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

2016 Dutch BxMO TST, 3
Tags: incircle , geometry , collinear , right triangle
Let $\vartriangle ABC$ be a right-angled triangle with $\angle A = 90^o$ and circumcircle $\Gamma$. The inscribed circle is tangent to $BC$ in point $D$. Let $E$ be the midpoint of the arc $AB$ of $\Gamma$ not containing $C$ and let $F$ be the midpoint of the arc $AC$ of $\Gamma$ not containing $B$. (a) Prove that $\vartriangle ABC \sim \vartriangle DEF$. (b) Prove that $EF$ goes through the points of tangency of the incircle to $AB$ and $AC$.

2003 China Second Round Olympiad, 2
Tags: integration , calculus , floor function , perimeter , modular arithmetic , algebra , geometry
Let the three sides of a triangle be $\ell, m, n$, respectively, satisfying $\ell>m>n$ and $\left\{\frac{3^\ell}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$, where $\{x\}=x-\lfloor{x}\rfloor$ and $\lfloor{x}\rfloor$ denotes the integral part of the number $x$. Find the minimum perimeter of such a triangle.

2007 Ukraine Team Selection Test, 8
Tags: algebra , polynomial , inequalities
$ F(x)$ is polynomial with real coefficients. $ F(x) \equal{} x^{4}\plus{}a_{1}x^{3}\plus{}a_{2}x^{2}\plus{}a_{1}x^{1}\plus{}a_{0}$. $ M$ is local maximum and $ m$ is minimum. Prove that $ \frac{3}{10}(\frac{a_{1}^{2}}{4}\minus{}\frac{2a_{2}}{3^{2}})^{2}< M\minus{}m < 3(\frac{a_{1}^{2}}{4}\minus{}\frac{2a_{2}}{3^{2}})^{2}$

1948 Moscow Mathematical Olympiad, 150
Tags: geometry , symmetry
Can a figure have a greater than $1$ and finite number of centers of symmetry?

2009 Canadian Mathematical Olympiad Qualification Repechage, 10
Tags: function , combinatorics
Ten boxes are arranged in a circle. Each box initially contains a positive number of golf balls. A move consists of taking all of the golf balls from one of the boxes and placing them into the boxes that follow it in a counterclockwise direction, putting one ball into each box. Prove that if the next move always starts with the box where the last ball of the previous move was placed, then after some number of moves, we get back to the initial distribution of golf balls in the boxes.

1969 Swedish Mathematical Competition, 6
Tags: combinatorics , combinatorial geometry , point , triangle
Given $3n$ points in the plane, no three collinear, is it always possible to form $n$ triangles (with vertices at the points), so that no point in the plane lies in more than one triangle?

2018 Korea Junior Math Olympiad, 2
Tags: number theory
Find all positive integer $N$ which has not less than $4$ positive divisors, such that the sum of squares of the $4$ smallest positive divisors of $N$ is equal to $N$.

1981 Czech and Slovak Olympiad III A, 3
Tags: square , locus , equilateral , triangle , geometry
Let $ABCD$ be a unit square. Consider an equilateral triangle $XYZ$ with $X,Y$ as (inner or boundary) points of the square. Determine the locus $M$ of vertices $Z$ of all these triangles $XYZ$ and compute the area of $M.$

2013 ELMO Shortlist, 2
Tags: combinatorics , induction , strong induction , logarithm
Let $n$ be a fixed positive integer. Initially, $n$ 1's are written on a blackboard. Every minute, David picks two numbers $x$ and $y$ written on the blackboard, erases them, and writes the number $(x+y)^4$ on the blackboard. Show that after $n-1$ minutes, the number written on the blackboard is at least $2^{\frac{4n^2-4}{3}}$. [i]Proposed by Calvin Deng[/i]

2012 Paraguay Mathematical Olympiad, 1
Tags: number theory
Define a list of number with the following properties: - The first number of the list is a one-digit natural number. - Each number (since the second) is obtained by adding $9$ to the number before in the list. - The number $2012$ is in that list. Find the first number of the list.

2011 Austria Beginners' Competition, 4
Tags: isosceles , geometry
Let $ABC$ be an isosceles triangle with $AC = BC$. On the arc $CA$ of its circumcircle, which does not contain $ B$, there is a point $ P$. The projection of $C$ on the line $AP$ is denoted by $E$, the projection of $C$ on the line $BP$ is denoted by $F$. Prove that the lines $AE$ and $BF$ have equal lengths. (W. Janous, WRG Ursulincn, Innsbruck)

2022/2023 Tournament of Towns, P3
Tags: pentagon , angle chasing , geometry
A pentagon $ABCDE$ is circumscribed about a circle. The angles at the vertices $A{}$, $C{}$ and $E{}$ of the pentagon are equal to $100^\circ$. Find the measure of the angle $\angle ACE$.

2019 MIG, 2
Tags:
A cup with a volume of $8$ fluid ounces is filled at the rate of $0.5$ ounces per second. However, a hole at the bottom of the cup also drains it at the rate of $0.3$ ounces per second. Once the cup is full, how many ounces of water will have drained out of the cup?

2025 CMIMC Team, 9
Tags: team
Given a triangle, $AB=78, BC=50, AC=112,$ construct squares $ABXY, BCPQ, ACMN$ outside the triangle. Let $L_1, L_2, L_3$ be the midpoints of $\overline{MP}, \overline{QX}, \overline{NY},$ respectively. Find the area of $L_1L_2L_3.$

Kyiv City MO 1984-93 - geometry, 1985.7.3
Tags: geometry , area
$O$ is the point of intersection of the diagonals of the convex quadrilateral $ABCD$. It is known that the areas of triangles $AOB, BOC, COD$ and $DOA$ are expressed in natural numbers. Prove that the product of these areas cannot end in $1985$.

2004 Switzerland Team Selection Test, 12
Tags: number theory , natural
Find all natural numbers which can be written in the form $\frac{(a+b+c)^2}{abc}$ , where $a,b,c \in N$.

2021/2022 Tournament of Towns, P5
Tags: geometry , hexagon
A parallelogram $ABCD$ is split by the diagonal $BD$ into two equal triangles. A regular hexagon is inscribed into the triangle $ABD$ so that two of its consecutive sides lie on $AB$ and $AD$ and one of its vertices lies on $BD$. Another regular hexagon is inscribed into the triangle $CBD{}$ so that two of its consecutive vertices lie on $CB$ and $CD$ and one of its sides lies on $BD$. Which of the hexagons is bigger? [i]Konstantin Knop[/i]

2000 Stanford Mathematics Tournament, 15
Tags: logarithm
Which is greater: $ (3^5)^{(5^3)}$ or $ (5^3)^{(3^5)}$?

2001 National Olympiad First Round, 2
Tags:
Each of the football teams Istanbulspor, Yesildirek, Vefa, Karagumruk, and Adalet, played exactly one match against the other four teams. Istanbulspor defeated all teams except Yesildirek; Yesildirek defeated Istanbulspor but lost to all the other teams. Vefa defeated all except Istanbulspor. The winner of the game Karagumruk-Adalet is Karagumruk. In how many ways one can order these five teams such that each team except the last, defeated the next team? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ \text{None of the preceding} $

2012 Mathcenter Contest + Longlist, 4
Tags: geometry , inequalities , geometric inequality , algebra
Let $a,b,c$ be the side lengths of any triangle. Prove that $$\frac{a}{\sqrt{2b^2+2c^2-a^2}}+\frac{b}{\sqrt{2c^2+2a^2-b^2 }}+\frac{c}{\sqrt{2a^2+2b^2-c^2}}\ge \sqrt{3}.$$ [i](Zhuge Liang)[/i]

2005 Thailand Mathematical Olympiad, 4
Tags: incenter , geometry
Triangle $\vartriangle ABC$ is inscribed in the circle with diameter $BC$. If $AB = 3$, $AC = 4$, and $O$ is the incenter of $\vartriangle ABC$, then find $BO \cdot OC$.

2014 Baltic Way, 16
Tags: modular arithmetic , number theory
Determine whether $712! + 1$ is a prime number.

2023 CMIMC Integration Bee, 1
Tags: calculus , integration
\[\int_2^0 x^2+3\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

1981 Yugoslav Team Selection Test, Problem 2
Tags: geometry
Suppose that there is a point $S$ inside a quadrilateral $ABCD$ such that segments $SA,SB,SC,SD$ divide the quadrilateral into four triangles of equal areas. Prove that one of the diagonals of the quadrilateral bisects the other one.

2012 Today's Calculation Of Integral, 776
Tags: calculus computations , calculus , integration
Evaluate $\int_{\frac{1-\sqrt{5}}{2}}^{\frac{1+\sqrt{5}}{2}} (2x^2-1)e^{2x}dx.$