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

2001 Mongolian Mathematical Olympiad, Problem 2
Tags: inequalities , sequence , algebra
For positive real numbers $b_1,b_2,\ldots,b_n$ define $$a_1=\frac{b_1}{b_1+b_2+\ldots+b_n}\enspace\text{ and }\enspace a_k=\frac{b_1+\ldots+b_k}{b_1+\ldots+b_{k-1}}\text{ for }k>1.$$Prove that $a_1+a_2+\ldots+a_n\le\frac1{a_1}+\frac1{a_2}+\ldots+\frac1{a_n}$

2015 India Regional MathematicaI Olympiad, 3
Tags:
3. Show that there are infinitely many triples (x,y,z) of integers such that $x^3 + y^4 = z^{31}$.

2012 Irish Math Olympiad, 2
Tags: geometry , angle bisector , circumcircle
Consider a triangle $ABC$ with $|AB|\neq |AC|$. The angle bisector of the angle $CAB$ intersects the circumcircle of $\triangle ABC$ at two points $A$ and $D$. The circle of center $D$ and radius $|DC|$ intersects the line $AC$ at two points $C$ and $B’$. The line $BB’$ intersects the circumcircle of $\triangle ABC$ at $B$ and $E$. Prove that $B’$ is the orthocenter of $\triangle AED$.

2020 MBMT, 35
Tags:
Tim has a multiset of positive integers. Let $c_i$ be the number of occurrences of numbers that are [i]at least[/i] $i$ in the multiset. Let $m$ be the maximum element of the multiset. Tim calls a multiset [i]spicy[/i] if $c_1, \dots, c_m$ is a sequence of strictly decreasing powers of $3$. Tim calls the [i]hotness[/i] of a spicy multiset the sum of its elements. Find the sum of the hotness of all spicy multisets that satisfy $c_1 = 3^{2020}$. Give your answer $\pmod{1000}$. (Note: a multiset is an unordered set of numbers that can have repeats) [i]Proposed by Timothy Qian[/i]

1985 IMO Shortlist, 6
Tags: algebra , sequence , inequality , calculus , recurrence relation , inequalities
Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that \[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]

2015 Princeton University Math Competition, 14
Tags:
Marie is painting a $4 \times 4$ grid of identical square windows. Initially, they are all orange but she wants to paint $4$ of them black. How many ways can she do this up to rotation and reflection?

2010 Today's Calculation Of Integral, 538
Tags: calculus , integration , calculus computations , logarithm
Evaluate $ \int_1^{\sqrt{2}} \frac{x^2\plus{}1}{x\sqrt{x^4\plus{}1}}\ dx$.

1969 IMO Longlists, 29
Tags: trigonometric equation , trigonometry , algebra
$(GDR 1)$ Find all real numbers $\lambda$ such that the equation $\sin^4 x - \cos^4 x = \lambda(\tan^4 x - \cot^4 x)$ $(a)$ has no solution, $(b)$ has exactly one solution, $(c)$ has exactly two solutions, $(d)$ has more than two solutions (in the interval $(0, \frac{\pi}{4}).$

2012 Gheorghe Vranceanu, 2
Tags: group theory , abstract algebra , superior algebra
A group $ G $ of order at least $ 4 $ has the property that there exists a natural number $ n\not\in\{ 1,|G| \} $ such that $ G $ admits exactly $ \binom{|G|-1}{n-1} $ subgroups of order $ n. $ Show that $ G $ is commutative. [i]Marius Tărnăuceanu[/i]

2009 Tournament Of Towns, 1
Tags: induction
One hundred pirates played cards. When the game was over, each pirate calculated the amount he won or lost. The pirates have a gold sand as a currency; each has enough to pay his debt. Gold could only change hands in the following way. Either one pirate pays an equal amount to every other pirate, or one pirate receives the same amount from every other pirate. Prove that after several such steps, it is possible for each winner to receive exactly what he has won and for each loser to pay exactly what he has lost. [i](4 points)[/i]

2022 Saint Petersburg Mathematical Olympiad, 5
Tags: geometry
Altitudes $AA_1, BB_1, CC_1$ of acute triangle $ABC$ intersect at point $H$. On the tangent drawn from point $C$ to the circle $(AB_1C_1)$, the perpendicular $HQ$ is drawn (the point $Q$ lies inside the triangle $ABC$). Prove that the circle passing through the point $B_1$ and touching the line $AB$ at point $A$ is also tangent to line $A_1Q$.

2018 Peru Iberoamerican Team Selection Test, P9
Tags: geometry , circumcircle , collinear , collinearity
Let $\Gamma$ be the circumcircle of a triangle $ABC$ with $AB <BC$, and let $M$ be the midpoint from the side $AC$ . The median of side $AC$ cuts $\Gamma$ at points $X$ and $Y$ ($X$ in the arc $ABC$). The circumcircle of the triangle $BMY$ cuts the line $AB$ at $P$ ($P \ne B$) and the line $BC$ in $Q$ ($Q \ne B$). The circumcircles of the triangles $PBC$ and $QBA$ are cut in $R$ ($R \ne B$). Prove that points $X, B$ and $R$ are collinear.

2019 Rioplatense Mathematical Olympiad, Level 3, 4
Tags: number theory
Prove that there are infinite triples $(a,b,c)$ of positive integers $a,b,c>1$, $gcd(a,b)=gcd(b,c)=gcd(c,a)=1$ such that $a+b+c$ divides $a^b+b^c+c^a$.

1994 APMO, 5
Tags: floor function , number theory , logarithm
You are given three lists $A$, $B$, and $C$. List $A$ contains the numbers of the form $10^k$ in base $10$, with $k$ any integer greater than or equal to $1$. Lists $B$ and $C$ contain the same numbers translated into base $2$ and $5$ respectively: $$\begin{array}{lll} A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}$$ Prove that for every integer $n > 1$, there is exactly one number in exactly one of the lists $B$ or $C$ that has exactly $n$ digits.

2013 Switzerland - Final Round, 8
Tags: algebra , inequalities
Let $a, b, c > 0$ be real numbers. Show the following inequality: $$a^2 \cdot \frac{a - b}{a + b}+ b^2\cdot \frac{b - c}{b + c}+ c^2\cdot \frac{c - a}{c + a} \ge 0 .$$ When does equality holds?

2009 Purple Comet Problems, 7
Tags:
How many distinct four letter arrangements can be formed by rearranging the letters found in the word [b]FLUFFY[/b]? For example, FLYF and ULFY are two possible arrangements.

2019 Sharygin Geometry Olympiad, 11
Tags: geometry
Morteza marks six points in the plane. He then calculates and writes down the area of every triangle with vertices in these points ($20$ numbers). Is it possible that all of these numbers are integers, and that they add up to $2019$?

PEN P Problems, 30
Tags: additive number theory
Let $a_{1}, a_{2}, a_{3}, \cdots$ be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form $a_{i}+2a_{j}+4a_{k}$, where $i, j, $ and $k$ are not necessarily distinct. Determine $a_{1998}$.

2022 Iran Team Selection Test, 1
Tags: combinatorics
Morteza Has $100$ sets. at each step Mahdi can choose two distinct sets of them and Morteza tells him the intersection and union of those two sets. Find the least steps that Mahdi can find all of the sets. Proposed by Morteza Saghafian

2010 IMO Shortlist, 3
Tags: matrix , combinatorics , chessboard , counting
2500 chess kings have to be placed on a $100 \times 100$ chessboard so that [b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex); [b](ii)[/b] each row and each column contains exactly 25 kings. Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.) [i]Proposed by Sergei Berlov, Russia[/i]

2009 Cuba MO, 4
Tags: algebra , functional
Determine all the functions $f : R \to R$ such that: $$x + f(xf(y)) = f(y) + yf(x)$$ for all $x, y \in R$.

1966 Polish MO Finals, 5
Tags: geometry , hexagon , concurrency
Each of the diagonals $AD$, $BE$, $CF$ of a convex hexagon $ABCDEF$ bisects the area of the hexagon. Prove that these three diagonals pass through the same point.

1992 Chile National Olympiad, 4
Tags: equilateral , parallel
Given three parallel lines, prove that there are three points, one on each line, which are the vertices of an equilateral triangle.

2008 IberoAmerican Olympiad For University Students, 2
Tags: polynomial , ratio , geometry , algebra
Prove that for each natural number $n$ there is a polynomial $f$ with real coefficients and degree $n$ such that $ p(x)=f(x^2-1)$ is divisible by $f(x)$ over the ring $\mathbb{R}[x]$.

2019 Iran Team Selection Test, 6
Tags: inequalities
$x,y$ and $z$ are real numbers such that $x+y+z=xy+yz+zx$. Prove that $$\frac{x}{\sqrt{x^4+x^2+1}}+\frac{y}{\sqrt{y^4+y^2+1}}+\frac{z}{\sqrt{z^4+z^2+1}}\geq \frac{-1}{\sqrt{3}}.$$ [i]Proposed by Navid Safaei[/i]