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

2015 IMO Shortlist, N5
Tags: number theory
Find all positive integers $(a,b,c)$ such that $$ab-c,\quad bc-a,\quad ca-b$$ are all powers of $2$. [i]Proposed by Serbia[/i]

2016 AMC 10, 5
Tags:
The mean age of Amanda's $4$ cousins is $8$, and their median age is $5$. What is the sum of the ages of Amanda's youngest and oldest cousins? $\textbf{(A)}\ 13\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 25$

2015 Polish MO Finals, 1
Tags: system of equations
Solve the system $$\begin{cases} x+y+z=1\\ x^5+y^5+z^5=1\end{cases}$$ in real numbers.

Taiwan TST 2015 Round 1, 2
Tags: function , functional equation , algebra
Given a positive integer $n \geq 3$. Find all $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for any $n$ positive reals $a_1,...,a_n$, the following condition is always satisfied: $\sum_{i=1}^{n}(a_i-a_{i+1})f(a_i+a_{i+1}) = 0$ where $a_{n+1} = a_1$.

2013 Online Math Open Problems, 6
Tags: rotation , geometry , geometric transformation
Circle $S_1$ has radius $5$. Circle $S_2$ has radius $7$ and has its center lying on $S_1$. Circle $S_3$ has an integer radius and has its center lying on $S_2$. If the center of $S_1$ lies on $S_3$, how many possible values are there for the radius of $S_3$? [i]Ray Li[/i]

2002 Poland - Second Round, 3
Tags: inequalities
Find all positive integers $n$ such that for all real numbers $x_1,x_2,\ldots ,x_n,y_1,y_2,\ldots ,y_n$ the following inequality holds: \[ x_1x_2\ldots x_n+y_1y_2\ldots y_n\le\sqrt{x_1^2+y_1^2}\cdot\sqrt{x_2^2+y_2^2}\cdot \cdots \sqrt{x_n^2+y_n^2}\cdot \]

1993 Turkey Team Selection Test, 4
Tags: combinatorics
Some towns are connected by roads, with at most one road between any two towns. Let $v$ be the number of towns and $e$ be the number of roads. Prove that $(a)$ if $e<v-1$, then there are two towns such that one cannot travel between them; $(b)$ if $2e>(v-1)(v-2)$, then one can travel between any two towns.

2022 Regional Olympiad of Mexico West, 3
Tags: equal angles , isosceles , geometry
In my isosceles triangle $\vartriangle ABC$ with $AB = CA$, we draw $D$ the midpoint of $BC$. Let $E$ be a point on $AC$ such that $\angle CDE = 60^o$ and $M$ the midpoint of $DE$. Prove that $\angle AME = \angle BMD$.

2019 Durer Math Competition Finals, 5
Tags: permutation , combinatorics
How many permutations $s$ does the set $\{1,2,..., 15\}$ have with the following properties: for every $1 \le k \le 13$ we have $s(k) < s(k+2)$ and for every $1 \le k \le 12$ we have $s(k) < s(k+3)$?

2019 Novosibirsk Oral Olympiad in Geometry, 2
Tags: geometry , distance , equilateral
Kikoriki live on the shores of a pond in the form of an equilateral triangle with a side of $600$ m, Krash and Wally live on the same shore, $300$ m from each other. In summer, Dokko to Krash walk $900$ m, and Wally to Rosa - also $900$ m. Prove that in winter, when the pond freezes and it will be possible to walk directly on the ice, Dokko will walk as many meters to Krash as Wally to Rosa. [url=https://en.wikipedia.org/wiki/Kikoriki]about Kikoriki/GoGoRiki / Smeshariki [/url]

2012 Singapore Senior Math Olympiad, 1
Tags: incenter , symmetry , geometry
A circle $\omega$ through the incentre$ I$ of a triangle $ABC$ and tangent to $AB$ at $A$, intersects the segment $BC$ at $D$ and the extension of$ BC$ at $E$. Prove that the line $IC$ intersects $\omega$ at a point $M$ such that $MD=ME$.

1995 All-Russian Olympiad, 2
Tags: function , symmetry , algebra
Prove that every real function, defined on all of $\mathbb R$, can be represented as a sum of two functions whose graphs both have an axis of symmetry. [i]D. Tereshin[/i]

1989 IMO Longlists, 17
Tags: function , limit , geometric series , algebra
Let $ a \in \mathbb{R}, 0 < a < 1,$ and $ f$ a continuous function on $ [0, 1]$ satisfying $ f(0) \equal{} 0, f(1) \equal{} 1,$ and \[ f \left( \frac{x\plus{}y}{2} \right) \equal{} (1\minus{}a) f(x) \plus{} a f(y) \quad \forall x,y \in [0,1] \text{ with } x \leq y.\] Determine $ f \left( \frac{1}{7} \right).$

2006 IMS, 2
Tags: induction , number theory
For each subset $C$ of $\mathbb N$, Suppose $C\oplus C=\{x+y|x,y\in C, x\neq y\}$. Prove that there exist a unique partition of $\mathbb N$ to sets $A$, $B$ that $A\oplus A$ and $B\oplus B$ do not have any prime numbers.

2018 MOAA, 9
Tags: geometry , team
Quadrilateral $ABCD$ with $AC = 800$ is inscribed in a circle, and $E, W, X, Y, Z$ are the midpoints of segments $BD$, $AB$, $BC$, $CD$, $DA$, respectively. If the circumcenters of $EW Z$ and $EXY$ are $O_1$ and $O_2$, respectively, determine $O_1O_2$.

2021 Denmark MO - Mohr Contest, 4
Tags: geometry , angle bisector , area
Given triangle $ABC$ with $|AC| > |BC|$. The point $M$ lies on the angle bisector of angle $C$, and $BM$ is perpendicular to the angle bisector. Prove that the area of triangle AMC is half of the area of triangle $ABC$. [img]https://cdn.artofproblemsolving.com/attachments/4/2/1b541b76ec4a9c052b8866acbfea9a0ce04b56.png[/img]

2022 Durer Math Competition Finals, 13
Tags: combinatorics
Write some positive integers in the following table such that $\cdot$ there is at most one number in each field $\cdot$ each number is equal to how many numbers there are in edge-adjacent fields, $\cdot$ edge-adjacent fields cannot have equal numbers. What is the sum of numbers in the resulting table? [img]https://cdn.artofproblemsolving.com/attachments/a/9/63a9c38762a4c895688fff049ed08c96b2c22c.png[/img]

PEN A Problems, 37
Tags: number theory , greatest common divisor , modular arithmetic , relatively prime , divisibility theory
If $n$ is a natural number, prove that the number $(n+1)(n+2)\cdots(n+10)$ is not a perfect square.

2019 Israel National Olympiad, 1
Tags: algebra
In kindergarden, there are 32 children and three classes: Judo, Agriculture, and Math. Every child is in exactly one class and every class has at least one participant. One day, the teacher gathered 6 children to clean up the classroom. The teacher counted and found that exactly 1/2 of the Judo students, 1/4 of the Agriculture students and 1/8 of the Math students are cleaning. How many children are in each class?

1987 Austrian-Polish Competition, 1
Tags: chord , geometry , 3d geometry , sphere
Three pairwise orthogonal chords of a sphere $S$ are drawn through a given point $P$ inside $S$. Prove that the sum of the squares of their lengths does not depend on their directions.

2024 Assara - South Russian Girl's MO, 8
Tags: combinatorics , set
Given a set $S$ of $2024$ natural numbers satisfying the following condition: if you select any $10$ (different) numbers from $S$, then you can select another number from $S$ so that the sum of all $11$ selected numbers is divisible by $10$. Prove that one of the numbers can be thrown out of $S$ so that the resulting set $S'$ of $2023$ numbers satisfies the condition: if you choose any $9$ (different) numbers from $S'$, then you can choose another number from $S'$ so that the sum of all $10$ selected numbers is divisible by $10$. [i]K.A.Sukhov[/i]

1990 Baltic Way, 18
Tags: inequalities
Numbers $1, 2,\dots , 101$ are written in the cells of a $101\times 101$ square board so that each number is repeated $101$ times. Prove that there exists either a column or a row containing at least $11$ different numbers.

2018 PUMaC Team Round, 15
Tags:
Aaron the Ant is somewhere on the exterior of a hollow cube of side length $2$ inches, and Fred the Flea is on the inside, at one of the vertices. At some instant, Fred flies in a straight line towards the opposite vertex, and simultaneously Aaron begins crawling on the exterior of the cube towards that same vertex. Fred moves at $\sqrt{3}$ inches per second and Aaron moves at $\sqrt{2}$ inches per second. If Aaron arrives before Fred, the area of the surface on the cube from which Aaron could have started can be written as $a\pi+\sqrt{b}+c$ where $a$, $b$, and $c$ are integers. Find $a+b+c.$

2011 Akdeniz University MO, 1
Tags: number theory , prime number
Let $m,n$ positive integers and $p$ prime number with $p=3k+2$. If $p \mid {(m+n)^2-mn}$ , prove that $$p \mid m,n$$

2008 China Team Selection Test, 3
Tags: probability , combinatorics
Let $ S$ be a set that contains $ n$ elements. Let $ A_{1},A_{2},\cdots,A_{k}$ be $ k$ distinct subsets of $ S$, where $ k\geq 2, |A_{i}| \equal{} a_{i}\geq 1 ( 1\leq i\leq k)$. Prove that the number of subsets of $ S$ that don't contain any $ A_{i} (1\leq i\leq k)$ is greater than or equal to $ 2^n\prod_{i \equal{} 1}^k(1 \minus{} \frac {1}{2^{a_{i}}}).$