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

2016 CMIMC, 2
Tags: combinatorics
Six people each flip a fair coin. Everyone who flipped tails then flips their coin again. Given that the probability that all the coins are now heads can be expressed as simplified fraction $\tfrac{m}{n}$, compute $m+n$.

2011 Romania Team Selection Test, 1
Tags: rotation , geometry
Suppose a square of sidelengh $l$ is inside an unit square and does not contain its centre. Show that $l\le 1/2.$ [i]Marius Cavachi[/i]

2000 All-Russian Olympiad Regional Round, 11.7
Tags: combinatorics , coloring
Given numbers $1, 2, . . .,N$, each of which is colored either black or white. It is allowed to repaint it in the opposite direction color any three numbers, one of which is equal to half the sum of the other two. At which $N$ numbers can always be made white?

2001 Mongolian Mathematical Olympiad, Problem 3
Tags: number theory
Let $k\ge0$ be a given integer. Suppose there exists positive integer $n,d$ and an odd integer $m>1$ with $d\mid m^{2^k}-1$ and $m\mid n^d+1$. Find all possible values of $\frac{m^{2^k}-1}d$.

2024 China Girls Math Olympiad, 6
Tags: number theory
Let $n,m,r$ be positive integers such that $n>m$ and both $n^2+r, m^2+r$ are powers of $2$. Show that $n>\frac{2m^2}{r}$.

2013 Stanford Mathematics Tournament, 6
Tags:
How many distinct sets of $5$ distinct positive integers $A$ satisfy the property that for any positive integer $x\le 29$, a subset of $A$ sums to $x$?

2017 Ecuador Juniors, 2
Tags: algebra , system of equations
Find all pairs of real numbers $x, y$ that satisfy the following system of equations $$\begin{cases} x^2 + 3y = 10 \\ 3 + y = \frac{10}{ x} \end{cases}$$

III Soros Olympiad 1996 - 97 (Russia), 11.4
Tags: inequalities , algebra , trigonometry
Find the smallest value of a function $$y = \cos 8x + 3\cos 4x +3\cos2x + 2\cos x.$$

2024 239 Open Mathematical Olympiad, 7
Tags: combinatorics
Prove that there exists a positive integer $k>100$, such that for any set $A$ of $k$ positive reals, there exists a subset $B$ of $100$ numbers, so that none of the sums of at least two numbers in $B$ is in the set $A$.

2005 All-Russian Olympiad, 1
Tags: linear algebra , matrix , inequalities , combinatorics
We select $16$ cells on an $8\times 8$ chessboard. What is the minimal number of pairs of selected cells in the same row or column?

1961 Putnam, A4
Tags: number theory , arithmetic function
Let $\Omega(n)$ be the number of prime factors of $n$. Define $f(1)=1$ and $f(n)=(-1)^{\Omega(n)}.$ Furthermore, let $$F(n)=\sum_{d|n} f(d).$$ Prove that $F(n)=0,1$ for all positive integers $n$. For which integers $n$ is $F(n)=1?$

2004 Miklós Schweitzer, 5
Tags: probability
Let $G$ be a non-solvable finite group and let $\varepsilon > 0$. Show that there exist a positive integer $k$ and a word $w\in F_k$ such that $w$ assumes the value $1$ with probability less than $\varepsilon$ when its $k$ arguments are considered to be independent and uniformly distributed random variables with values in $G$. (We write $F_k$ for the free group generated by $k$ elements.)

1989 All Soviet Union Mathematical Olympiad, 499
Tags: irrational , sum , sum of powers , rational , algebra
Do there exist two reals whose sum is rational, but the sum of their $n$ th powers is irrational for all $n > 1$? Do there exist two reals whose sum is irrational, but the sum of whose $n$ th powers is rational for all $n > 1$?

KoMaL A Problems 2022/2023, A. 834
Tags: geometry , octagon , conic section
Let $A_1A_2\ldots A_8$ be a convex cyclic octagon, and for $i=1,2\ldots,8$ let $B_i=A_iA_{i+3}\cap A_{i+1}A_{i+4}$ (indices are meant modulo 8). Prove that points $B_1,\ldots, B_8$ lie on the same conic section.

PEN A Problems, 51
Tags: divisibility theory
Let $a,b,c$ and $d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^{k}$ and $b+c=2^{m}$ for some integers $k$ and $m$, then $a=1$.

1999 Slovenia National Olympiad, Problem 4
Tags: combinatorics
Let be given three-element subsets $A_1,A_2,\ldots,A_6$ of a six-element set $X$. Prove that the elements of $X$ can be colored with two colors in such a way that none of the given subsets are monochromatic.

2019 PUMaC Algebra A, 3
Tags: quadratic , polynomial , algebra
Let $Q$ be a quadratic polynomial. If the sum of the roots of $Q^{100}(x)$ (where $Q^i(x)$ is defined by $Q^1(x)=Q(x)$, $Q^i(x)=Q(Q^{i-1}(x))$ for integers $i\geq 2$) is $8$ and the sum of the roots of $Q$ is $S$, compute $|\log_2(S)|$.

2021 MOAA, 7
Tags: accuracy
Jeffrey rolls fair three six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

1989 IMO Longlists, 32
Tags: geometry
Given an acute triangle find a point inside the triangle such that the sum of the distances from this point to the three vertices is the least.

2016 Fall CHMMC, 3
Tags: probability
A gambler offers you a $2$ dollar ticket to play the following game: First, you pick a real number $0 \leq p \leq 1$, then you are given a weighted coin that comes up heads with probability $p$. If you receive $1$ dollar the [i]first[/i] time you flip a tail, and if you receive $2$ dollars [i]first[/i] time you flip a head, what is the optimal expected net winning of flipping the coin twice?

2014 Harvard-MIT Mathematics Tournament, 10
Tags: algebra , polynomial , function
For an integer $n$, let $f_9(n)$ denote the number of positive integers $d\leq 9$ dividing $n$. Suppose that $m$ is a positive integer and $b_1,b_2,\ldots,b_m$ are real numbers such that $f_9(n)=\textstyle\sum_{j=1}^mb_jf_9(n-j)$ for all $n>m$. Find the smallest possible value of $m$.

2014 Contests, 2
Tags: number theory , divisor
Find the least natural number $n$, which has at least 6 different divisors $1=d_1<d_2<d_3<d_4<d_5<d_6<...$, for which $d_3+d_4=d_5+6$ and $d_4+d_5=d_6+7$.

2012 NIMO Problems, 1
Tags:
Compute the largest integer $N \le 2012$ with four distinct digits. [i]Proposed by Evan Chen[/i]

2002 Italy TST, 3
Tags: algebra , polynomial , vieta , relatively prime , number theory
Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that $(\text{i})$ $x$ and $y$ are relatively prime; $(\text{ii})$ $x$ divides $y^2+m;$ $(\text{iii})$ $y$ divides $x^2+m.$

Fractal Edition 2, P1
Tags:
Viorel claims that for any natural number $n$ greater than $2024$, the number $2024^n + 1$ is prime. Is Viorel's statement true?