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.

AND:
OR:
NO:

Found problems: 85335

2011 Saudi Arabia Pre-TST, 2.2
Tags: power of 2 , sum , number theory
Consider the sequence $x_n = 2^n-n$, $n = 0,1 ,2 ,...$. Find all integers $m \ge 0$ such that $s_m = x_0 + x_1 + x_2 + ... + x_m$ is a power of $2$.

2021 JHMT HS, 3
Tags: number theory , greatest common divisor
Let $B=\{2^1,2^2,2^3,\dots,2^{21}\}.$ Find the remainder when \[ \sum_{m, n \in B: \ m<n}\gcd(m,n) \] is divided by $1000,$ where the sum is taken over all pairs of elements $(m,n)$ of $B$ such that $m<n.$

LMT Team Rounds 2010-20, A22 B24
Tags:
In a game of Among Us, there are $10$ players and $12$ colors. Each player has a "default" color that they will automatically get if nobody else has that color. Otherwise, they get a random color that is not selected. If $10$ random players with random default colors join a game one by one, the expected number of players to get their default color can be expressed as $\frac{m}{n}$. Compute $m+n$. Note that the default colors are not necessarily distinct. [i]Proposed by Jeff Lin[/i]

1984 Bulgaria National Olympiad, Problem 6
Tags: geometry , 3d geometry , pyramid , parallelogram , locus
Let there be given a pyramid $SABCD$ whose base $ABCD$ is a parallelogram. Let $N$ be the midpoint of $BC$. A plane $\lambda$ intersects the lines $SC,SA,AB$ at points $P,Q,R$ respectively such that $\overline{CP}/\overline{CS}=\overline{SQ}/\overline{SA}=\overline{AR}/\overline{AB}$. A point $M$ on the line $SD$ is such that the line $MN$ is parallel to $\lambda$. Show that the locus of points $M$, when $\lambda$ takes all possible positions, is a segment of the length $\frac{\sqrt5}2SD$.

2020 Kosovo National Mathematical Olympiad, 4
Tags: algebra , number theory , sequence
Let $a_0$ be a fixed positive integer. We define an infinite sequence of positive integers $\{a_n\}_{n\ge 1}$ in an inductive way as follows: if we are given the terms $a_0,a_1,...a_{n-1}$ , then $a_n$ is the smallest positive integer such that $\sqrt[n]{a_0\cdot a_1\cdot ...\cdot a_n}$ is a positive integer. Show that the sequence $\{a_n\}_{n\ge 1}$ is eventually constant. [b]Note:[/b] The sequence $\{a_n\}_{n\ge 1}$ is eventually constant if there exists a positive integer $k$ such that $a_n=c$, for every $n\ge k$.

2025 Turkey Team Selection Test, 5
Tags: combinatorics
Let $P$ be a polygon formed by the edges of an infinite chessboard, which does not intersect itself. Let the numbers $a_1,a_2,a_3$ represent the number of unit squares that have exactly $1,2\text{ or } 3$ edges on the boundary of $P$ respectively. Find the largest real number $k$ such that the inequality $a_1+a_2>ka_3$ holds for each polygon constructed with these conditions.

2023 Harvard-MIT Mathematics Tournament, 10
Tags: algebra , complex number
Let $\zeta= e^{2\pi i/99}$ and $\omega e^{2\pi i/101}$. The polynomial $$x^{9999} + a_{9998}x^{9998} + ...+ a_1x + a_0$$ has roots $\zeta^m + \omega^n$ for all pairs of integers $(m, n)$ with $0 \le m < 99$ and $0 \le n < 101$. Compute $a_{9799} + a_{9800} + ...+ a_{9998}$.

1985 Austrian-Polish Competition, 7
Tags: inequalities , maximization , ratio , four variables
Find an upper bound for the ratio $$\frac{x_1x_2+2x_2x_3+x_3x_4}{x_1^2+x_2^2+x_3^2+x_4^2}$$ over all quadruples of real numbers $(x_1,x_2,x_3,x_4)\neq (0,0,0,0)$. [i]Note.[/i] The smaller the bound, the better the solution.

1994 Spain Mathematical Olympiad, 2
Tags: geometry , circumcenter , sphere , 3-dimensional geometry , locus , 3d geometry
Let $Oxyz$ be a trihedron whose edges $x,y, z$ are mutually perpendicular. Let $C$ be the point on the ray $z$ with $OC = c$. Points $P$ and $Q$ vary on the rays $x$ and $y$ respectively in such a way that $OP+OQ = k$ is constant. For every $P$ and $Q$, the circumcenter of the sphere through $O,C,P,Q$ is denoted by $W$. Find the locus of the projection of $W$ on the plane O$xy$. Also find the locus of points $W$.

PEN E Problems, 28
Tags:
Show that $n^{\pi(2n)-\pi(n)}<4^{n}$ for all positive integer $n$.

2021 Turkey MO (2nd round), 4
Tags: angle bisector , z -orthocenter , geometry
Points $D$ and $E$ are taken on $[BC]$ and $[AC]$ of acute angled triangle $ABC$ such that $BD$ and $CE$ are angle bisectors. Projections of $D$ onto $BC$ and $BA$ are $P$ and $Q$, projections of $E$ onto $CA$ and $CB$ are $R$ and $S$. Let $AP \cap CQ=X$, $AS \cap BR=Y$ and $BX \cap CY=Z$. Show that $AZ \perp BC$.

1977 IMO Longlists, 45
Tags: geometry , 3d geometry , pyramid , combinatorics
Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds: (i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$ (ii) some plane contains exactly three points from $E.$

2021 Middle European Mathematical Olympiad, 3
Tags: combinatorics , strategy , induction
Let $n, b$ and $c$ be positive integers. A group of $n$ pirates wants to fairly split their treasure. The treasure consists of $c \cdot n$ identical coins distributed over $b \cdot n$ bags, of which at least $n-1$ bags are initially empty. Captain Jack inspects the contents of each bag and then performs a sequence of moves. In one move, he can take any number of coins from a single bag and put them into one empty bag. Prove that no matter how the coins are initially distributed, Jack can perform at most $n-1$ moves and then split the bags among the pirates such that each pirate gets $b$ bags and $c$ coins.

2018 IMO Shortlist, C7
Tags: combinatorics
Consider $2018$ pairwise crossing circles no three of which are concurrent. These circles subdivide the plane into regions bounded by circular $edges$ that meet at $vertices$. Notice that there are an even number of vertices on each circle. Given the circle, alternately colour the vertices on that circle red and blue. In doing so for each circle, every vertex is coloured twice- once for each of the two circle that cross at that point. If the two colours agree at a vertex, then it is assigned that colour; otherwise, it becomes yellow. Show that, if some circle contains at least $2061$ yellow points, then the vertices of some region are all yellow. Proposed by [i]India[/i]

2001 IMO Shortlist, 1
Tags: number theory , decimal representation , digit , factorial
Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.

2009 SDMO (Middle School), 5
Tags:
Let $A=33\cdots3$, where $A$ contains $2009$ $3$s. Let $B=11\cdots1088\cdots89$, where $B$ contains $2008$ $1$s and $2008$ $8$s. Prove that $A^2=B$.

2010 Contests, 3
Tags: induction , floor function , combinatorics , logarithm
There are $ n$ websites $ 1,2,\ldots,n$ ($ n \geq 2$). If there is a link from website $ i$ to $ j$, we can use this link so we can move website $ i$ to $ j$. For all $ i \in \left\{1,2,\ldots,n - 1 \right\}$, there is a link from website $ i$ to $ i+1$. Prove that we can add less or equal than $ 3(n - 1)\log_{2}(\log_{2} n)$ links so that for all integers $ 1 \leq i < j \leq n$, starting with website $ i$, and using at most three links to website $ j$. (If we use a link, website's number should increase. For example, No.7 to 4 is impossible). Sorry for my bad English.

2007 Iran Team Selection Test, 3
Tags: analytic geometry , graphing lines , slope , trigonometry , geometry , geometric transformation , reflection
Let $P$ be a point in a square whose side are mirror. A ray of light comes from $P$ and with slope $\alpha$. We know that this ray of light never arrives to a vertex. We make an infinite sequence of $0,1$. After each contact of light ray with a horizontal side, we put $0$, and after each contact with a vertical side, we put $1$. For each $n\geq 1$, let $B_{n}$ be set of all blocks of length $n$, in this sequence. a) Prove that $B_{n}$ does not depend on location of $P$. b) Prove that if $\frac{\alpha}{\pi}$ is irrational, then $|B_{n}|=n+1$.

2015 IFYM, Sozopol, 1
Tags: parallelogram , inequalities , trigonometry , trig identities , law of cosines , geometry
Let ABCD be a convex quadrilateral such that $AB + CD = \sqrt{2}AC$ and $BC + DA = \sqrt{2}BD$. Prove that ABCD is a parallelogram.

1997 Poland - Second Round, 5
Tags: sum , probability , dice , remainder , number theory , combinatorics
We have thrown $k$ white dice and $m$ black dice. Find the probability that the remainder modulo $7$ of the sum of the numbers on the white dice is equal to the remainder modulo $7$ of the sum of the numbers on the black dice.

2021 Durer Math Competition Finals, 1
Tags: number theory , prime
Show that if the difference of two positive cube numbers is a positive prime, then this prime number has remainder $1$ after division by $6$.

2015 District Olympiad, 4
Tags: function , algebra
Find the functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ that satisfy the following relation: $$ \gcd\left( x,f(y)\right)\cdot\text{lcm}\left(f(x), y\right) = \gcd (x,y)\cdot\text{lcm}\left( f(x), f(y)\right) ,\quad\forall x,y\in\mathbb{N} . $$

2017 Kosovo National Mathematical Olympiad, 4
Tags:
4. Find all triples of consecutive numbers ,whose sum of squares is equal to some fourdigit number with all four digits being equal.

1953 Moscow Mathematical Olympiad, 258
Tags: combinatorics , chessboard , infinite board
A knight stands on an infinite chess board. Find all places it can reach in exactly $2n$ moves.

1997 Tournament Of Towns, (558) 3
Tags: number theory , diophantine , diophantine equation
Prove that the equation $$xy(x -y) + yz(y-z) + zx(z-x) = 6$$ has infinitely many solutions in integers $x, y$ and $z$. (N Vassiliev)