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

1969 IMO Longlists, 61
Tags: function , algebra , binomial theorem , number theory , power of 2
$(SWE 4)$ Let $a_0, a_1, a_2, \cdots$ be determined with $a_0 = 0, a_{n+1} = 2a_n + 2^n$. Prove that if $n$ is power of $2$, then so is $a_n$

2022 Baltic Way, 10
Tags: number theory , combinatorics
A natural number $a$ is said [i]to be contained[/i] in the natural number $b$ if it is possible to obtain a by erasing some digits from $b$ (in their decimal representations). For example, $123$ is contained in $901523$, but not contained in $3412$. Does there exist an infinite set of natural numbers such that no number in the set is contained in any other number from the set?

May Olympiad L1 - geometry, 2003.2
Tags: geometry , area of a triangle , right triangle , area
The triangle $ABC$ is right in $A$ and $R$ is the midpoint of the hypotenuse $BC$ . On the major leg $AB$ the point $P$ is marked such that $CP = BP$ and on the segment $BP$ the point $Q$ is marked such that the triangle $PQR$ is equilateral. If the area of triangle $ABC$ is $27$, calculate the area of triangle $PQR$ .

2010 IMO Shortlist, 4
Tags: modular arithmetic , divisibility , number theory
Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$ [list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good. [*][b](b)[/b] Show that all 2010-good pairs are very good.[/list] [i]Proposed by Okan Tekman, Turkey[/i]

2019 Stanford Mathematics Tournament, 2
Tags: geometry
Let $ABCD$ be a rectangle with $AB = 8$ and $BC = 6$. Point $E$ is outside of the rectangle such that $CE = DE$. Point $D$ is reflected over line $AE$ so that its image, $D'$ , lies on the interior of the rectangle. Point $D'$ is then reflected over diagonal $AC$, and its image lies on side $AB$. What is the length of $DE$?

2017 Princeton University Math Competition, A3/B5
Tags: geometry , 3d geometry , prism
A right regular hexagonal prism has bases $ABCDEF$, $A'B'C'D'E'F'$ and edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, $FF'$, each of which is perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$. The plane $P$ passes through points $A$, $C'$, and $E$. The area of the portion of $P$ contained in the prism can be expressed as $m\sqrt{n}$, where $n$ is not divisible by the square of any prime. Find $m+n$.

2006 Purple Comet Problems, 19
Tags: probability , expected value
There is a very popular race course where runners frequently go for a daily run. Assume that all runners randomly select a start time, a starting position on the course, and a direction to run. Also assume that all runners make exactly one complete circuit of the race course, all runners run at the same speed, and all runners complete the circuit in one hour. Suppose that one afternoon you go for a run on this race course, and you count $300$ runners which you pass going in the opposite direction, although some of those runners you count twice since you pass them twice. What is the expected value of the number of different runners that you pass not counting duplicates?

2018 Saint Petersburg Mathematical Olympiad, 5
Tags: geometry
Can we draw $\triangle ABC$ and points $X,Y$, such that $AX=BY=AB$, $ BX = CY = BC$, $CX = AY = CA$?

2008 AIME Problems, 1
Tags:
Of the students attending a school party, $ 60\%$ of the students are girls, and $ 40\%$ of the students like to dance. After these students are joined by $ 20$ more boy students, all of whom like to dance, the party is now $ 58\%$ girls. How many students now at the party like to dance?

2021 NICE Olympiad, 7
Tags:
Find all functions $f\colon \mathbb Z \to \mathbb Z$ for which \[ f(x)+f(y)+xy \quad \text{divides} \quad xf(x)-y^3 \] for all pairs of integers $(x, y)$. Here, we use the convention that $a$ divides $b$ if and only if there exists some integer $c$ such that $ac=b$. [i]Dennis Chen and Andrew Wen[/i]

2025 Kyiv City MO Round 1, Problem 1
Tags: number theory , algebra , bruteforce
How many three-digit numbers are there, which do not have a zero in their decimal representation and whose sum of digits is $7$?

2009 Croatia Team Selection Test, 4
Tags: number theory
Determine all natural $ n$ for which there exists natural $ m$ divisible by all natural numbers from 1 to $ n$ but not divisible by any of the numbers $ n \plus{} 1$, $ n \plus{} 2$, $ n \plus{} 3$.

1988 IMO Longlists, 73
Tags: combinatorics
A two-person game is played with nine boxes arranged in a $3 \times 3$ square and with white and black stones. At each move a player puts three stones, not necessarily of the same colour, in three boxes in either a horizontal or a vertical line. No box can contain stones of different colours: if, for instance, a player puts a white stone in a box containing black stones the white stone and one of the black stones are removed from the box. The game is over when the centrebox and the cornerboxes contain one black stone and the other boxes are empty. At one stage of a game $x$ boxes contained one black stone each and the other boxes were empty. Determine all possible values for $x.$

2014 PUMaC Geometry A, 7
Tags: geometry , trigonometry , analytic geometry , trig identities , law of cosines
Let $O$ be the center of a circle of radius $26$, and let $A$, $B$ be two distinct points on the circle, with $M$ being the midpoint of $AB$. Consider point $C$ for which $CO=34$ and $\angle COM=15^\circ$. Let $N$ be the midpoint of $CO$. Suppose that $\angle ACB=90^\circ$. Find $MN$.

2018 BMT Spring, Tie 3
Tags: combinatorics
Alice and Bob are playing rock paper scissors. Alice however is cheating, so in each round, she has a $\frac35$ chance of winning, $\frac25$ chance of drawing, and $\frac25$ chance of losing. The first person to win $5$ more rounds than the other person wins the match. What is the probability Alice wins?

2021 Taiwan TST Round 1, A
Tags: algebra , inequality , 4-variable inequality , inequalities
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

2017 CMIMC Computer Science, 10
Tags: computer science
How many distinct spanning trees does the graph below have? Recall that a $\emph{spanning tree}$ of a graph $G$ is a subgraph of $G$ that is a tree and containing all the vertices of $G$. [center][img]http://i.imgur.com/NMF12pE.png[/img][/center]

2024 All-Russian Olympiad, 8
Tags: number theory
Prove that there exists $c>0$ such that for any odd prime $p=2k+1$, the numbers $1^0, 2^1,3^2,\dots,k^{k-1}$ give at least $c\sqrt{p}$ distinct residues modulo $p$. [i]Proposed by M. Turevsky, I. Bogdanov[/i]

Swiss NMO - geometry, 2012.6
Tags: geometry , circumcircle , equal circles , parallelogram
Let $ABCD$ be a parallelogram with at least an angle not equal to $90^o$ and $k$ the circumcircle of the triangle $ABC$. Let $E$ be the diametrically opposite point of $B$. Show that the circumcircle of the triangle $ADE$ and $k$ have the same radius.

2024/2025 TOURNAMENT OF TOWNS, P6
Tags: combinatorics
Let us name a move of the chess knight horizontal if it moves two cells horizontally and one vertically, and vertical otherwise. It is required to place the knight on a cell of a ${46} \times {46}$ board and alternate horizontal and vertical moves. Prove that if each cell is visited not more than once then the number of moves does not exceed 2024. Alexandr Gribalko

2017 Poland - Second Round, 1
Tags: number theory
Prove that for each prime $p>2$ there exists exactly one positive integer $n$, such that $n^2+np$ is a perfect square.

2018 IFYM, Sozopol, 4
Tags: algebra , inequality , inequalities
Find all real numbers $k$ for which the inequality $(1+t)^k (1-t)^{1-k} \leq 1$ is true for every real number $t \in (-1, 1)$.

2005 Estonia National Olympiad, 2
Tags: equal angles , polygon , convex , geometry
Consider a convex $n$-gon in the plane with $n$ being odd. Prove that if one may find a point in the plane from which all the sides of the $n$-gon are viewed at equal angles, then this point is unique. (We say that segment $AB$ is viewed at angle $\gamma$ from point $O$ iff $\angle AOB =\gamma$ .)

1950 Poland - Second Round, 1
Tags: algebra , system of equations , system
Solve the system of equations $$\begin{cases} x^2+x+y=8\\ y^2+2xy+z=168\\ z^2+2yz+2xz=12480 \end{cases}$$

2022 Vietnam National Olympiad, 3
Tags: geometry , circumcircle
Let $ABC$ be an acute triangle, $B,C$ fixed, $A$ moves on the big arc $BC$ of $(ABC)$. Let $O$ be the circumcenter of $(ABC)$ $(B,O,C$ are not collinear, $AB \ne AC)$, $(I)$ is the incircle of triangle $ABC$. $(I)$ tangents to $BC$ at $D$. Let $I_a$ be the $A$-excenter of triangle $ABC$. $I_aD$ cuts $OI$ at $L$. Let $E$ lies on $(I)$ such that $DE \parallel AI$. a) $LE$ cuts $AI$ at $F$. Prove that $AF=AI$. b) Let $M$ lies on the circle $(J)$ go through $I_a,B,C$ such that $I_aM \parallel AD$. $MD$ cuts $(J)$ again at $N$. Prove that the midpoint $T$ of $MN$ lies on a fixed circle.