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

1989 IMO Longlists, 93
Tags: modular arithmetic , number theory , prime number , sequence , power of number
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.

1986 IMO Longlists, 40
Tags: function , algebra
Find the maximum value that the quantity $2m+7n$ can have such that there exist distinct positive integers $x_i \ (1 \leq i \leq m), y_j \ (1 \leq j \leq n)$ such that the $x_i$'s are even, the $y_j$'s are odd, and $\sum_{i=1}^{m} x_i +\sum_{j=1}^{n} y_j=1986.$

2012 Czech-Polish-Slovak Junior Match, 3
Tags: projection , circumcircle , equilateral , fixed , area of a triangle , geometry
Different points $A, B, C, D$ lie on a circle with a center at the point $O$ at such way that $\angle AOB$ $= \angle BOC =$ $\angle COD =$ $60^o$. Point $P$ lies on the shorter arc $BC$ of this circle. Points $K, L, M$ are projections of $P$ on lines $AO, BO, CO$ respectively . Show that (a) the triangle $KLM$ is equilateral, (b) the area of triangle $KLM$ does not depend on the choice of the position of point $P$ on the shorter arc $BC$

2021 Taiwan TST Round 2, N
Tags: number theory
Let $S$ be a set of positive integers such that for every $a,b\in S$, there always exists $c\in S$ such that $c^2$ divides $a(a+b)$. Show that there exists an $a\in S$ such that $a$ divides every element of $S$. [i]Proposed by usjl[/i]

2024 ELMO Shortlist, A4
Tags: algebra
The number $2024$ is written on a blackboard. Each second, if there exist positive integers $a,b,k$ such that $a^k+b^k$ is written on the blackboard, you may write $a^{k'}+b^{k'}$ on the blackboard for any positive integer $k'.$ Find all positive integers that you can eventually write on the blackboard. [i]Srinivas Arun[/i]

1996 Bundeswettbewerb Mathematik, 2
Tags: combinatorics , sum , board
The cells of an $n \times n$ board are labelled with the numbers $1$ through $n^2$ in the usual way. Let $n$ of these cells be selected, no two of which are in the same row or column. Find all possible values of the sum of their labels.

1994 Flanders Math Olympiad, 4
Tags: function , algebra , domain , quadratic
Let $(f_i)$ be a sequence of functions defined by: $f_1(x)=x, f_n(x) = \sqrt{f_{n-1}(x)}-\dfrac14$. ($n\in \mathbb{N}, n\ge2$) (a) Prove that $f_n(x) \le f_{n-1}(x)$ for all x where both functions are defined. (b) Find for each $n$ the points of $x$ inside the domain for which $f_n(x)=x$.

1976 IMO Longlists, 6
Tags: function , inequalities
For each point $X$ of a given polytope, denote by $f(X)$ the sum of the distances of the point $X$ from all the planes of the faces of the polytope. Prove that if $f$ attains its maximum at an interior point of the polytope, then $f$ is constant.

2018 Middle European Mathematical Olympiad, 6
Tags: circumcircle , geometry
Let $ABC$ be a triangle . The internal bisector of $ABC$ intersects the side $AC$ at $ L$ and the circumcircle of $ABC$ again at $W \neq B.$ Let $K$ be the perpendicular projection of $L$ onto $AW.$ the circumcircle of $BLC$ intersects line $CK$ again at $P \neq C.$ Lines $BP$ and $AW$ meet at point $T.$ Prove that $$AW=WT.$$

2025 Polish MO Finals, 2
Tags: number theory , prime number
Positive integers $k, m, n ,p $ integers are such that $p=2^{2^n}+1$ is prime and $p\mid 2^k-m$. Prove that there exists a positive integer $l$ such that $p^2\mid 2^l-m$.

2002 Italy TST, 2
Tags: floor function , combinatorics
On a soccer tournament with $n\ge 3$ teams taking part, several matches are played in such a way that among any three teams, some two play a match. $(a)$ If $n=7$, find the smallest number of matches that must be played. $(b)$ Find the smallest number of matches in terms of $n$.

2023 Middle European Mathematical Olympiad, 3
Tags: geometry
Let $ABC$ be a triangle with incenter $I$, and the incircle touches $BC$ at $D$. The points $E, F$ are such that $BE \parallel AI \parallel CF$ and $\angle BEI=\angle CFI=90^{\circ}$. If $DE, DF$ meet the incircle at $E', F'$, show that $E'F' \perp AI$.

2014-2015 SDML (High School), 7
Tags:
The names of $10$ people are inside $10$ boxes. Each box is labeled with someone's name. Unfortunately, there was a mix-up and not everyone's name is in his or her own box. Each person looks through $5$ boxes in the following way. First, they look in their own box. After looking in a box, they look at the box labeled with the name they found in the previous box. (For example, if someone looks in a box and finds the name "Joe," he or she will then look at the box labeled "Joe.") What is the probability that every person finds his or her own name in a box?

2022 Stanford Mathematics Tournament, 2
Tags:
Let $ABC$ be an acute, scalene triangle. Let $H$ be the orthocenter. Let the circle going through $B$, $H$, and $C$ intersect $CA$ again at $D$. Given that $\angle ABH=20^\circ$, find, in degrees, $\angle BDC$.

1986 Miklós Schweitzer, 9
Tags: geometry , parallelogram , algebra , function , domain
Consider a latticelike packing of translates of a convex region $K$. Let $t$ be the area of the fundamental parallelogram of the lattice defining the packing, and let $t_{\min} (K)$ denote the minimal value of $t$ taken for all latticelike packings. Is there a natural number $N$ such that for any $n>N$ and for any $K$ different from a parallelogram, $nt_{\min} (K)$ is smaller that the area of any convex domain in which $n$ translates to $K$ can be placed without overlapping? (By a [i]latticelike packing[/i] of $K$ we mean a set of nonoverlapping translates of $K$ obtained from $K$ by translations with all vectors of a lattice.) [G. and L. Fejes-Toth]

2021 CMIMC, 2.8 1.4
Tags: geometry
Let $ABCDEF$ be an equilateral heaxagon such that $\triangle ACE \cong \triangle DFB$. Given that $AC = 7$, $CE=8$, and $EA=9$, what is the side length of this hexagon? [i]Proposed by Thomas Lam[/i]

1988 India National Olympiad, 6
Tags: polynomial , modular arithmetic , algebra
If $ a_0,a_1,\dots,a_{50}$ are the coefficients of the polynomial \[ \left(1\plus{}x\plus{}x^2\right)^{25}\] show that $ a_0\plus{}a_2\plus{}a_4\plus{}\cdots\plus{}a_{50}$ is even.

2015 ASDAN Math Tournament, 4
Tags:
Given a positive integer $x>1$ with $n$ divisors, define $f(x)$ to be the product of the smallest $\lceil\tfrac{n}{2}\rceil$ divisors of $x$. Let $a$ be the least value of $x$ such that $f(x)$ is a multiple of $X$, and $b$ be the least value of $n$ such that $f(y)$ is a multiple of $y$ for some $y$ that has exactly $n$ factors. Compute $a+b$.

2003 South africa National Olympiad, 1
Tags:
You have five pieces of paper. You pick one or more of them and cut each of them into five smaller pieces. Now you take one or more of the pieces from this lot and cut each of these into five smaller pieces. And so on. Prove that you will never have 2003 pieces.

2020 SIME, 3
Tags:
Real numbers $x, y > 1$ are chosen such that the three numbers \[\log_4x, \; 2\log_xy, \; \log_y2\] form a geometric progression in that order. If $x + y = 90$, then find the value of $xy$.

1999 Irish Math Olympiad, 1
Tags: inequalities
Find all real numbers $ x$ which satisfy: $ \frac{x^2}{(x\plus{}1\minus{}\sqrt{x\plus{}1})^2}<\frac{x^2\plus{}3x\plus{}18}{(x\plus{}1)^2}.$

1991 Tournament Of Towns, (300) 1
Tags: geometry , equal segments , isosceles
The centre of circle $1$ lies on circle $2$. $A$ and $B$ are the intersection points of the circles. The tangent line to circle $2$ at point $B$ intersects circle $1$ at point $C$. Prove that $AB = BC$. (V. Prasovov, Moscow)

1991 Polish MO Finals, 2
Tags: geometry
Two noncongruent circles $k_1$ and $k_2$ are exterior to each other. Their common tangents intersect the line through their centers at points $A$ and $B$. Let $P$ be any point of $k_1$. Prove that there is a diameter of $k_2$ with one endpoint on line $PA$ and the other on $PB$.

1954 Miklós Schweitzer, 5
Tags: probability
[b]5.[/b] Let $\xi _{1},\xi _{2},\dots ,\xi _{n},... $ be independent random variables of uniform distribution in $(0,1)$. Show that the distribution of the random variable $\eta _{n}= \sqrt[]{n}\prod_{k=1}^{n}(1-\frac{\xi _{k}}{k}) (n= 1,2,...)$ tends to a limit distribution for $n \to \infty $. [b](P. 6)[/b]

2007 Purple Comet Problems, 23
Tags: geometry , trigonometry , trig identities , law of sines
Two circles with radius $2$ and radius $4$ have a common center at P. Points $A, B,$ and $C$ on the larger circle are the vertices of an equilateral triangle. Point $D$ is the intersection of the smaller circle and the line segment $PB$. Find the square of the area of triangle $ADC$.