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

PEN A Problems, 75
Tags: modular arithmetic , divisibility theory
Find all triples $(a,b,c)$ of positive integers such that $2^{c}-1$ divides $2^{a}+2^{b}+1$.

2002 Polish MO Finals, 3
Tags: combinatorics
Three non-negative integers are written on a blackboard. A move is to replace two of the integers $k,m$ by $k+m$ and $|k-m|$. Determine whether we can always end with triplet which has at least two zeros

2023 IFYM, Sozopol, 5
Tags: algebra , polynomial
Let $a$ and $b$ be natural numbers. Prove that the number of polynomials $P(x)$ with integer coefficients such that $|P(n)| \leq a^n$ for every natural number $n \geq b$ is finite.

1988 Nordic, 3
Tags: geometry , 3d geometry , sphere
Two concentric spheres have radii $r$ and $R,r < R$. We try to select points $A, B$ and $C$ on the surface of the larger sphere such that all sides of the triangle $ABC$ would be tangent to the surface of the smaller sphere. Show that the points can be selected if and only if $R \le 2r$.

2022 CCA Math Bonanza, I6
Tags:
Let regular tetrahedron $ABCD$ have center $O$. Find $\tan^2(\angle AOB)$. [i]2022 CCA Math Bonanza Individual Round #6[/i]

1989 Flanders Math Olympiad, 2
Tags: ratio , trigonometry , geometry
When drawing all diagonals in a regular pentagon, one gets an smaller pentagon in the middle. What's the ratio of the areas of those pentagons?

2009 Iran Team Selection Test, 5
Tags: angle bisector , geometry
$ ABC$ is a triangle and $ AA'$ , $ BB'$ and $ CC'$ are three altitudes of this triangle . Let $ P$ be the feet of perpendicular from $ C'$ to $ A'B'$ , and $ Q$ is a point on $ A'B'$ such that $ QA \equal{} QB$ . Prove that : $ \angle PBQ \equal{} \angle PAQ \equal{} \angle PC'C$

2014 JBMO TST - Macedonia, 3
Tags: number theory , digit , divisibility
Find all positive integers $n$ which are divisible by 11 and satisfy the following condition: all the numbers which are generated by an arbitrary rearrangement of the digits of $n$, are also divisible by 11.

1999 Croatia National Olympiad, Problem 4
Tags: combinatorics , tournament
In a basketball competition, $n$ teams took part. Each pair of teams played exactly one match, and there were no draws. At the end of the competition the $i$-th team had $x_i$ wins and $y_i$ defeats $(i=1,\ldots,n)$. Prove that $x_1^2+x_2^2+\ldots+x_n^2=y_1^2+y_2^2+\ldots+y_n^2$.

2014 AMC 8, 19
Tags: geometry , 3d geometry , probability
A cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are colored red and $6$ are colored white. If the $3$-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white? $\textbf{(A) }\frac{5}{54}\qquad\textbf{(B) }\frac{1}{9}\qquad\textbf{(C) }\frac{5}{27}\qquad\textbf{(D) }\frac{2}{9}\qquad \textbf{(E) }\frac{1}{3}$

2011 Romanian Masters In Mathematics, 2
Tags: algebra , polynomial , modular arithmetic , function , number theory , greatest common divisor , system of equations
Determine all positive integers $n$ for which there exists a polynomial $f(x)$ with real coefficients, with the following properties: (1) for each integer $k$, the number $f(k)$ is an integer if and only if $k$ is not divisible by $n$; (2) the degree of $f$ is less than $n$. [i](Hungary) Géza Kós[/i]

I Soros Olympiad 1994-95 (Rus + Ukr), 11.5
Tags: function , inequalities , algebra
Function $f(x)$. which is defined on the set of non-negative real numbers, acquires real values. It is known that $f(0)\le 0$ and the function $f(x)/x$ is increasing for $x>0$. Prove that for arbitrary $x\ge 0$ and $y\ge 0$, holds the inequality $f(x+y)\ge f(x)+ f(y)$ .

2006 MOP Homework, 2
Tags: combinatorics
Mykolka the numismatist possesses $241$ coins, each worth an integer number of turgiks. The total value of the coins is $360$ turgiks. Is it necessarily true that the coins can be divided into three groups of equal total value?

2006 Nordic, 4
Tags: combinatorics
Each square of a $100\times 100$ board is painted with one of $100$ different colours, so that each colour is used exactly $100$ times. Show that there exists a row or column of the chessboard in which at least $10$ colours are used.

2021 New Zealand MO, 6
Tags: combinatorics , consecutive , divisor , number theory
Is it possible to place a positive integer in every cell of a $10 \times 10$ array in such a way that both the following conditions are satisfied? $\bullet$ Each number (not in the top row) is a proper divisor of the number immediately above. $\bullet$ Each row consists of 1$0$ consecutive positive integers (but not necessarily in order).

2024 JHMT HS, 9
Tags: number theory
Let $N \in \{10, 11, \ldots, 99\}$ be a two-digit positive integer. Compute the number of values of $N$ for which the last two digits in the decimal expansion of $N^{21}$ are the digits of $N$ in the same order.

2010 District Olympiad, 2
Tags: superior algebra , abstract algebra , group theory
Let $ G$ be a group such that if $ a,b\in \mathbb{G}$ and $ a^2b\equal{}ba^2$, then $ ab\equal{}ba$. i)If $ G$ has $ 2^n$ elements, prove that $ G$ is abelian. ii) Give an example of a non-abelian group with $ G$'s property from the enounce.

2004 Junior Balkan Team Selection Tests - Moldova, 7
Tags: circumcircle , angle bisector , geometry
Let the triangle $ABC$ have area $1$. The interior bisectors of the angles $\angle BAC,\angle ABC, \angle BCA$ intersect the sides $(BC), (AC), (AB) $ and the circumscribed circle of the respective triangle $ABC$ at the points $L$ and $G, N$ and $F, Q$ and $E$. The lines $EF, FG,GE$ intersect the bisectors $(AL), (CQ) ,(BN)$ respectively at points $P, M, R$. Determine the area of the hexagon $LMNPR$.

2004 Junior Balkan Team Selection Tests - Romania, 4
Tags: coloring , combinatorics , regular
A regular polygon with $1000$ sides has the vertices colored in red, yellow or blue. A move consists in choosing to adjiacent vertices colored differently and coloring them in the third color. Prove that there is a sequence of moves after which all the vertices of the polygon will have the same color. Marius Ghergu

2015 Sharygin Geometry Olympiad, 7
Tags: geometry , cyclic quadrilateral , tangential quadrilateral
Point $M$ on side $AB$ of quadrilateral $ABCD$ is such that quadrilaterals $AMCD$ and $BMDC$ are circumscribed around circles centered at $O_1$ and $O_2$ respectively. Line $O_1O_2$ cuts an isosceles triangle with vertex M from angle $CMD$. Prove that $ABCD$ is a cyclic quadrilateral. (M. Kungozhin)

2017 Princeton University Math Competition, A6/B8
Tags: number theory
Find the least positive integer $N$ such that the only values of $n$ for which $1 + N \cdot 2^n$ is prime are multiples of $12$.

2010 Indonesia TST, 3
Tags: function , induction , number theory
Let $ \mathbb{Z}$ be the set of all integers. Define the set $ \mathbb{H}$ as follows: (1). $ \dfrac{1}{2} \in \mathbb{H}$, (2). if $ x \in \mathbb{H}$, then $ \dfrac{1}{1\plus{}x} \in \mathbb{H}$ and also $ \dfrac{x}{1\plus{}x} \in \mathbb{H}$. Prove that there exists a bijective function $ f: \mathbb{Z} \rightarrow \mathbb{H}$.

2019 Czech and Slovak Olympiad III A, 5
Tags: number theory
Prove that there are infinitely many integers which cannot be expressed as $2^a+3^b-5^c$ for non-negative integers $a,b,c$.

1960 AMC 12/AHSME, 9
Tags: algebra , polynomial
The fraction $\frac{a^2+b^2-c^2+2ab}{a^2+c^2-b^2+2ac}$ is (with suitable restrictions of the values of $a$, $b$, and $c$): $ \textbf{(A) }\text{irreducible}\qquad\textbf{(B) }\text{reducible to negative 1}\qquad$ $\textbf{(C) }\text{reducible to a polynomial of three terms} \qquad\textbf{(D) }\text{reducible to} \frac{a-b+c}{a+b-c} \qquad\textbf{(E) }\text{reducible to} \frac{a+b-c}{a-b+c} $

2020 Sharygin Geometry Olympiad, 12
Tags: geometry
Let $H$ be the orthocenter of a nonisosceles triangle $ABC$. The bisector of angle $BHC$ meets $AB$ and $AC$ at points $P$ and $Q$ respectively. The perpendiculars to $AB$ and $AC$ from $P$ and $Q$ meet at $K$. Prove that $KH$ bisects the segment $BC$.