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

2023 Romania Team Selection Test, P4
Tags: combinatorics
Consider a $4\times 4$ array of pairwise distinct positive integers such that on each column, respectively row, one of the numbers is equal to the sum of the other three. Determine the least possible value of the largest number such an array may contain. [i]The Problem Selection Committee[/i]

2024 ELMO Shortlist, G6
Tags: geometry , water balloon
In triangle $ABC$ with $AB<AC$ and $AB+AC=2BC$, let $M$ be the midpoint of $\overline{BC}$. Choose point $P$ on the extension of $\overline{BA}$ past $A$ and point $Q$ on segment $\overline{AC}$ such that $M$ lies on $\overline{PQ}$. Let $X$ be on the opposite side of $\overline{AB}$ from $C$ such that $\overline{AX} \parallel \overline{BC}$ and $AX=AP=AQ$. Let $\overline{BX}$ intersect the circumcircle of $BMQ$ again at $Y \neq B$, and let $\overline{CX}$ intersect the circumcircle of $CMP$ again at $Z \neq C$. Prove that $A$, $Y$, and $Z$ are collinear. [i]Tiger Zhang[/i]

2006 Estonia Team Selection Test, 5
Tags: inequalities , mean , algebra
Let $a_1, a_2, a_3, ...$ be a sequence of positive real numbers. Prove that for any positive integer $n$ the inequality holds $\sum_{i=1}^n b_i^2 \le 4 \sum_{i=1}^n a_i^2$ where $b_i$ is the arithmetic mean of the numbers $a_1, a_2, ..., a_n$

2009 Today's Calculation Of Integral, 444
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate $ \int_0^{\frac {\pi}{6}} \frac {\sin x \plus{} \cos x}{1 \minus{} \sin 2x}\ln\ (2 \plus{} \sin 2x)\ dx.$

2013 China Northern MO, 3
Tags: geometry , fixed
As shown in figure , $A,B$ are two fixed points of circle $\odot O$, $C$ is the midpoint of the major arc $AB$, $D$ is any point of the minor arc $AB$. Tangent at $D$ intersects tangents at $A,B$ at points $E,F$ respectively. Segments $CE$ and $CF$ intersect chord $AB$ at points $G$ and $H$ respectively. Prove that the length of line segment $GH$ has a fixed value. [img]https://cdn.artofproblemsolving.com/attachments/9/2/85227f169193f61e313293e9128f6ece2ff1f7.png[/img]

2016 Canada National Olympiad, 2
Tags: algebra , system of equations
Consider the following system of $10$ equations in $10$ real variables $v_1, \ldots, v_{10}$: \[v_i = 1 + \frac{6v_i^2}{v_1^2 + v_2^2 + \cdots + v_{10}^2} \qquad (i = 1, \ldots, 10).\] Find all $10$-tuples $(v_1, v_2, \ldots , v_{10})$ that are solutions of this system.

2002 Romania Team Selection Test, 3
Tags: pigeonhole principle , number theory
Let $a,b$ be positive real numbers. For any positive integer $n$, denote by $x_n$ the sum of digits of the number $[an+b]$ in it's decimal representation. Show that the sequence $(x_n)_{n\ge 1}$ contains a constant subsequence. [i]Laurentiu Panaitopol[/i]

2011 India National Olympiad, 5
Tags: cyclic quadrilateral , geometry
Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\Gamma.$ Let $E,F,G,H$ be the midpoints of arcs $AB,BC,CD,AD$ of $\Gamma,$ respectively. Suppose that $AC\cdot BD=EG\cdot FH.$ Show that $AC,BD,EG,FH$ are all concurrent.

2009 Balkan MO Shortlist, A1
Tags:
Let $N \in \mathbb{N}$ and $x_k \in [-1,1]$, $1 \le k \le N$ such that $\sum_{k=1}^N x_k =s$. Find all possible values of $\sum_{k=1}^N |x_k|$

1978 IMO Longlists, 22
Tags: special factorizations , number theory
Let $x$ and $y$ be two integers not equal to $0$ such that $x+y$ is a divisor of $x^2+y^2$. And let $\frac{x^2+y^2}{x+y}$ be a divisor of $1978$. Prove that $x = y$. [i]German IMO Selection Test 1979, problem 2[/i]

2008 Puerto Rico Team Selection Test, 2
Tags:
Using digits $ 1, 2, 3, 4, 5, 6$, without repetition, $ 3$ two-digit numbers are formed. The numbers are then added together. Through this procedure, how many different sums may be obtained?

1985 IMO Longlists, 15
Tags: combinatorics
[i]Superchess[/i] is played on on a $12 \times 12$ board, and it uses [i]superknights[/i], which move between opposite corner cells of any $3\times4$ subboard. Is it possible for a [i]superknight[/i] to visit every other cell of a superchessboard exactly once and return to its starting cell ?

2018 Malaysia National Olympiad, A6
Tags: number theory , prime
How many integers $n$ are there such that $n^4 + 2n^3 + 2n^2 + 2n + 1$ is a prime number?

2016 AMC 10, 13
Tags:
Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up? $\textbf{(A) }1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$

Brazil L2 Finals (OBM) - geometry, 1999.1
Tags: geometry
Let $ABCDE$ be a regular pentagon. The star $ACEBD$ has area 1. $AC$ and $BE$ meet at $P$, while $BD$ and $CE$ meet at $Q$. Find the area of $APQD$.

2011 Math Prize For Girls Problems, 8
Tags: geometry
In the figure below, points $A$, $B$, and $C$ are distance 6 from each other. Say that a point $X$ is [i]reachable[/i] if there is a path (not necessarily straight) connecting $A$ and $X$ of length at most 8 that does not intersect the interior of $\overline{BC}$. (Both $X$ and the path must lie on the plane containing $A$, $B$, and $C$.) Let $R$ be the set of reachable points. What is the area of $R$? [asy] unitsize(40); pair A = dir(90); pair B = dir(210); pair C = dir(330); dot(A); dot(B); dot(C); draw(B -- C); label("$A$", A, N); label("$B$", B, W); label("$C$", C, E); [/asy]

2007 China Team Selection Test, 2
Tags: floor function , number theory
A rational number $ x$ is called [i]good[/i] if it satisfies: $ x\equal{}\frac{p}{q}>1$ with $ p$, $ q$ being positive integers, $ \gcd (p,q)\equal{}1$ and there exists constant numbers $ \alpha$, $ N$ such that for any integer $ n\geq N$, \[ |\{x^n\}\minus{}\alpha|\leq\dfrac{1}{2(p\plus{}q)}\] Find all the good numbers.

2019 PUMaC Combinatorics B, 4
Tags: combinatorics , binary
Keith has $10$ coins labeled $1$ through $10$, where the $i$th coin has weight $2^i$. The coins are all fair, so the probability of flipping heads on any of the coins is $\tfrac{1}{2}$. After flipping all of the coins, Keith takes all of the coins which land heads and measures their total weight, $W$. If the probability that $137\le W\le 1061$ is $\tfrac{m}{n}$ for coprime positive integers $m,n$, determine $m+n$.

2010 Tournament Of Towns, 3
Tags: rectangle , exterior angle , geometry
An angle is given in a plane. Using only a compass, one must find out $(a)$ if this angle is acute. Find the minimal number of circles one must draw to be sure. $(b)$ if this angle equals $31^{\circ}$.(One may draw as many circles as one needs).

1962 IMO, 6
Tags: geometry , circumcircle , trigonometry , circles , isosceles
Consider an isosceles triangle. let $R$ be the radius of its circumscribed circle and $r$ be the radius of its inscribed circle. Prove that the distance $d$ between the centers of these two circle is \[ d=\sqrt{R(R-2r)} \]

Ukrainian TYM Qualifying - geometry, I.17
Tags: geometry , 3d geometry , cone , right triangle
A right triangle when rotating around a large leg forms a cone with a volume of $100\pi$. Calculate the length of the path that passes through each vertex of the triangle at rotation of $180^o$ around the point of intersection of its bisectors, if the sum of the diameters of the circles, inscribed in the triangle and circumscribed around it, are equal to $17$.

2019 Yasinsky Geometry Olympiad, p1
Tags: geometry , inradius , angle chasing , angle
It is known that in the triangle $ABC$ the distance from the intersection point of the angle bisector to each of the vertices of the triangle does not exceed the diameter of the circle inscribed in this triangle. Find the angles of the triangle $ABC$. (Grigory Filippovsky)

LMT Team Rounds 2010-20, B5
Tags: algebra
Given the following system of equations $a_1 + a_2 + a_3 = 1$ $a_2 + a_3 + a_4 = 2$ $a_3 + a_4 + a_5 = 3$ $...$ $a_{12} + a_{13} + a_{14} = 12$ $a_{13} + a_{14} + a_1 = 13$ $a_{14 }+ a_1 + a_2 = 14$ find the value of $a_{14}$.

1952 Putnam, A3
Tags:
Develop necessary and sufficient conditions which ensure that $r_1, r_2, r_3$ and $r_1^2, r_2^2, r_3^2$ are simultaneously roots of the equation $x^3 + ax^2 + bx + c = 0.$

2018-2019 Fall SDPC, 2
Tags: number theory
Find all pairs of positive integers $(m,n)$ such that $2^m-n^2$ is the square of an integer.