. Prove that it is always possible to cover three of these points by a round coin of diameter $ \sqrt{3}$, part of which may lie outside the triangle.