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

1983 All Soviet Union Mathematical Olympiad, 360

Given natural $n,m,k$. It is known that $m^n$ is divisible by $n^m$, and $n^k$ is divisible by $k^n$. Prove that $m^k$ is divisible by $k^m$.

2008 Portugal MO, 2

Tags: geometry
Let $AEBC$ be a cyclic quadrilateral. Let $D$ be a point on the ray $AE$ which is outside the circumscribed circumference of $AEBC$. Suppose that $\angle CAB=\angle BAE$. Prove that $AB=BD$ if and only if $DE=AC$.

1978 Yugoslav Team Selection Test, Problem 2

Let $k_0$ be a unit semi-circle with diameter $AB$. Assume that $k_1$ is a circle of radius $r_1=\frac12$ that is tangent to both $k_0$ and $AB$. The circle $k_{n+1}$ of radius $r_{n+1}$ touches $k_n,k_0$, and $AB$. Prove that: (a) For each $n\in\{2,3,\ldots\}$ it holds that $\frac1{r_{n+1}}+\frac1{r_{n-1}}=\frac6{r_n}-4$. (b) $\frac1{r_n}$ is either a square of an even integer, or twice a square of an odd integer.

2019 Online Math Open Problems, 28

Tags:
Let $ABC$ be a triangle. There exists a positive real number $x$ such that $AB=6x^2+1$ and $AC = 2x^2+2x$, and there exist points $W$ and $X$ on segment $AB$ along with points $Y$ and $Z$ on segment $AC$ such that $AW=x$, $WX=x+4$, $AY=x+1$, and $YZ=x$. For any line $\ell$ not intersecting segment $BC$, let $f(\ell)$ be the unique point $P$ on line $\ell$ and on the same side of $BC$ as $A$ such that $\ell$ is tangent to the circumcircle of triangle $PBC$. Suppose lines $f(WY)f(XY)$ and $f(WZ)f(XZ)$ meet at $B$, and that lines $f(WZ)f(WY)$ and $f(XY)f(XZ)$ meet at $C$. Then the product of all possible values for the length of $BC$ can be expressed in the form $a + \dfrac{b\sqrt{c}}{d}$ for positive integers $a,b,c,d$ with $c$ squarefree and $\gcd (b,d)=1$. Compute $100a+b+c+d$. [i]Proposed by Vincent Huang[/i]

2016 ASDAN Math Tournament, 5

Tags:
$ABCD$ is a four digit number ($A\neq0$) such that both $ABC$ and $BCD$ are divisible by $9$ ($ABCD$ is not necessarily divisible by $9$, and $B,C,D$ may be $0$). Compute the number of four digit numbers satisfying this property.

2000 Romania Team Selection Test, 2

Tags: inequalities
Let $n\ge 1$ be a positive integer and $x_1,x_2\ldots ,x_n$ be real numbers such that $|x_{k+1}-x_k|\le 1$ for $k=1,2,\ldots ,n-1$. Prove that \[\sum_{k=1}^n|x_k|-\left|\sum_{k=1}^nx_k\right|\le\frac{n^2-1}{4}\] [i]Gh. Eckstein[/i]

2017 IMO, 1

For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as $$a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\ a_n + 3 & \text{otherwise.} \end{cases} $$ Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$. [i]Proposed by Stephan Wagner, South Africa[/i]

Russian TST 2016, P1

The positive numbers $a, b, c$ are such that $a^2<16bc, b^2<16ca$ and $c^2<16ab$. Prove that \[a^2+b^2+c^2<2(ab+bc+ca).\]

2004 Romania National Olympiad, 2

Let $n \in \mathbb N$, $n \geq 2$. (a) Give an example of two matrices $A,B \in \mathcal M_n \left( \mathbb C \right)$ such that \[ \textrm{rank} \left( AB \right) - \textrm{rank} \left( BA \right) = \left\lfloor \frac{n}{2} \right\rfloor . \] (b) Prove that for all matrices $X,Y \in \mathcal M_n \left( \mathbb C \right)$ we have \[ \textrm{rank} \left( XY \right) - \textrm{rank} \left( YX \right) \leq \left\lfloor \frac{n}{2} \right\rfloor . \] [i]Ion Savu[/i]

2014 India Regional Mathematical Olympiad, 1

In an acute-angled triangle $ABC, \angle ABC$ is the largest angle. The perpendicular bisectors of $BC$ and $BA$ intersect AC at $X$ and $Y$ respectively. Prove that circumcentre of triangle $ABC$ is incentre of triangle $BXY$ .

KoMaL A Problems 2019/2020, A. 775

Tags: geometry
Let $H\subseteq\mathbb{R}^3$ such that if we reflect any point in $H$ across another point of $H$, the resulting point is also in $H$. Prove that either $H$ is dense in ${R}^3$ or one can find equidistant parallel planes which cover $H$

1996 Rioplatense Mathematical Olympiad, Level 3, 5

There is a board with $n$ rows and $4$ columns, and white, yellow and light blue chips. Player $A$ places four tokens on the first row of the board and covers them so Player $B$ doesn't know them. How should player $B$ do to fill the minimum number of rows with chips that will ensure that in any of the rows he will have at least three hits? Clarification: A hit by player $B$ occurs when he places a token of the same color and in the same column as $A$.

2017 Bosnia and Herzegovina Team Selection Test, Problem 2

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2017 AMC 12/AHSME, 25

Tags:
A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $5$-player teams, but no two teams may have exactly the same $5$ members. The site statistics show a curious fact: The average, over all subsets of size $9$ of the set of $n$ participants, of the number of complete teams whose members are among those $9$ people is equal to the reciprocal of the average, over all subsets of size $8$ of the set of $n$ participants, of the number of complete teams whose members are among those $8$ people. How many values $n$, $9\leq n\leq 2017$, can be the number of participants? $\textbf{(A) } 477 \qquad \textbf{(B) } 482 \qquad \textbf{(C) } 487 \qquad \textbf{(D) } 557 \qquad \textbf{(E) } 562$

1987 China Team Selection Test, 1

a.) For all positive integer $k$ find the smallest positive integer $f(k)$ such that $5$ sets $s_1,s_2, \ldots , s_5$ exist satisfying: [b]i.[/b] each has $k$ elements; [b]ii.[/b] $s_i$ and $s_{i+1}$ are disjoint for $i=1,2,...,5$ ($s_6=s_1$) [b]iii.[/b] the union of the $5$ sets has exactly $f(k)$ elements. b.) Generalisation: Consider $n \geq 3$ sets instead of $5$.

1967 AMC 12/AHSME, 9

Let $K$, in square units, be the area of a trapezoid such that the shorter base, the altitude, and the longer base, in that order, are in arithmetic progression. Then: $\textbf{(A)}\ K \; \text{must be an integer} \qquad \textbf{(B)}\ K \; \text{must be a rational fraction} \\ \textbf{(C)}\ K \; \text{must be an irrational number} \qquad \textbf{(D)}\ K\; \text{must be an integer or a rational fraction} \qquad$ $\textbf{(E)}\ \text{taken alone neither} \; \textbf{(A)} \; \text{nor} \; \textbf{(B)} \; \text{nor} \; \textbf{(C)} \; \text{nor} \; \textbf{(D)} \; \text{is true}$

2013 AMC 10, 7

Tags: geometry
Six points are equally spaced around a circle of radius 1. Three of these points are the vertices of a triangle that is neither equilateral nor isosceles. What is the area of this triangle? $ \textbf{(A) }\frac{\sqrt3}3\qquad\textbf{(B) }\frac{\sqrt3}2\qquad\textbf{(C) }1\qquad\textbf{(D) }\sqrt2\qquad\textbf{(E) }2$

1977 Bulgaria National Olympiad, Problem 2

In the space are given $n$ points and no four of them belongs to a common plane. Some of the points are connected with segments. It is known that four of the given points are vertices of tetrahedron which edges belong to the segments given. It is also known that common number of the segments, passing through vertices of tetrahedron is $2n$. Prove that there exists at least two tetrahedrons every one of which have a common face with the first (initial) tetrahedron. [i]N. Nenov, N. Hadzhiivanov[/i]

1991 Putnam, A4

Tags: geometry
Can we find an (infinite) sequence of disks in the Euclidean plane such that: $(1)$ their centers have no (finite) limit point in the plane; $(2)$ the total area of the disks is finite; and $(3)$ every line in the plane intersects at least one of the disks?

2003 Romania National Olympiad, 3

Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the property that $$ xf(x)\ge \int_0^x f(t)dt , $$ for all real numbers $ x. $ Prove that [b]a)[/b] the mapping $ x\mapsto \frac{1}{x}\int_0^x f(t) dt $ is nondecreasing on the restrictions $ \mathbb{R}_{<0 } $ and $ \mathbb{R}_{>0 } . $ [b]b)[/b] if $ \int_x^{x+1} f(t)dt=\int_{x-1}^x f(t)dt , $ for any real number $ x, $ then $ f $ is constant. [i]Mihai Piticari[/i]

2000 National Olympiad First Round, 12

Tags:
$(a_n)$ is a sequence with $a_1=1$ and $|a_n| = |a_{n-1}+2|$ for every positive integer $n\geq 2$. What is the minimum possible value of $\sum_{i = 1}^{2000}a_{i}$? $ \textbf{(A)}\ -4000 \qquad\textbf{(B)}\ -3000 \qquad\textbf{(C)}\ -2000 \qquad\textbf{(D)}\ -1000 \qquad\textbf{(E)}\ \text{None} $

2015 Saudi Arabia BMO TST, 3

Let $ABC$ be a triangle, $\Gamma$ its circumcircle, $I$ its incenter, and $\omega$ a tangent circle to the line $AI$ at $I$ and to the side $BC$. Prove that the circles $\Gamma$ and $\omega$ are tangent. Malik Talbi

1999 Junior Balkan Team Selection Tests - Romania, 3

Tags: algebra
Consider the set $ \mathcal{M}=\left\{ \gcd(2n+3m+13,3n+5m+1,6n+6m-1) | m,n\in\mathbb{N} \right\} . $ Show that there is a natural $ k $ such that the set of its positive divisors is $ \mathcal{M} . $ [i]Dan Brânzei[/i]

1998 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 1

Tags:
Let $ a \geq b$ be real number such that $ a^2\plus{}b^2 \equal{} 31$ and $ ab \equal{} 3.$ Then $ a\minus{}b$ equals $ \text{(A)}\ 5 \qquad \text{(B)}\ \frac{31}{6} \qquad \text{(C)}\ 2 \sqrt{6} \qquad \text{(D)}\ \frac{5}{6} \sqrt{31} \qquad \text{(E)}\ \frac{5}{6} \sqrt{37}$

1986 Traian Lălescu, 2.4

Show that there is an unique group $ G $ (up to isomorphism) of order $ 1986 $ which has the property that there is at most one subgroup of it having order $ n, $ for every natural number $ n. $