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

1982 USAMO, 1
Tags: combinatorics
In a party with $1982$ persons, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else?

2024 CMIMC Geometry, 1
Tags: geometry
Let $ABCD$ be a rectangle with $AB=5$. Let $E$ be on $\overline{AB}$ and $F$ be on $\overline {CD}$ such that $AE=CF=4$. Let $P$ and $Q$ lie inside $ABCD$ such that triangles $AEP$ and $CFQ$ are equilateral. If $E$, $P$, $Q$, and $F$ lie on a single line, find $\overline{BC}$. [i]Proposed by Connor Gordon[/i]

1974 IMO Longlists, 32
Tags: inequalities
Let $a_1,a_2,\ldots ,a_n$ be $n$ real numbers such that $0<a\le a_k\le b$ for $k=1,2,\ldots ,n$. If $m_1=\frac{1}{n}(a_1+a_2+\cdots+a_n)$ and $m_2=\frac{1}{n}(a_1^2+a_2^2+\cdots + a_n^2)$, prove that $m_2\le\frac{(a+b)^2}{4ab}m_1^2$ and find a necessary and sufficient condition for equality.

VMEO III 2006, 12.4
Tags: combinatorics , symmetry
Given a binary serie $A=a_1a_2...a_k$ is called "symmetry" if $a_i=a_{k+1-i}$ for all $i=1,2,3,...,k$, and $k$ is the length of that binary serie. If $A=11...1$ or $A=00...0$ then it is called "special". Find all positive integers $m$ and $n$ such that there exist non "special" binary series $A$ (length $m$) and $B$ (length $n$) satisfying when we place them next to each other, we receive a "symmetry" binary serie $AB$

2005 Today's Calculation Of Integral, 56
Tags: limit , integration , calculus , calculus computations
Evaluate \[\lim_{n\to\infty} \sum_{k=1}^n \frac{[\sqrt{2n^2-k^2}\ ]}{n^2}\] $[x]$ is the greatest integer $\leq x$.

2004 Silk Road, 4
Tags: group theory , abstract algebra , combinatorics
Natural $n \geq 2$ is given. Group of people calls $n-compact$, if for any men from group, we can found $n$ people (without he), each two of there are familiar. Find maximum $N$ such that for any $n-compact$ group, consisting $N$ people contains subgroup from $n+1$ people, each of two of there are familiar.

2011 Bogdan Stan, 3
Tags: unique solution , function , algebra
Solve in $ \mathbb{R} $ the equation $ 4^{x^2-x}=\log_2 x+\sqrt{x-1} +14. $ [i]Marin Tolosi[/i]

2013 Saudi Arabia Pre-TST, 3.2
Tags: divide , divisible , number theory
Let $a_1, a_2,..., a_9$ be integers. Prove that if $19$ divides $a_1^9+a_2^9+...+a_9^9$ then $19$ divides the product $a_1a_2...a_9$.

2023 IFYM, Sozopol, 3
Tags: number theory
Let $n \geq 2$ be an integer such that $6^n + 11^n$ is divisible by $n$. Prove that $n^{100} + 6^n + 11^n$ is divisible by $17n$ and not divisible by $289n$.

1993 Tournament Of Towns, (380) 2
Tags: geometry , collinear
Vertices $A$, $B$ and $C$ of a triangle are connected with points $A'$ , $B'$ and $C'$ lying in the opposite sides of the triangle (not at vertices). Can the midpoints of the segments $AA'$, $BB'$ and $CC'$ lie in a straight line? (Folklore)

2023 Belarus Team Selection Test, 3.1
Tags: algebra
Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

2022 New Zealand MO, 2
Tags: perfect power , number theory , power of number
Is it possible to pair up the numbers $0, 1, 2, 3,... , 61$ in such a way that when we sum each pair, the product of the $31$ numbers we get is a perfect f ifth power?

2022 Taiwan TST Round 1, N
Tags: number theory
Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2021 CCA Math Bonanza, I5
Tags:
If digits $A$, $B$, and $C$ (between $0$ and $9$ inclusive) satisfy \begin{tabular}{c@{\,}c@{\,}c@{\,}c} & $C$ & $C$ & $A$ \\ + & $B$ & $2$ & $B$ \\\hline & $A$ & $8$ & $8$ \\ \end{tabular} what is $A \cdot B \cdot C$? [i]2021 CCA Math Bonanza Individual Round #5[/i]

2014 NIMO Problems, 9
Tags: geometry , rectangle , induction
Two players play a game involving an $n \times n$ grid of chocolate. Each turn, a player may either eat a piece of chocolate (of any size), or split an existing piece of chocolate into two rectangles along a grid-line. The player who moves last loses. For how many positive integers $n$ less than $1000$ does the second player win? (Splitting a piece of chocolate refers to taking an $a \times b$ piece, and breaking it into an $(a-c) \times b$ and a $c \times b$ piece, or an $a \times (b-d)$ and an $a \times d$ piece.) [i]Proposed by Lewis Chen[/i]

2022 AMC 10, 19
Tags: least common multiple
Define $L_n$ as the least common multiple of all the integers from $1$ to $n$ inclusive. There is a unique integer $h$ such that $\frac{1}{1}+\frac{1}{2}+\frac{1}{3} \ldots +\frac{1}{17}=\frac{h}{L_{17}}$. What is the remainder when $h$ is divided by $17?$ $\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$

1984 AMC 12/AHSME, 28
Tags:
The number of distinct pairs of integers $(x,y)$ such that \[0 < x < y\quad \text{and}\quad \sqrt{1984} = \sqrt{x} + \sqrt{y}\] is $\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }3\qquad \textbf{(E) }7$

1991 National High School Mathematics League, 11
Tags: complex number
For two complex numbers $z_1,z_2$ satisfy that $|z_1|=|z_1+z_2|=3,|z_1-z_2|=3\sqrt3$, then $\log_3|(z_1\overline{z_2})^{2000}+(\overline{z_1}z_2)^{2000}|=$________.

2022 Saudi Arabia IMO TST, 2
Tags: geometry , cyclic quadrilateral , concurrency , angle chasing
Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

2023 CCA Math Bonanza, L1.1
Tags: probability
If 100 dice are rolled, what is the probability that the sum of the numbers rolled is even? [i]Lightning 1.1[/i]

2013 India Regional Mathematical Olympiad, 2
Tags: divisor , prime
Determine the smallest prime that does not divide any five-digit number whose digits are in a strictly increasing order.

2009 Balkan MO Shortlist, A4
Tags: function , induction , algebra
Denote by $ S$ the set of all positive integers. Find all functions $ f: S \rightarrow S$ such that \[ f (f^2(m) \plus{} 2f^2(n)) \equal{} m^2 \plus{} 2 n^2\] for all $ m,n \in S$. [i]Bulgaria[/i]

2010 IFYM, Sozopol, 3
Tags: geometry , construction , constructive geometry
Two circles are intersecting in points $P$ and $Q$. Construct two points $A$ and $B$ on these circles so that $P\in AB$ and the product $AP.PB$ is maximal.

2012 Kazakhstan National Olympiad, 2
Tags: combinatorics
We call a $6\times 6$ table consisting of zeros and ones [i]right[/i] if the sum of the numbers in each row and each column is equal to $3$. Two right tables are called [i]similar[/i] if one can get from one to the other by successive interchanges of rows and columns. Find the largest possible size of a set of pairwise similar right tables.

2013 Romania National Olympiad, 2
Tags: complex number , inequalities
To be considered the following complex and distinct $a,b,c,d$. Prove that the following affirmations are equivalent: i)For every $z\in \mathbb{C}$ the inequality takes place :$\left| z-a \right|+\left| z-b \right|\ge \left| z-c \right|+\left| z-d \right|$; ii)There is $t\in \left( 0,1 \right)$ so that $c=ta+\left( 1-t \right)b$ si $d=\left( 1-t \right)a+tb$