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 Poland - Second Round, 4
Tags: algebra , system of equations
Given pairwise different real numbers $a,b,c,d,e$ such that $$ \left\{ \begin{array}{ll} ab + b = ac + a, \\ bc + c = bd + b, \\ cd + d = ce + c, \\ de + e = da + d. \end{array} \right. $$ Prove that $abcde=1$.

1987 China Team Selection Test, 2
Tags: polygon , analytic geometry , induction , geometry , coordinates
A closed recticular polygon with 100 sides (may be concave) is given such that it's vertices have integer coordinates, it's sides are parallel to the axis and all it's sides have odd length. Prove that it's area is odd.

2012-2013 SDML (Middle School), 1
Tags:
What is the least positive integer $n$ for which $9n$ is a perfect square and $12n$ is a perfect cube?

2018 BMT Spring, Tie 3
Tags: geometry
Consider a regular polygon with $2^n$ sides, for $n \ge 2$, inscribed in a circle of radius $1$. Denote the area of this polygon by $A_n$. Compute $\prod_{i=2}^{\infty}\frac{A_i}{A_{i+1}}$

2004 AMC 10, 3
Tags:
Alicia earns $ \$20$ per hour, of which $ 1.45\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes? $ \textbf{(A)}\ 0.0029\qquad \textbf{(B)}\ 0.029\qquad \textbf{(C)}\ 0.29\qquad \textbf{(D)}\ 2.9\qquad \textbf{(E)}\ 29$

2006 Silk Road, 3
Tags: quadratic , number theory
A subset $S$ of the set $M=\{1,2,.....,p-1\}$,where $p$ is a prime number of the kind $12n+11$,is [i]essential[/i],if the product ${\Pi}_s$ of all elements of the subset is not less than the product $\bar{{\Pi}_s}$ of all other elements of the set.The [b]difference[/b] $\bigtriangleup_s=\Pi_s-\bar{{\Pi}_s}$ is called [i]the deviation[/i] of the subset $S$.Define the least possible remainder of division by $p$ of the deviation of an essential subset,containing $\frac{p-1}{2}$ elements.

2012 Argentina Cono Sur TST, 1
Tags: pigeonhole principle , ceiling function , rectangle , geometry
Sofía colours $46$ cells of a $9 \times 9$ board red. If Pedro can find a $2 \times 2$ square from the board that has $3$ or more red cells, he wins; otherwise, Sofía wins. Determine the player with the winning strategy.

2019 India Regional Mathematical Olympiad, 3
Tags: algebra , inequality , system of equations
Find all triples of non-negative real numbers $(a,b,c)$ which satisfy the following set of equations $$a^2+ab=c$$ $$b^2+bc=a$$ $$c^2+ca=b$$

2017 Romania National Olympiad, 2
Tags: function , algebra
A function $ f:\mathbb{Q}_{>0}\longrightarrow\mathbb{Q} $ has the following property: $$ f(xy)=f(x)+f(y),\quad x,y\in\mathbb{Q}_{>0} $$ [b]a)[/b] Demonstrate that there are no injective functions with this property. [b]b)[/b] Do exist surjective functions having this property?

2000 AMC 12/AHSME, 2
Tags:
$ 2000(2000^{2000}) \equal{}$ $ \textbf{(A)}\ 2000^{2001} \qquad \textbf{(B)}\ 4000^{2000} \qquad \textbf{(C)}\ 2000^{4000}\qquad \textbf{(D)}\ 4,000,000^{2000} \qquad \textbf{(E)}\ 2000^{4,000,000}$

2021 AMC 12/AHSME Fall, 9
Tags: geometry
Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A$, $O$, and $C$? $\textbf{(A)}\ 9\pi \qquad\textbf{(B)}\ 12\pi \qquad\textbf{(C)}\ 18\pi \qquad\textbf{(D)}\ 24\pi \qquad\textbf{(E)}\ 27\pi$

2004 Putnam, A5
Tags: probability , induction , perimeter , expected value , geometry
An $m\times n$ checkerboard is colored randomly: each square is independently assigned red or black with probability $\frac12.$ we say that two squares, $p$ and $q$, are in the same connected monochromatic region if there is a sequence of squares, all of the same color, starting at $p$ and ending at $q,$ in which successive squares in the sequence share a common side. Show that the expected number of connected monochromatic regions is greater than $\frac{mn}8.$

2006 Bulgaria Team Selection Test, 3
Tags: algebra , polynomial , combinatorics
[b]Problem 6.[/b] Let $p>2$ be prime. Find the number of the subsets $B$ of the set $A=\{1,2,\ldots,p-1\}$ such that, the sum of the elements of $B$ is divisible by $p.$ [i] Ivan Landgev[/i]

2013 European Mathematical Cup, 3
Tags: combinatorics
We call a sequence of $n$ digits one or zero a code. Subsequence of a code is a palindrome if it is the same after we reverse the order of its digits. A palindrome is called nice if its digits occur consecutively in the code. (Code $(1101)$ contains $10$ palindromes, of which $6$ are nice.) a) What is the least number of palindromes in a code? b) What is the least number of nice palindromes in a code?

2005 Grigore Moisil Urziceni, 1
Tags: monotone , inequalities
Prove that $ 5^x+6^x\le 4^x+8^x, $ for any nonegative real numbers $ x. $

2014 Cuba MO, 7
Tags: diophantine equation , diophantine , number theory
Find all pairs of integers $(a, b)$ that satisfy the equation $$(a + 1)(b- 1) = a^2b^2.$$

1994 Romania TST for IMO, 4:
Tags: triangle inequality , geometry , inequalities
Let be given two concentric circles of radii $R$ and $R_1 > R$. Let quadrilateral $ABCD$ is inscribed in the smaller circle and let the rays $CD, DA, AB, BC$ meet the larger circle at $A_1, B_1, C_1, D_1$ respectively. Prove that $$ \frac{\sigma(A_1B_1C_1D_1)}{\sigma(ABCD)} \geq \frac{R_1^2}{R^2}$$ where $\sigma(P)$ denotes the area of a polygon $P.$

III Soros Olympiad 1996 - 97 (Russia), 11.7
Tags: algebra , polynomial
Let us assume that each of the equations $x^7 + x^2 + 1= 0$ and $x^5- x^4 + x^2- x + 1.001 = 0$ has a single root. Which of these roots is larger?

2014 Baltic Way, 11
Tags: circumcircle , geometric transformation , reflection , geometry
Let $\Gamma$ be the circumcircle of an acute triangle $ABC.$ The perpendicular to $AB$ from $C$ meets $AB$ at $D$ and $\Gamma$ again at $E.$ The bisector of angle $C$ meets $AB$ at $F$ and $\Gamma$ again at $G.$ The line $GD$ meets $\Gamma$ again at $H$ and the line $HF$ meets $\Gamma$ again at $I.$ Prove that $AI = EB.$

1990 Bundeswettbewerb Mathematik, 4
Tags: inequalities , function , 3d geometry , sphere , geometry
In the plane there is a worm of length 1. Prove that it can be always covered by means of half of a circular disk of diameter 1. [i]Note.[/i] Under a "worm", we understand a continuous curve. The "half of a circular disk" is a semicircle including its boundary.

2023 Stanford Mathematics Tournament, 8
Tags:
Define the Fibonacci numbers via $F_0=0$, $f_1=1$, and $F_{n-1}+F_{n-2}$. Olivia flips two fair coins at the same time, repeatedly, until she has flipped a tails on both, not necessarily on the same throw. She records the number of pairs of flips $c$ until this happens (not including the last pair, so if on the last flip both coins turned up tails $c$ would be $0$). What is the expected value $F_c$?

2014 Junior Balkan Team Selection Tests - Moldova, 2
Tags: diophantine equation , diophantine , number theory
Determine all pairs of integers $(x, y)$ that satisfy equation $(y - 2) x^2 + (y^2 - 6y + 8) x = y^2 - 5y + 62$.

2022 MOAA, 3
Tags: geometry
The area of the figure enclosed by the $x$-axis, $y$-axis, and line $7x + 8y = 15$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2003 Bulgaria National Olympiad, 2
Tags: inequalities , number theory
Let $a,b,c$ be rational numbers such that $a+b+c$ and $a^2+b^2+c^2$ are [b]equal[/b] integers. Prove that the number $abc$ can be written as the ratio of a perfect cube and a perfect square which are relatively prime.

1993 All-Russian Olympiad Regional Round, 11.2
Tags: number theory , divide , divisible
Prove that, for every integer $n > 2$, the number $$\left[\left( \sqrt[3]{n}+\sqrt[3]{n+2}\right)^3\right]+1$$ is divisible by $8$.