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

1976 Chisinau City MO, 131
Tags: algebra , inequalities
The sum of the real numbers $x_1, x_2, ...,x_n$ belonging to the segment $[a, b]$ is equal to zero. Prove that $$x_1^2+ x_2^2+ ...+x_n^2 \le - nab.$$

1975 AMC 12/AHSME, 27
Tags:
If $p, q$ and $r$ are distinct roots of $x^3-x^2+x-2=0$, then $p^3+q^3+r^3$ equals $ \textbf{(A)}\ -1 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{none of these} $

2013 NIMO Problems, 6
Tags: symmetry , probability , expected value
Tom has a scientific calculator. Unfortunately, all keys are broken except for one row: 1, 2, 3, + and -. Tom presses a sequence of $5$ random keystrokes; at each stroke, each key is equally likely to be pressed. The calculator then evaluates the entire expression, yielding a result of $E$. Find the expected value of $E$. (Note: Negative numbers are permitted, so 13-22 gives $E = -9$. Any excess operators are parsed as signs, so -2-+3 gives $E=-5$ and -+-31 gives $E = 31$. Trailing operators are discarded, so 2++-+ gives $E=2$. A string consisting only of operators, such as -++-+, gives $E=0$.) [i]Proposed by Lewis Chen[/i]

2008 Bundeswettbewerb Mathematik, 1
Tags: inequalities , algebra
Determine all real $ x$ satisfying the equation \[ \sqrt[5]{x^3 \plus{} 2x} \equal{} \sqrt[3]{x^5\minus{}2x}.\] Odd roots for negative radicands shall be included in the discussion.

2000 Harvard-MIT Mathematics Tournament, 6
Tags: geometry
What is the area of the largest circle contained in an equilateral triangle of area $8\sqrt3$?

2020 Germany Team Selection Test, 1
Tags: combinatorics , induction , local argument
You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

1995 Baltic Way, 6
Tags: search , inequalities
Prove that for positive $a,b,c,d$ \[\frac{a+c}{a+b}+\frac{b+d}{b+c}+\frac{c+a}{c+d}+\frac{d+b}{d+a}\ge 4\]

1989 China Team Selection Test, 2
Tags: trigonometry , function , angle bisector , geometry
$AD$ is the altitude on side $BC$ of triangle $ABC$. If $BC+AD-AB-AC = 0$, find the range of $\angle BAC$. [i]Alternative formulation.[/i] Let $AD$ be the altitude of triangle $ABC$ to the side $BC$. If $BC+AD=AB+AC$, then find the range of $\angle{A}$.

2016 Middle European Mathematical Olympiad, 7
Tags: parity , digit , number theory
A positive integer $n$ is [i]Mozart[/i] if the decimal representation of the sequence $1, 2, \ldots, n$ contains each digit an even number of times. Prove that: 1. All Mozart numbers are even. 2. There are infinitely many Mozart numbers.

2021 Peru Iberoamerican Team Selection Test, P7
Tags: combinatorics
Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right. Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors. [i]Proposed by Gurgen Asatryan, Armenia[/i]

2014 Saudi Arabia Pre-TST, 1.1
Tags: algebra , inequalities
Let $a_1, a_2,...,a_{2n}$ be positive real numbers such that $a_i + a_{n+i} = 1$, for all $i = 1,...,n$. Prove that there exist two different integers $1 \le j, k \le 2n$ for which $$\sqrt{a^2_j-a^2_k} < \frac{1}{\sqrt{n} +\sqrt{n - 1}}$$

2008 China National Olympiad, 1
Tags: circumcircle , incenter , geometric transformation , reflection , perpendicular bisector , geometry
Suppose $\triangle ABC$ is scalene. $O$ is the circumcenter and $A'$ is a point on the extension of segment $AO$ such that $\angle BA'A = \angle CA'A$. Let point $A_1$ and $A_2$ be foot of perpendicular from $A'$ onto $AB$ and $AC$. $H_{A}$ is the foot of perpendicular from $A$ onto $BC$. Denote $R_{A}$ to be the radius of circumcircle of $\triangle H_{A}A_1A_2$. Similiarly we can define $R_{B}$ and $R_{C}$. Show that: \[\frac{1}{R_{A}} + \frac{1}{R_{B}} + \frac{1}{R_{C}} = \frac{2}{R}\] where R is the radius of circumcircle of $\triangle ABC$.

2013 SEEMOUS, Problem 3
Tags: calculus , integration , inequalities
Find the maximum value of $$\int^1_0|f'(x)|^2|f(x)|\frac1{\sqrt x}dx$$over all continuously differentiable functions $f:[0,1]\to\mathbb R$ with $f(0)=0$ and $$\int^1_0|f'(x)|^2dx\le1.$$

2014 Taiwan TST Round 3, 6
Tags: combinatorics , game , invariant
Players $A$ and $B$ play a "paintful" game on the real line. Player $A$ has a pot of paint with four units of black ink. A quantity $p$ of this ink suffices to blacken a (closed) real interval of length $p$. In every round, player $A$ picks some positive integer $m$ and provides $1/2^m $ units of ink from the pot. Player $B$ then picks an integer $k$ and blackens the interval from $k/2^m$ to $(k+1)/2^m$ (some parts of this interval may have been blackened before). The goal of player $A$ is to reach a situation where the pot is empty and the interval $[0,1]$ is not completely blackened. Decide whether there exists a strategy for player $A$ to win in a finite number of moves.

2021 Dutch IMO TST, 3
Tags: number theory , divide , divisible
Prove that for every positive integer $n$ there are positive integers $a$ and $b$ exist with $n | 4a^2 + 9b^2 -1$.

1985 Balkan MO, 4
Tags: combinatorics
There are $1985$ participants to an international meeting. In any group of three participants there are at least two who speak the same language. It is known that each participant speaks at most five languages. Prove that there exist at least $200$ participans who speak the same language.

2020 Taiwan APMO Preliminary, P2
Tags: combinatorics
A and B two people are throwing n fair coins.X and Y are the times they get heads. If throwing coins are mutually independent events, (1)When n=5, what is the possibility of X=Y? (2)When n=6, what is the possibility of X=Y+1?

2005 USAMTS Problems, 2
Tags: probability , expected value
George has six ropes. He chooses two of the twelve loose ends at random (possibly from the same rope), and ties them together, leaving ten loose ends. He again chooses two loose ends at random and joins them, and so on, until there are no loose ends. Find, with proof, the expected value of the number of loops George ends up with.

STEMS 2021 Math Cat C, Q5
Tags: combinatorics
Find the largest constant $c$, such that if there are $N$ discs in the plane such that every two of them intersect, then there must exist a point which lies in the common intersection of $cN + O(1)$ discs

2012 Putnam, 6
Tags: function , calculus , integration , rectangle , real analysis
Let $f(x,y)$ be a continuous, real-valued function on $\mathbb{R}^2.$ Suppose that, for every rectangular region $R$ of area $1,$ the double integral of $f(x,y)$ over $R$ equals $0.$ Must $f(x,y)$ be identically $0?$

1966 IMO Longlists, 37
Tags: parallelogram , cyclic quadrilateral , geometry
Show that the four perpendiculars dropped from the midpoints of the sides of a cyclic quadrilateral to the respective opposite sides are concurrent. [b]Note by Darij:[/b] A [i]cyclic quadrilateral [/i]is a quadrilateral inscribed in a circle.

2007 Hong Kong TST, 5
Tags: algebra
[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url] Problem 5 The sequence $\{a_{n}\}$ is defined by $a_{1}=0$ and $(n+1)^{3}a_{n+1}=2n^{2}(2n+1)a_{n}+2(3n+1)$ for all integers $\geq 1$. Show that infintely many members of the sequence are positive integers.

2012 Purple Comet Problems, 28
Tags: probability , number theory , relatively prime
A bag contains $8$ green candies and $4$ red candies. You randomly select one candy at a time to eat. If you eat five candies, there are relatively prime positive integers $m$ and $n$ so that $\frac{m}{n}$ is the probability that you do not eat a green candy after you eat a red candy. Find $m+n$.

2023 Switzerland - Final Round, 5
Tags: algebra , functional equation , function , domain
Let $D$ be the set of real numbers excluding $-1$. Find all functions $f: D \to D$ such that for all $x,y \in D$ satisfying $x \neq 0$ and $y \neq -x$, the equality $$(f(f(x))+y)f \left(\frac{y}{x} \right)+f(f(y))=x$$ holds.

2023 MOAA, 10
Tags:
If $x,y,z$ satisfy the system of equations \[xy+yz+zx=23\] \[\frac{y}{x+y}+\frac{z}{y+z}+\frac{x}{z+x}=-1\] \[\frac{z^2x}{x+y}+\frac{x^2y}{y+z}+\frac{y^2z}{z+x}=202\] Find the value of $x^2+y^2+z^2$. [i]Proposed by Harry Kim[/i]