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

2008 Indonesia MO, 2
Tags: function , inequalities
Prove that for $ x,y\in\mathbb{R^ \plus{} }$, $ \frac {1}{(1 \plus{} \sqrt {x})^{2}} \plus{} \frac {1}{(1 \plus{} \sqrt {y})^{2}} \ge \frac {2}{x \plus{} y \plus{} 2}$

1993 AMC 8, 19
Tags:
$(1901+1902+1903+\cdots + 1993) - (101+102+103+\cdots + 193) = $ $\text{(A)}\ 167,400 \qquad \text{(B)}\ 172,050 \qquad \text{(C)}\ 181,071 \qquad \text{(D)}\ 199,300 \qquad \text{(E)}\ 362,142$

2016 Serbia Additional Team Selection Test, 1
Tags: polynomial , algebra
Let $P_0(x)=x^3-4x$. Sequence of polynomials is defined as following:\\ $P_{n+1}=P_n(1+x)P_n(1-x)-1$.\\ Prove that $x^{2016}|P_{2016}(x)$.

1994 Czech And Slovak Olympiad IIIA, 5
Tags: altitude , equal area , geometry , equilateral , triangle area
In an acute-angled triangle $ABC$, the altitudes $AA_1,BB_1,CC_1$ intersect at point $V$. If the triangles $AC_1V, BA_1V, CB_1V$ have the same area, does it follow that the triangle $ABC$ is equilateral?

2011 Middle European Mathematical Olympiad, 7
Tags: modular arithmetic , number theory , case view
Let $A$ and $B$ be disjoint nonempty sets with $A \cup B = \{1, 2,3, \ldots, 10\}$. Show that there exist elements $a \in A$ and $b \in B$ such that the number $a^3 + ab^2 + b^3$ is divisible by $11$.

2011 AIME Problems, 13
Tags: geometry , 3d geometry , vector , analytic geometry , graphing lines , slope , ratio
A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\tfrac{r-\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positive integers, and $r+s+t<1000$. Find $r+s+t$.

2023 Korea Junior Math Olympiad, 5
Tags: combinatorics
For a positive integer $n(\geq 5)$, there are $n$ white stones and $n$ black stones (total $2n$ stones) lined up in a row. The first $n$ stones from the left are white, and the next $n$ stones are black. $$\underbrace{\Circle \Circle \cdots \Circle}_n \underbrace{\CIRCLE \CIRCLE \cdots \CIRCLE}_n $$ You can swap the stones by repeating the following operation. [b](Operation)[/b] Choose a positive integer $k (\leq 2n - 5)$, and swap $k$-th stone and $(k+5)$-th stone from the left. Find all positive integers $n$ such that we can make first $n$ stones to be black and the next $n$ stones to be white in finite number of operations.

2004 National High School Mathematics League, 10
Tags: number theory
$p$ is a give odd prime, if $\sqrt{k^2-pk}$ is a positive integer, then the value of positive integer $k$ is________.

2007 Princeton University Math Competition, 2
Tags: geometry , 3d geometry
A black witch's hat is in the classic shape of a cone on top of a circular brim. The cone has a slant height of $18$ inches and a base radius of $3$ inches. The brim has a radius of $5$ inches. What is the total surface area of the hat?

2004 India IMO Training Camp, 1
Tags: trigonometry , inequalities , geometry
Prove that in any triangle $ABC$, \[ 0 < \cot { \left( \frac{A}{4} \right)} - \tan{ \left( \frac{B}{4} \right) } - \tan{ \left( \frac{C}{4} \right) } - 1 < 2 \cot { \left( \frac{A}{2} \right) }. \]

2023 LMT Spring, 7
Tags: geometry
In $\vartriangle ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$ be a point on $BC$ such that $BD = 6$. Let $E$ be a point on $CA$ such that $CE = 6$. Finally, let $F$ be a point on $AB$ such that $AF = 6$. Find the area of $\vartriangle DEF$.

2019 MIG, 8
Tags:
Greg plays a game in which he is given three random $1$ digit numbers, each between $0$ and $9$, inclusive, with repeats allowed. He is to put these three numbers into any order. Exactly one ordering of the three numbers is correct, and if he guesses the correct ordering, he wins $\$150$. What are Greg's expected winnings for this game, given that he randomly guesses one valid ordering when he plays?

2011 IFYM, Sozopol, 8
Tags: fraction , algebra
The fraction $\frac{1}{p}$, where $p$ is a prime number coprime with 10, is presented as an infinite periodic fraction. Prove that, if the number of digits in the period is even, then the arithmetic mean of the digits in the period is equal to $\frac{9}{2}$.

2023 CMIMC Integration Bee, 15
Tags: calculus , integration
\[\int_0^\infty \left(1-e^{-\pi/x^2}\right)^2\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

2024 Junior Balkan Team Selection Tests - Moldova, 2
Tags: number theory
Prove that the number $ \underbrace{88\dots8}_\text{2024\; \textrm{times}}$ is divisible by 2024.

2017 Harvard-MIT Mathematics Tournament, 12
Tags: combinatorics
In a certain college containing $1000$ students, students may choose to major in exactly one of math, computer science, finance, or English. The [i]diversity ratio[/i] $d(s)$ of a student $s$ is the defined as number of students in a different major from $s$ divided by the number of students in the same major as $s$ (including $s$). The [i]diversity[/i] $D$ of the college is the sum of all the diversity ratios $d(s)$. Determine all possible values of $D$.

2011 Balkan MO Shortlist, G4
Tags: geometric inequality , geometry , incircle
Given a triangle $ABC$, the line parallel to the side $BC$ and tangent to the incircle of the triangle meets the sides $AB$ and $AC$ at the points $A_1$ and $A_2$ , the points $B_1, B_2$ and $C_1, C_2$ are de ned similarly. Show that $$AA_1 \cdot AA_2 + BB_1 \cdot BB_2 + CC_1 \cdot CC_2 \ge \frac19 (AB^2 + BC^2 + CA^2)$$

2022 Belarusian National Olympiad, 10.7
Tags: algebra , polynomial
Find all positive integers $a$ for which there exists a polynomial $p(x)$ with integer coefficients such that $p(\sqrt{2}+1)=2-\sqrt{2}$ and $p(\sqrt{2}+2)=a$

1998 APMO, 2
Tags: number theory
Show that for any positive integers $a$ and $b$, $(36a+b)(a+36b)$ cannot be a power of $2$.

2020 Lusophon Mathematical Olympiad, 3
Tags: geometry , geometric inequality , inequalities
Let $ABC$ be a triangle and on the sides we draw, externally, the squares $BADE, CBFG$ and $ACHI$. Determine the greatest positive real constant $k$ such that, for any triangle $\triangle ABC$, the following inequality is true: $[DEFGHI]\geq k\cdot [ABC]$ Note: $[X]$ denotes the area of polygon $X$.

2020 Romania EGMO TST, P1
Tags: number theory , divisibility
Determine if for any positive integers $a,b,c$ there exist pairwise distinct non-negative integers $A,B,C$ which are greater than $2019$ such that $a+A,b+B$ and $c+C$ divide $ABC$.

1960 Poland - Second Round, 1
Tags: sum , algebra , inequalities
Prove that if the real numbers $ a $ and $ b $ are not both equal to zero, then for every natural $ n $ $$ a^{2n} + a^{2n-1}b + a^{2n-2} b^2 + \ldots + ab^{2n-1} + b^{2n} > 0. $$

2019 Romanian Masters In Mathematics, 3
Tags: cycle , combinatorics , graph theory , probability
Given any positive real number $\varepsilon$, prove that, for all but finitely many positive integers $v$, any graph on $v$ vertices with at least $(1+\varepsilon)v$ edges has two distinct simple cycles of equal lengths. (Recall that the notion of a simple cycle does not allow repetition of vertices in a cycle.) [i]Fedor Petrov, Russia[/i]

2023 MOAA, 21
Tags:
In obtuse triangle $ABC$ where $\angle B > 90^\circ$ let $H$ and $O$ be its orthocenter and circumcenter respectively. Let $D$ be the foot of the altitude from $A$ to $HC$ and $E$ be the foot of the altitude from $B$ to $AC$ such that $O,E,D$ lie on a line. If $OC=8$ and $OE=4$, find the area of triangle $HAB$. [i]Proposed by Harry Kim[/i]

1999 National High School Mathematics League, 5
Tags:
In a ping-pong game, it was planned to have a competition between any two players. But three players quit the game after having 2 competitions. In the end, the number of competitions played is 50. So the number of competitions between the three players is $\text{(A)}0\qquad\text{(B)}1\qquad\text{(C)}2\qquad\text{(D)}3$