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

2017 Saudi Arabia JBMO TST, 4
Tags: combinatorics , chessboard
Find the number of ways one can put numbers $1$ or $2$ in each cell of an $8\times 8$ chessboard in such a way that the sum of the numbers in each column and in each row is an odd number. (Two ways are considered different if the number in some cell in the first way is different from the number in the cell situated in the corresponding position in the second way)

2002 Greece JBMO TST, 3
Tags: geometry , angle bisector , perpendicular , angle
Let $ABC$ be a triangle with $\angle A=60^o, AB\ne AC$ and let $AD$ be the angle bisector of $\angle A$. Line $(e)$ that is perpendicular on the angle bisector $AD$ at point $A$, intersects the extension of side $BC$ at point $E$ and also $BE=AB+AC$. Find the angles $\angle B$ and $\angle C$ of the triangle $ABC$.

1948 Moscow Mathematical Olympiad, 151
Tags: midpoint , parallel , distance , geometry , convex
The distance between the midpoints of the opposite sides of a convex quadrilateral is equal to a half sum of lengths of the other two sides. Prove that the first pair of sides is parallel.

2010 Stanford Mathematics Tournament, 9
Tags: geometry , circumcircle , incenter , trigonometry , system of equations , algebra
For an acute triangle $ABC$ and a point $X$ satisfying $\angle{ABX}+\angle{ACX}=\angle{CBX}+\angle{BCX}$. Fi nd the minimum length of $AX$ if $AB=13$, $BC=14$, and $CA=15$.

2017 QEDMO 15th, 5
Tags: number theory
Let $F$ be a finite subset of the integer numbers. We define a new subset $s(F)$ in that $a\in Z$ lies in $s (F)$ if and only if exactly one of the numbers $a$ and $a -1$ in $F$. In the same way one gets from $s (F)$ the set $s^2(F) = s (s (F))$ and by $n$-fold application of $s$ then iteratively further subsets $s^n (F)$. Prove there are infinitely many natural numbers $n$ for which $s^n (F) = F\cup \{a + n|a \in F\}$.

2011 AMC 10, 24
Tags: geometry , 3d geometry , tetrahedron , ratio , octahedron , rhombus
Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra? $ \textbf{(A)}\ \frac{1}{12}\qquad\textbf{(B)}\ \frac{\sqrt{2}}{12}\qquad\textbf{(C)}\ \frac{\sqrt{3}}{12}\qquad\textbf{(D)}\ \frac{1}{6}\qquad\textbf{(E)}\ \frac{\sqrt{2}}{6} $

LMT Guts Rounds, 13
Tags:
A circle with center $O$ has radius $5,$ and has two points $A,B$ on the circle such that $\angle AOB = 90^{\circ}.$ Rays $OA$ and $OB$ are extended to points $C$ and $D,$ respectively, such that $AB$ is parallel to $CD,$ and the length of $CD$ is $200\%$ more than the radius of circle $O.$ Determine the length of $AC.$

2006 Kazakhstan National Olympiad, 3
Tags: combinatorics
The racing tournament has $12$ stages and $ n $ participants. After each stage, all participants, depending on the occupied place $ k $, receive points $ a_k $ (the numbers $ a_k $ are natural and $ a_1> a_2> \dots> a_n $). For what is the smallest $ n $ the tournament organizer can choose the numbers $ a_1 $, $ \dots $, $ a_n $ so that after the penultimate stage for any possible distribution of places at least two participants had a chance to take first place.

2020 Serbia National Math Olympiad, 3
Tags: circumcircle , triangle , geometry , angle
We are given a triangle $ABC$. Points $D$ and $E$ on the line $AB$ are such that $AD=AC$ and $BE=BC$, with the arrangment of points $D - A - B - E$. The circumscribed circles of the triangles $DBC$ and $EAC$ meet again at the point $X\neq C$, and the circumscribed circles of the triangles $DEC$ and $ABC$ meet again at the point $Y\neq C$. Find the measure of $\angle ACB$ given the condition $DY+EY=2XY$.

2005 MOP Homework, 6
Tags: combinatorics
A computer network is formed by connecting $2004$ computers by cables. A set $S$ of these computers is said to be independent if no pair of computers of $S$ is connected by a cable. Suppose that the number of cables used is the minimum number possible such that the size of any independent set is at most $50$. Let $c(L)$ be the number of cables connected to computer $L$. Show that for any distinct computers $A$ and $B$, $c(A)=c(B)$ if they are connected by a cable and $|c(A)-c(B)| \le 1$ otherwise. Also, find the number of cables used in the network.

2013 Polish MO Finals, 4
Tags: 3d geometry , tetrahedron , rotation , trapezoid , geometry
Given is a tetrahedron $ABCD$ in which $AB=CD$ and the sum of measures of the angles $BAD$ and $BCD$ equals $180$ degrees. Prove that the measure of the angle $BAD$ is larger than the measure of the angle $ADC$.

2019 Romania National Olympiad, 2
Tags: group theory , abstract algebra , superior algebra
Let $n \geq 4$ be an even natural number and $G$ be a subgroup of $GL_2(\mathbb{C})$ with $|G| = n.$ Prove that there exists $H \leq G$ such that $\{ I_2 \} \neq H$ and $H \neq G$ such that $XYX^{-1} \in H, \: \forall X \in G$ and $\forall Y \in H$

2019 Paraguay Mathematical Olympiad, 1
Tags: algebra , trinomial , quadratic
Elías and Juanca solve the same problem by posing a quadratic equation. Elijah is wrong when writing the independent term and gets as results of the problem $-1$ and $-3$. Juanca is wrong only when writing the coefficient of the first degree term and gets as results of the problem $16$ and $-2$. What are the correct results of the problem?

2011 NIMO Problems, 2
Tags:
The sum of three consecutive integers is $15$. Determine their product.

2015 Iran Geometry Olympiad, 2
Tags: midpoint , reflection , circumcircle , geometry
In acute-angled triangle $ABC$, $BH$ is the altitude of the vertex $B$. The points $D$ and $E$ are midpoints of $AB$ and $AC$ respectively. Suppose that $F$ be the reflection of $H$ with respect to $ED$. Prove that the line $BF$ passes through circumcenter of $ABC$. by Davood Vakili

1963 Poland - Second Round, 6
Tags: geometry , 3d geometry
From the point $ S $ of space arise $ 3 $ half-lines: $ SA $, $ SB $ and $ SC $, none of which is perpendicular to both others. Through each of these rays, a plane is drawn perpendicular to the plane containing the other two rays. Prove that the drawn planes intersect along one line $ d $.

2014 India IMO Training Camp, 2
Tags: number theory
Find all positive integers $x$ and $y$ such that $x^{x+y}=y^{3x}$.

2009 AMC 8, 11
Tags:
The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $ \$1.43$. Some of the $ 30$ sixth graders each bought a pencil, and they paid a total of $ \$1.95$. How many more sixth graders than seventh graders bought a pencil? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

2005 Finnish National High School Mathematics Competition, 3
Tags: absolute value , algebra
Solve the group of equations: \[\begin{cases} (x + y)^3 = z \\ (y + z)^3 = x \\ (z + x)^3 = y \end{cases}\]

2023 CMIMC Integration Bee, 13
Tags: calculus , integration
\[\int_0^1 2^{\sqrt x}\log^2(2)+\log^2(1+x)\,\mathrm dx\] [i]Proposed by Thomas Lam[/i]

2010 AMC 8, 9
Tags: percent
Ryan got $80\%$ of the problems on a $25$-problem test, $90\%$ on a $40$-problem test, and $70\%$ on a $10$-problem test. What percent of all problems did Ryan answer correctly? $ \textbf{(A)}\ 64 \qquad\textbf{(B)}\ 75\qquad\textbf{(C)}\ 80\qquad\textbf{(D)}\ 84\qquad\textbf{(E)}\ 86 $

2013 India National Olympiad, 3
Tags: calculus , integration , inequalities , function , algebra , polynomial
Let $a,b,c,d \in \mathbb{N}$ such that $a \ge b \ge c \ge d $. Show that the equation $x^4 - ax^3 - bx^2 - cx -d = 0$ has no integer solution.

2017 CMIMC Computer Science, 6
Tags: computer science
Define a self-balanced tree to be a tree such that for any node, the size of the left subtree is within 1 of the size of the right subtree. How many balanced trees are there of size 2046?

1985 Traian Lălescu, 2.1
Tags: equation , algebra , floor
Solve $ \quad 5\lfloor x^2\rfloor -2\lfloor x\rfloor +2=0. $

1992 National High School Mathematics League, 7
Tags: arithmetic sequence , geometric series
For real numbers $x,y,z$, $3x,4y,5z$ are geometric series, $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ are arithmetic sequence. Then $\frac{x}{z}+\frac{z}{x}=$________.