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

1972 Miklós Schweitzer, 4
Tags: superior algebra , abstract algebra , group theory , search
Let $ G$ be a solvable torsion group in which every Abelian subgroup is finitely generated. Prove that $ G$ is finite. [i]J. Pelikan[/i]

2007 Harvard-MIT Mathematics Tournament, 10
Tags: calculus , integration
Compute \[\int_0^\infty \dfrac{e^{-x}\sin(x)}{x}dx\]

2022 JBMO Shortlist, C6
Tags: combinatorics , board , shortlist
Let $n \ge 2$ be an integer. In each cell of a $4n \times 4n$ table we write the sum of the cell row index and the cell column index. Initially, no cell is colored. A move consists of choosing two cells which are not colored and coloring one of them in red and one of them in blue. Show that, however Alex perfors $n^2$ moves, Jane can afterwards perform a number of moves (eventually none) after which the sum of the numbers written in the red cells is the same as the sum of the numbers written in the blue ones.

2014 PUMaC Geometry A, 3
Tags: geometry , circumcircle , trigonometry , perpendicular bisector
Let $O$ be the circumcenter of triangle $ABC$ with circumradius $15$. Let $G$ be the centroid of $ABC$ and let $M$ be the midpoint of $BC$. If $BC=18$ and $\angle MOA=150^\circ$, find the area of $OMG$.

2000 Bundeswettbewerb Mathematik, 2
Tags: number theory
Prove that for every integer $n \geq 2$ there exist $n$ different positive integers such that for any two of these integers $a$ and $b$ their sum $a+b$ is divisible by their difference $a - b.$

2005 Taiwan TST Round 1, 2
Tags: 3d geometry , tetrahedron , geometry
Show that for any tetrahedron, the condition that opposite edges have the same length is equivalent to the condition that the segment drawn between the midpoints of any two opposite edges is perpendicular to the two edges.

2017 Peru Iberoamerican Team Selection Test, P6
Tags: number theory
For each positive integer $k$, let $S(k)$ be the sum of the digits of $k$ in the decimal system. Prove that there exists a positive integer $k$, which does not have the digit $9$ in its decimal representation, such that: $$S(2^{24^{2017}}k)=S(k)$$

2017 Miklós Schweitzer, 9
Tags: normed vector space , functional analysis , analysis , advanced fields
Let $N$ be a normed linear space with a dense linear subspace $M$. Prove that if $L_1,\ldots,L_m$ are continuous linear functionals on $N$, then for all $x\in N$ there exists a sequence $(y_n)$ in $M$ converging to $x$ satisfying $L_j(y_n)=L_j(x)$ for all $j=1,\ldots,m$ and $n\in \mathbb{N}$.

2024 Argentina Cono Sur TST, 4
Tags: algebra
Determine the least possible value of $\dfrac{(x^2+1)(4y^2+1)(9z^2+1)}{6xyz}$ if $x$, $y$ and $z$ are positive real numbers.

2000 Portugal MO, 1
Tags: algebra , combinatorics
Consider the following table where initially all squares contain zeros: $ \begin{tabular}{ | l | c | r| } \hline 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ \hline \end{tabular} $ To change the table, the following operation is allowed: a $2 \times 2$ square formed by adjacent squares is chosen, and a unit is added to all its numbers. Complete the following table, knowing that it was obtained by a sequence of permitted operations $ \begin{tabular}{ | l | c | r| } \hline 14 & & \\ \hline 19 & 36 & \\ \hline & 16 & \\ \hline \end{tabular} $

1984 Swedish Mathematical Competition, 6
Tags: sum , number theory , combinatorics
Assume $a_1,a_2,...,a_{14}$ are positive integers such that $\sum_{i=1}^{14}3^{a_i} = 6558$. Prove that the numbers $a_1,a_2,...,a_{14}$ consist of the numbers $1,...,7$, each taken twice.

1973 Kurschak Competition, 3
Tags: combinatorics , combinatorial geometry
$n > 4$ planes are in general position (so every $3$ planes have just one common point, and no point belongs to more than $3$ planes). Show that there are at least $\frac{2n-3}{ 4}$ tetrahedra among the regions formed by the planes.

2001 ITAMO, 1
Tags: geometry
A hexagon has all its angles equal, and the lengths of four consecutive sides are $5$, $3$, $6$ and $7$, respectively. Find the lengths of the remaining two edges.

2009 Germany Team Selection Test, 1
Tags: prime number , number theory
For which $ n \geq 2, n \in \mathbb{N}$ are there positive integers $ A_1, A_2, \ldots, A_n$ which are not the same pairwise and have the property that the product $ \prod^n_{i \equal{} 1} (A_i \plus{} k)$ is a power for each natural number $ k.$

2024 USAJMO, 3
Tags: awesomeness_in_a_bun
Let $a(n)$ be the sequence defined by $a(1)=2$ and $a(n+1)=(a(n))^{n+1}-1$ for each integer $n\geq 1$. Suppose that $p>2$ is a prime and $k$ is a positive integer. Prove that some term of the sequence $a(n)$ is divisible by $p^k$. [i]Proposed by John Berman[/i]

2016 Middle European Mathematical Olympiad, 5
Tags: circumcircle , law of cosines , geometry
Let $ABC$ be an acute triangle for which $AB \neq AC$, and let $O$ be its circumcenter. Line $AO$ meets the circumcircle of $ABC$ again in $D$, and the line $BC$ in $E$. The circumcircle of $CDE$ meets the line $CA$ again in $P$. The lines $PE$ and $AB$ intersect in $Q$. Line passing through $O$ parallel to the line $PE$ intersects the $A$-altitude of $ABC$ in $F$. Prove that $FP = FQ$.

2011 NIMO Problems, 11
Tags: number theory , least common multiple , greatest common divisor
How many ordered pairs of positive integers $(m, n)$ satisfy the system \begin{align*} \gcd (m^3, n^2) & = 2^2 \cdot 3^2, \\ \text{LCM} [m^2, n^3] & = 2^4 \cdot 3^4 \cdot 5^6, \end{align*} where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$, respectively?

1990 Romania Team Selection Test, 3
Tags: polynomial , algebra , functional , functional equation
Find all polynomials $P(x)$ such that $2P(2x^2 -1) = P(x)^2 -1$ for all $x$.

2004 Purple Comet Problems, 3
Tags: algebra , polynomial
How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x?\]

2022 Putnam, A5
Tags:
Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that cover two adjacent squares, with Alice going first. By rule, a tile must not cover a square that is already covered by another tile. The game ends when no tile can be placed according to this rule. Alice's goal is to maximize the number of uncovered squares when the game ends; Bob's goal is to minimize it. What is the greatest number of uncovered squares that Alice can ensure at the end of the game, no matter how Bob plays?

2018 Purple Comet Problems, 1
Tags: algebra
Find the positive integer $n$ such that $\frac12 \cdot \frac34 + \frac56 \cdot \frac78 + \frac{9}{10}\cdot \frac{11}{12 }= \frac{n}{1200}$ .

1952 AMC 12/AHSME, 28
Tags:
In the table shown, the formula relating $ x$ and $ y$ is: \[ \begin{tabular}{|c|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 \\ \hline y & 3 & 7 & 13 & 21 & 31 \\ \hline \end{tabular} \]$ \textbf{(A)}\ y \equal{} 4x \minus{} 1 \qquad\textbf{(B)}\ y \equal{} x^3 \minus{} x^2 \plus{} x \plus{} 2 \qquad\textbf{(C)}\ y \equal{} x^2 \plus{} x \plus{} 1$ $ \textbf{(D)}\ y \equal{} (x^2 \plus{} x \plus{} 1)(x \minus{} 1) \qquad\textbf{(E)}\ \text{none of these}$

2001 Putnam, 1
Tags:
Let $n$ be an even positive integer. Write the numbers $1, 2, \cdots, n^2$ in the squares of an $n \times n$ grid so that the $k$th row, from left to right, is \[ (k-1)n + 1, \ (k-1)n + 2, \ \cdots, \ (k-1)n + n. \] Color the squares of the grid so that half of the squares in each row and in each column are red and the other half are black (a checkerboard coloring is one possibility). Prove that for each coloring, the sum of the numbers on the red squares is equal to the sum of the numbers on the black squares.

2017 Serbia National Math Olympiad, 3
Tags: geometry , excircle
Let $k$ be the circumcircle of $\triangle ABC$ and let $k_a$ be A-excircle .Let the two common tangents of $k,k_a$ cut $BC$ in $P,Q$.Prove that $\measuredangle PAB=\measuredangle CAQ$.

2023 CMIMC Team, 13
Tags: team
Suppose that the sequence of real numbers $a_1,a_2,\ldots$ satisfies $a_1 = - \sqrt{1}, a_2 = \sqrt{2}$, and for all $k > 1$, \[ \frac{a_{k+1}+a_{k-1}}{a_k} = \frac{\sqrt{3} + \sqrt{1}}{\sqrt{2}}. \] Find $a_{2023}$. [i]Proposed by Kevin You[/i]