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

2012 Nordic, 2
Tags: circumcircle , geometric transformation , reflection , geometry
Given a triangle $ABC$, let $P$ lie on the circumcircle of the triangle and be the midpoint of the arc $BC$ which does not contain $A$. Draw a straight line $l$ through $P$ so that $l$ is parallel to $AB$. Denote by $k$ the circle which passes through $B$, and is tangent to $l$ at the point $P$. Let $Q$ be the second point of intersection of $k$ and the line $AB$ (if there is no second point of intersection, choose $Q = B$). Prove that $AQ = AC$.

1993 ITAMO, 3
Tags: combinatorics , infinite chessboard , sequence
Consider an infinite chessboard whose rows and columns are indexed by positive integers. At most one coin can be put on any cell of the chessboard. Let be given two arbitrary sequences ($a_n$) and ($b_n$) of positive integers ($n \in N$). Assuming that infinitely many coins are available, prove that they can be arranged on the chessboard so that there are $a_n$ coins in the $n$-th row and $b_n$ coins in the $n$-th column for all $n$.

2006 AMC 12/AHSME, 24
Tags: inequalities , geometry , analytic geometry , binomial theorem , algebra
The expression \[ (x \plus{} y \plus{} z)^{2006} \plus{} (x \minus{} y \minus{} z)^{2006} \]is simplified by expanding it and combining like terms. How many terms are in the simplified expression? $ \textbf{(A) } 6018 \qquad \textbf{(B) } 671,676 \qquad \textbf{(C) } 1,007,514 \qquad \textbf{(D) } 1,008,016 \qquad \textbf{(E) } 2,015,028$

1981 National High School Mathematics League, 4
Tags:
In the four figures, which one has the largest area? $\text{(A)}\triangle ABC: \angle A=\frac{\pi}{3},\angle B=\frac{\pi}{4},|AC|=\sqrt2$ $\text{(B)}$trapezium: two diagonals are $\sqrt2$ and $\sqrt3$, intersection angle is $\frac{5\pi}{12}$. $\text{(C)}$Circle: with a radius of $1$. $\text{(D)}$Square: the length of a diagonal is $2.5$.

2019 Swedish Mathematical Competition, 2
Tags: geometry , concyclic , diameter , tangent
Segment $AB$ is the diameter of a circle. Points $C$ and $D$ lie on the circle. The rays $AC$ and $AD$ intersect the tangent to the circle at point $B$ at points $P$ and $Q$, respectively. Show that points $C, D, P$ and $Q$ lie on a circle.

1979 All Soviet Union Mathematical Olympiad, 275
Tags: chessboard , minimum , combinatorics
What is the least possible number of the checkers being required a) for the $8\times 8$ chess-board, b) for the $n\times n$ chess-board, to provide the property: [i]Every line (of the chess-board fields) parallel to the side or diagonal is occupied by at least one checker[/i] ?

1989 USAMO, 1
Tags: number theory
For each positive integer $n$, let \begin{eqnarray*} S_n &=& 1 + \frac 12 + \frac 13 + \cdots + \frac 1n, \\ T_n &=& S_1 + S_2 + S_3 + \cdots + S_n, \\ U_n &=& \frac{T_1}{2} + \frac{T_2}{3} + \frac{T_3}{4} + \cdots + \frac{T_n}{n+1}. \end{eqnarray*} Find, with proof, integers $0 < a, b,c, d < 1000000$ such that $T_{1988} = a S_{1989} - b$ and $U_{1988} = c S_{1989} - d$.

1988 India National Olympiad, 3
Tags: combinatorics , knights and knaves
Five men, $ A$, $ B$, $ C$, $ D$, $ E$ are wearing caps of black or white colour without each knowing the colour of his cap. It is known that a man wearing black cap always speaks the truth while the ones wearing white always tell lies. If they make the following statements, find the colour worn by each of them: $ A$ : I see three black caps and one white cap. $ B$ : I see four white caps $ C$ : I see one black cap and three white caps $ D$ : I see your four black caps.

1984 IMO Longlists, 4
Tags: geometry theorems , geometry
Given a triangle $ABC$, three equilateral triangles $AEB, BFC$, and $CGA$ are constructed in the exterior of $ABC$. Prove that: $(a) CE = AF = BG$; $(b) CE, AF$, and $BG$ have a common point. I could not find a separate topic for this question and I need one. http://en.wikipedia.org/wiki/Fermat_point of course.

1979 Bulgaria National Olympiad, Problem 1
Tags: number theory , diophantine equation , diophantine
Show that there are no integers $x$ and $y$ satisfying $x^2 + 5 = y^3$. Daniel Harrer

2007 District Olympiad, 4
Tags: number theory
Let $n$ be a positive integer which is not prime. Prove that there exist $k, a_{1},a_{2},...a_{k}>1$ positive integers such that $a_{1}+a_{2}+\cdots+a_{k}=n(\frac1{a_{1}}+\frac1{a_{2}}+\cdots+\frac1{a_{k}})$ Edit: the $a_{i}'s$ have to be grater than 1. Sorry, my mistake :blush:

1955 Putnam, B6
Tags:
Prove: If $f(x) > 0$ for all $x$ and $f(x) \rightarrow 0$ as $x \rightarrow \infty,$ then there exists at most a finite number of solutions of \[ f(m) + f(n) + f(p) = 1 \] in positive integers $m, n,$ and $p.$

2024 CMIMC Integration Bee, 3
Tags: calculus , integration
\[\int_0^1 \frac{\log(x)}{\sqrt x}\mathrm dx\] [i]Proposed by Robert Trosten[/i]

2005 Poland - Second Round, 1
Tags: algebra , polynomial , modular arithmetic , quadratic , number theory
The polynomial $W(x)=x^2+ax+b$ with integer coefficients has the following property: for every prime number $p$ there is an integer $k$ such that both $W(k)$ and $W(k+1)$ are divisible by $p$. Show that there is an integer $m$ such that $W(m)=W(m+1)=0$.

2014 Harvard-MIT Mathematics Tournament, 23
Tags: probability , expected value , absolute value
Let $S=\{-100,-99,-98,\ldots,99,100\}$. Choose a $50$-element subset $T$ of $S$ at random. Find the expected number of elements of the set $\{|x|:x\in T\}$.

2010 F = Ma, 15
Tags:
A small block moving with initial speed $v_\text{0}$ moves smoothly onto a sloped big block of mass $M$. After the small block reaches the height $h$ on the slope, it slides down. Find the height $h$. (A) $h=\frac{v_\text{0}^2}{2g}$ (B) $h=\frac{1}{g}\frac{Mv_\text{0}^2}{m+M}$ (C) $h=\frac{1}{2g}\frac{Mv_\text{0}^2}{m+M}$ (D) $h=\frac{1}{2g}\frac{mv_\text{0}^2}{m+M}$ (e) $h=\frac{v_\text{0}^2}{g}$

2018 Moscow Mathematical Olympiad, 3
Tags: geometry , circumcircle
$O$ is circumcircle and $AH$ is the altitude of $\triangle ABC$. $P$ is the point on line $OC$ such that $AP \perp OC$. Prove, that midpoint of $AB$ lies on the line $HP$.

2020 Saint Petersburg Mathematical Olympiad, 4.
Tags: inequalities , number theory
Let $m$ be a given positive integer. Prove that there exists a positive integer $k$ such that it holds $$1\leq \frac{1^m+2^m+3^m+\ldots +(k-1)^m}{k^m}<2.$$

1999 Slovenia National Olympiad, Problem 3
Tags: geometry
A semicircle with diameter $AB$ is given. Two non-intersecting circles $k_1$ and $k_2$ with different radii touch the diameter $AB$ and touch the semicircle internally at $C$ and $D$, respectively. An interior common tangent $t$ of $k_1$ and $k_2$ touches $k_1$ at $E$ and $k_2$ at $F$. Prove that the lines $CE$ and $DF$ intersect on the semicircle.

2018 Taiwan TST Round 1, 2
Tags: geometry , circumcircle
In a plane, we are given $ 100 $ circles with radius $ 1 $ so that the area of any triangle whose vertices are circumcenters of those circles is at most $ 100 $. Prove that one may find a line that intersects at least $ 10 $ circles.

2010 Contests, 523
Tags: calculus , integration , calculus computations , logarithm
Prove the following inequality. \[ \ln \frac {\sqrt {2009} \plus{} \sqrt {2010}}{\sqrt {2008} \plus{} \sqrt {2009}} < \int_{\sqrt {2008}}^{\sqrt {2009}} \frac {\sqrt {1 \minus{} e^{ \minus{} x^2}}}{x}\ dx < \sqrt {2009} \minus{} \sqrt {2008}\]

2019 Regional Olympiad of Mexico Southeast, 1
Tags: number theory
Found the smaller multiple of $2019$ of the form $abcabc\dots abc$, where $a,b$ and $c$ are digits.

1962 Czech and Slovak Olympiad III A, 1
Tags: perfect square , prime , algebra , quadratic polynomial
Determine all integers $x$ such that $2x^2-x-36$ is a perfect square of a prime.

1997 All-Russian Olympiad Regional Round, 8.3
Tags: geometry , equilateral , angle
On sides $AB$ and $BC$ of an equilateral triangle $ABC$ are taken points $D$ and $K$, and on the side $AC$ , points $E$ and $M$ so that $DA + AE = KC +CM = AB$. Prove that the angle between lines $DM$ and $KE$ is equal to $60^o$.

2011 Tournament of Towns, 5
Tags: geometry
$AD$ and $BE$ are altitudes of an acute triangle $ABC$. From $D$, perpendiculars are dropped to $AB$ at $G$ and $AC$ at $K$. From $E$, perpendiculars are dropped to $AB$ at $F$ and $BC$ at $H$. Prove that $FG$ is parallel to $HK$ and $FK = GH$.