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

2020 MBMT, 4
Tags:
Ken has a six sided die. He rolls the die, and if the result is not even, he rolls the die one more time. Find the probability that he ends up with an even number. [i]Proposed by Gabriel Wu[/i]

2016 NIMO Problems, 1
Tags: arithmetic sequence
Find the value of $\lfloor 1 \rfloor + \lfloor 1.7 \rfloor +\lfloor 2.4 \rfloor +\lfloor 3.1 \rfloor +\cdots+\lfloor 99 \rfloor$. [i]Proposed by Jack Cornish[/i]

2018 CMIMC Team, 3-1/3-2
Tags: team
Let $\Omega$ be a semicircle with endpoints $A$ and $B$ and diameter 3. Points $X$ and $Y$ are located on the boundary of $\Omega$ such that the distance from $X$ to $AB$ is $\frac{5}{4}$ and the distance from $Y$ to $AB$ is $\frac{1}{4}$. Compute \[(AX+BX)^2 - (AY+BY)^2.\] Let $T = TNYWR$. $T$ people each put a distinct marble into a bag; its contents are mixed randomly and one marble is distributed back to each person. Given that at least one person got their own marble back, what is the probability that everyone else also received their own marble?

2013 Moldova Team Selection Test, 4
Tags: number theory
$p$ is a 4k+3 prime. Prove that there are infinite $p$ which satisfies $p|2^ny+1$. $y$ is an random integer.

2007 China Team Selection Test, 3
Tags: function , induction , combinatorics
Let $ n$ be positive integer, $ A,B\subseteq[0,n]$ are sets of integers satisfying $ \mid A\mid \plus{} \mid B\mid\ge n \plus{} 2.$ Prove that there exist $ a\in A, b\in B$ such that $ a \plus{} b$ is a power of $ 2.$

1949-56 Chisinau City MO, 35
Tags: arithmetic sequence , algebra
The numbers $a^2, b^2, c^2$ form an arithmetic progression. Show that the numbers $\frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b}$ also form arithmetic progression.

2002 Kazakhstan National Olympiad, 1
Tags: geometry , incenter , equal segments , angle
Let $ O $ be the center of the inscribed circle of the triangle $ ABC $, tangent to the side of $ BC $. Let $ M $ be the midpoint of $ AC $, and $ P $ be the intersection point of $ MO $ and $ BC $. Prove that $ AB = BP $ if $ \angle BAC = 2 \angle ACB $.

2015 ASDAN Math Tournament, 8
Tags:
Lynnelle and Moor love toy cars, and together, they have $27$ red cars, $27$ purple cars, and $27$ green cars. The number of red cars Lynnelle has individually is the same as the number of green cars Moor has individually. In addition, Lynnelle has $17$ more cars of any color than Moor has of any color. How many purple cars does Lynnelle have?

2025 Belarusian National Olympiad, 8.3
Tags: number theory
A positive integer with three digits is written on the board. Each second the number $n$ on the board gets replaced by $n+\frac{n}{p}$, where $p$ is the largest prime divisor of $n$. Prove that either after 999 seconds or 1000 second the number on the board will be a power of two. [i]A. Voidelevich[/i]

2021 Nigerian Senior MO Round 3, 3
Tags: number theory
Find all pairs of natural numbers $(p,n)$ with $p$ prime such that $p^6+p^5+n^3+n=n^5+n^2$

2015 Online Math Open Problems, 2
Tags:
At a national math contest, students are being housed in single rooms and double rooms; it is known that $75\%$ of the students are housed in double rooms. What percentage of the rooms occupied are double rooms? [i]Proposed by Evan Chen[/i]

2005 Olympic Revenge, 6
Tags: combinatorics
Zé Roberto and Humberto are playing the Millenium Game! There are 30 empty boxes in a queue, and each box have a capacity of one blue stome. Each player takes a blue stone and places it in a box (and it is a [i]move[/i]). The winner is who, in its move, obtain three full consecutive boxes. If Zé Roberto is the first player, who has the winner strategy?

2016 Polish MO Finals, 3
Tags: combinatorics , counting
Let $a, \ b \in \mathbb{Z_{+}}$. Denote $f(a, b)$ the number sequences $s_1, \ s_2, \ ..., \ s_a$, $s_i \in \mathbb{Z}$ such that $|s_1|+|s_2|+...+|s_a| \le b$. Show that $f(a, b)=f(b, a)$.

2014 Harvard-MIT Mathematics Tournament, 4
Tags: probability , expected value
[4] Let $D$ be the set of divisors of $100$. Let $Z$ be the set of integers between $1$ and $100$, inclusive. Mark chooses an element $d$ of $D$ and an element $z$ of $Z$ uniformly at random. What is the probability that $d$ divides $z$?

1998 Dutch Mathematical Olympiad, 5
Tags: function
Find all real solutions of the following equation: \[ (x + 1995)(x + 1997)(x + 1999)(x + 2001) + 16 = 0. \]

2010 Singapore Senior Math Olympiad, 1
Tags: geometry
In the $\triangle ABC$ with $AC>BC$ and $\angle B<90^{\circ}$, $D$ is the foot of the perpendicular from $A$ onto $BC$ and $E$ is the foot of perpendicular from $D$ onto $AC$. Let $F$ be the point on the segment $DE$ such that $EF \cdot DC=BD \cdot DE$. Prove that $AF$ is perpendicular to $BF$.

1995 Tournament Of Towns, (454) 3
Tags: perpendicular , geometry
Triangle $ABC$ is inscribed in a circle with center $O$. Let $q$ be the circle passing through $A$, $O$ and $B$. The lines $CA$ and $CB$ intersect $q$ at the points $D$ and $E$ (different from $A$ and $B$). Prove that the lines $CO$ and $DE$ are perpendicular to each other. (S Markelov)

1972 All Soviet Union Mathematical Olympiad, 166
Tags: combinatorial geometry , geometry , square
Each of the $9$ straight lines divides the given square onto two quadrangles with the areas ratio as $2:3$. Prove that there exist three of them intersecting in one point

2011 Bosnia And Herzegovina - Regional Olympiad, 2
Tags: number theory , prime number
If $p>2$ is prime number and $m$ and $n$ are positive integers such that $$\frac{m}{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1}$$ Prove that $p$ divides $m$

2002 Baltic Way, 18
Tags: number theory
Find all integers $n>1$ such that any prime divisor of $n^6-1$ is a divisor of $(n^3-1)(n^2-1)$.

2011 Turkey Team Selection Test, 2
Tags: inequalities
Let $a,b,c$ be positive real numbers satisfying $a^2+b^2+c^2 \geq 3.$ Prove that \[ \frac{(a+1)(b+2)}{(b+1)(b+5)} + \frac{(b+1)(c+2)}{(c+1)(c+5)}+\frac{(c+1)(a+2)}{(a+1)(a+5)} \geq \frac{3}{2} \]

2010 Philippine MO, 5
Tags: number theory
Determine, with proof, the smallest positive integer $n$ with the following property: For every choice of $n$ integers, there exist at least two whose sum or difference is divisible by $2009$.

2012 Pan African, 1
Tags: induction , sfft , geometry , 3d geometry , invariant , special factorizations , combinatorics
The numbers $\frac{1}{1}, \frac{1}{2}, \cdots , \frac{1}{2012}$ are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say $x$ and $y$, erases them and she writes instead the number $x + y + xy$. She continues to do this until only one number is left on the board. What are the possible values of the final number?

2010 Saint Petersburg Mathematical Olympiad, 1
Tags: algebra , polynomial
$f(x)$ is square trinomial. Is it always possible to find polynomial $g(x)$ with fourth degree, such that $f(g(x))=0$ has not roots?

2024 AIME, 11
Tags:
Each vertex of a regular octagon is coloured either red or blue with equal probability. The probability that the octagon can then be rotated in such a way that all of the blue vertices end up at points that were originally red is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?