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

2009 Baltic Way, 8

Determine all positive integers $n$ for which there exists a partition of the set \[\{n,n+1,n+2,\ldots ,n+8\}\] into two subsets such that the product of all elements of the first subset is equal to the product of all elements of the second subset.

2011 Junior Balkan Team Selection Tests - Romania, 5

Consider $n$ persons, each of them speaking at most $3$ languages. From any $3$ persons there are at least two which speak a common language. i) For $n \le 8$, exhibit an example in which no language is spoken by more than two persons. ii) For $n \ge 9$, prove that there exists a language which is spoken by at least three persons

2019 CHKMO, 3

Tags: geometry , incenter
The incircle of $\triangle{ABC}$, with incentre $I$, meets $BC, CA$, and $AB$ at $D,E$, and $F$, respectively. The line $EF$ cuts the lines $BI$, $CI, BC$, and $DI$ at $K,L,M$, and $Q$, respectively. The line through the midpoint of $CL$ and $M$ meets $CK$ at $P$. (a) Determine $\angle{BKC}$. (b) Show that the lines $PQ$ and $CL$ are parallel.

1990 IMO Longlists, 19

Given an initial integer $ n_0 > 1$, two players, $ {\mathcal A}$ and $ {\mathcal B}$, choose integers $ n_1$, $ n_2$, $ n_3$, $ \ldots$ alternately according to the following rules : [b]I.)[/b] Knowing $ n_{2k}$, $ {\mathcal A}$ chooses any integer $ n_{2k \plus{} 1}$ such that \[ n_{2k} \leq n_{2k \plus{} 1} \leq n_{2k}^2. \] [b]II.)[/b] Knowing $ n_{2k \plus{} 1}$, $ {\mathcal B}$ chooses any integer $ n_{2k \plus{} 2}$ such that \[ \frac {n_{2k \plus{} 1}}{n_{2k \plus{} 2}} \] is a prime raised to a positive integer power. Player $ {\mathcal A}$ wins the game by choosing the number 1990; player $ {\mathcal B}$ wins by choosing the number 1. For which $ n_0$ does : [b]a.)[/b] $ {\mathcal A}$ have a winning strategy? [b]b.)[/b] $ {\mathcal B}$ have a winning strategy? [b]c.)[/b] Neither player have a winning strategy?

2004 German National Olympiad, 3

Prove that for every positive integer $n$ there is an $n$-digit number $z$ with none of its digits $0$ and such that $z$ is divisible by its sum of digits.

2018 OMMock - Mexico National Olympiad Mock Exam, 5

Let $ABC$ be a triangle with circumcirle $\Gamma$, and let $M$ and $N$ be the respective midpoints of the minor arcs $AB$ and $AC$ of $\Gamma$. Let $P$ and $Q$ be points such that $AB=BP$, $AC=CQ$, and $P$, $B$, $C$, $Q$ lie on $BC$ in that order. Prove that $PM$ and $QN$ meet at a point on $\Gamma$. [i]Proposed by Victor Domínguez[/i]

2023 New Zealand MO, 5

Let $x, y$ and $z$ be real numbers such that: $x^2 = y + 2$, and $y^2 = z + 2$, and $z^2 = x + 2$. Prove that $x + y + z$ is an integer.

2017 BMT Spring, 11

Ben picks a positive number $n$ less than $2017$ uniformly at random. Then Rex, starting with the number $ 1$, repeatedly multiplies his number by $n$ and then finds the remainder when dividing by $2017$. Rex does this until he gets back to the number $ 1$. What is the probability that, during this process, Rex reaches every positive number less than $2017$ before returning back to $ 1$?

1977 IMO Longlists, 12

Let $z$ be an integer $> 1$ and let $M$ be the set of all numbers of the form $z_k = 1+z + \cdots+ z^k, \ k = 0, 1,\ldots$. Determine the set $T$ of divisors of at least one of the numbers $z_k$ from $M.$

2005 Alexandru Myller, 4

Prove that there exists an undirected graph having $ 2004 $ vertices such that for any $ \in\{ 1,2,\ldots ,1002 \} , $ there exists at least two vertices whose orders are $ n. $

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 2

Justify your answer whether $A=\left( \begin{array}{ccc} -4 & -1& -1 \\ 1 & -2& 1 \\ 0 & 0& -3 \end{array} \right)$ is similar to $B=\left( \begin{array}{ccc} -2 & 1& 0 \\ -1 & -4& 1 \\ 0 & 0& -3 \end{array} \right),\ A,\ B\in{M(\mathbb{C})}$ or not.

2025 Harvard-MIT Mathematics Tournament, 6

Tags: geometry
Trapezoid $ABCD,$ with $AB \parallel CD,$ has side lengths $AB=11, BC=8, CD=19,$ and $DA=4.$ Compute the area of the convex quadrilateral whose vertices are the circumcenters of $\triangle{ABC}, \triangle{BCD}, \triangle{CDA},$ and $\triangle{DAB}.$

2001 India IMO Training Camp, 3

Tags: geometry , inradius
Points $B = B_1 , B_2, \cdots , B_n , B_{n+1} = C$ are chosen on side $BC$ of a triangle $ABC$ in that order. Let $r_j$ be the inradius of triangle $AB_jB_{j+1}$ for $j = 1, \cdots, n$ , and $r$ be the inradius of $\triangle ABC$. Show that there is a constant $\lambda$ independent of $n$ such that : \[(\lambda -r_1)(\lambda -r_2)\cdots (\lambda -r_n) =\lambda^{n-1}(\lambda -r)\]

2004 IMO, 5

In a convex quadrilateral $ABCD$, the diagonal $BD$ bisects neither the angle $ABC$ nor the angle $CDA$. The point $P$ lies inside $ABCD$ and satisfies \[\angle PBC=\angle DBA\quad\text{and}\quad \angle PDC=\angle BDA.\] Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP=CP$.

2006 CHKMO, 1

On a planet there are $3\times2005!$ aliens and $2005$ languages. Each pair of aliens communicates with each other in exactly one language. Show that there are $3$ aliens who communicate with each other in one common language.

2008 May Olympiad, 1

How many different numbers with $6$ digits and multiples of $45$ can be written by adding one digit to the left and one to the right of $2008$?

2018 Singapore Senior Math Olympiad, 4

Let $a,b,c,d$ be positive integers such that $a+c=20$ and $\frac{a}{b}+\frac{c}{d}<1$. Find the maximum possible value of $\frac{a}{b}+\frac{c}{d}$.

2023 Princeton University Math Competition, B1

Tags: algebra
Consider the equations $x^2+y^2=16$ and $xy=\tfrac{9}{2}.$ Find the sum, over all ordered pairs $(x,y)$ satisfying these equations, of $|x+y|.$

2024 Princeton University Math Competition, A1 / B3

Tags: geometry
The following three squares are inscribed within each other such that they all share the same center, and the largest and smallest squares have parallel sides. If the largest square has side length $17$ and the middle square has side length $13,$ the side length of the smallest square can be expressed in the form $\tfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. Find $a + b.$ [center] [img]https://cdn.artofproblemsolving.com/attachments/c/e/86948ff8c3941fa125784a1ca0d53ac769b169.png[/img] [/center]

2006 Singapore MO Open, 4

Let $n$ be positive integer. Let $S_1,S_2,\cdots,S_k$ be a collection of $2n$-element subsets of $\{1,2,3,4,...,4n-1,4n\}$ so that $S_{i}\cap S_{j}$ contains at most $n$ elements for all $1\leq i<j\leq k$. Show that $$k\leq 6^{(n+1)/2}$$

1961 All-Soviet Union Olympiad, 4

Given are arbitrary integers $a,b,p$. Prove that there always exist relatively prime integers $k$ and $\ell$ such that $ak+b\ell$ is divisible by $p$.

2024 Harvard-MIT Mathematics Tournament, 30

Tags: guts
Let $ABC$ be an equilateral triangle with side length $1.$ Points $D, E,$ and $F$ lie inside triangle $ABC$ such that $A, E, F$ are collinear, $B, F, D$ are collinear, $C, D, E$ are collinear, and triangle $DEF$ is equilateral. Suppose that there exists a unique equilateral triangle $XYZ$ with $X$ on side $\overline{BC},$ $Y$ is on side $\overline{AB},$ and $Z$ is on side $\overline{AC}$ such that $D$ lies on side $\overline{XZ},$ $E$ lies on side $\overline{YZ},$ and $F$ lies on side $\overline{XY}.$ Compute $AZ.$

2011 Princeton University Math Competition, A1 / B2

Tags: geometry
Two logs of length 10 are laying on the ground touching each other. Their radii are 3 and 1, and the smaller log is fastened to the ground. The bigger log rolls over the smaller log without slipping, and stops as soon as it touches the ground again. The volume of the set of points swept out by the larger log as it rolls over the smaller one can be expressed as $n \pi$, where $n$ is an integer. Find $n$.

2017 NZMOC Camp Selection Problems, 2

Let $ABCD$ be a parallelogram with an acute angle at $A$. Let $G$ be the point on the line $AB$, distinct from $B$, such that $CG = CB$. Let $H$ be the point on the line $BC$, distinct from $B$, such that $AB = AH$. Prove that triangle $DGH$ is isosceles.

2013 India Regional Mathematical Olympiad, 4

Find the number of $10$-tuples $(a_1,a_2,\dots,a_9,a_{10})$ of integers such that $|a_1|\leq 1$ and \[a_1^2+a_2^2+a_3^2+\cdots+a_{10}^2-a_1a_2-a_2a_3-a_3a_4-\cdots-a_9a_{10}-a_{10}a_1=2.\]