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

2018 Iran MO (1st Round), 3
Tags: number theory , divisibility tests
How many $8$-digit numbers in base $4$ formed of the digits $1,2, 3$ are divisible by $3$?

1965 AMC 12/AHSME, 34
Tags: quadratics , calculus , function
For $ x \ge 0$ the smallest value of $ \frac {4x^2 \plus{} 8x \plus{} 13}{6(1 \plus{} x)}$ is: $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \frac {25}{12} \qquad \textbf{(D)}\ \frac {13}{6} \qquad \textbf{(E)}\ \frac {34}{5}$

1968 AMC 12/AHSME, 2
Tags: AMC
The real value of $x$ such that $64^{x-1}$ divided by $4^{x-1}$ equals $256^{2x}$ is: $\textbf{(A)}\ -\dfrac{2}{3} \qquad \textbf{(B)}\ -\dfrac{1}{3} \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{3}{8} $

2022 Assam Mathematical Olympiad, 12
Tags:
A particle is in the origin of the Cartesian plane. In each step the particle can go $1$ unit in any of the directions, left, right, up or down. Find the number of ways to go from $(0, 0)$ to $(0, 2)$ in $6$ steps. (Note: Two paths where identical set of points is traversed are considered different if the order of traversal of each point is different in both paths.)

1997 Finnish National High School Mathematics Competition, 1
Tags: parameterization , quadratics , logarithms , algebra , polynomial , product of roots , algebra unsolved
Determine the real numbers $a$ such that the equation $a 3^x + 3^{-x} = 3$ has exactly one solution $x.$

2010 Germany Team Selection Test, 1
Tags: geometry unsolved , geometry
In the plane we have points $P,Q,A,B,C$ such triangles $APQ,QBP$ and $PQC$ are similar accordantly (same direction). Then let $A'$ ($B',C'$ respectively) be the intersection of lines $BP$ and $CQ$ ($CP$ and $AQ;$ $AP$ and $BQ,$ respectively.) Show that the points $A,B,C,A',B',C'$ lie on a circle.

2015 Paraguayan Mathematical Olympiad, Problem 5
Tags: combinatorics
In the figure, the rectangle is formed by $4$ smaller equal rectangles. If we count the total number of rectangles in the figure we find $10$. How many rectangles in total will there be in a rectangle that is formed by $n$ smaller equal rectangles?

2023 Belarusian National Olympiad, 11.5
Tags: combinatorics , Sequence
A sequence of positive integers is given such that the sum of any $6$ consecutive terms does not exceed $11$. Prove that for any positive integer $a$ in the sequence one can find consecutive terms with sum $a$

2012 Serbia Team Selection Test, 1
Tags: algebra , polynomial , function , inequalities , absolute value , algebra proposed
Let $P(x)$ be a polynomial of degree $2012$ with real coefficients satisfying the condition \[P(a)^3 + P(b)^3 + P(c)^3 \geq 3P(a)P(b)P(c),\] for all real numbers $a,b,c$ such that $a+b+c=0$. Is it possible for $P(x)$ to have exactly $2012$ distinct real roots?

2010 Contests, 3
Tags: geometry , rectangle , combinatorics unsolved , combinatorics
We call a rectangle of the size $1 \times 2$ a domino. Rectangle of the $2 \times 3$ removing two opposite (under center of rectangle) corners we call tetramino. These figures can be rotated. It requires to tile rectangle of size $2008 \times 2010$ by using dominoes and tetraminoes. What is the minimal number of dominoes should be used?

1996 Tournament Of Towns, (511) 4
Tags: combinatorics , tiles , Tiling , combinatorial geometry
(a) A square is cut into right triangles with legs of lengths $3$ and $4$. Prove that the total number of the triangles is even. (b) A rectangle is cut into right triangles with legs of lengths $1$ and $2$. Prove that the total number of the triangles is even. (A Shapovalov)

1949-56 Chisinau City MO, 60
Tags: geometry , distance , regular tetrahedron , tetrahedron , 3D geometry
Show that the sum of the distances from any point of a regular tetrahedron to its faces is equal to the height of this tetrahedron.

2021 Turkey MO (2nd round), 5
Tags: number theory , prime numbers
There are finitely many primes dividing the numbers $\{ a \cdot b^n + c\cdot d^n : n=1, 2, 3,... \}$ where $a, b, c, d$ are positive integers. Prove that $b=d$.

2018 Iran MO (1st Round), 21
Tags: geometry , geometric transformation , reflection
The point $P$ is chosen inside or on the equilateral triangle $ABC$ of side length $1$. The reflection of $P$ with respect to $AB$ is $K$, the reflection of $K$ about $BC$ is $M$, and the reflection of $M$ with respect to $AC$ is $N$. What is the maximum length of $NP$? $\textbf{(A)}\ 2\sqrt 3\qquad\textbf{(B)}\ \sqrt 3\qquad\textbf{(C)}\ \frac{\sqrt 3}{2} \qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 1$

1995 China Team Selection Test, 1
Tags: modular arithmetic , number theory
Find the smallest prime number $p$ that cannot be represented in the form $|3^{a} - 2^{b}|$, where $a$ and $b$ are non-negative integers.

2019 Math Prize for Girls Problems, 10
Tags:
A $1 \times 5$ rectangle is split into five unit squares (cells) numbered 1 through 5 from left to right. A frog starts at cell 1. Every second it jumps from its current cell to one of the adjacent cells. The frog makes exactly 14 jumps. How many paths can the frog take to finish at cell 5?

2023 HMNT, 31
Tags:
Let $s(n)$ denote the sum of the digits (in base ten) of a positive integer $n.$ Compute the number of positive integers $n$ at most $10^4$ that satisfy $$s(11n)=2s(n).$$

2008 Saint Petersburg Mathematical Olympiad, 4
Tags: induction , inequalities proposed , inequalities
The numbers $x_1,...x_{100}$ are written on a board so that $ x_1=\frac{1}{2}$ and for every $n$ from $1$ to $99$, $x_{n+1}=1-x_1x_2x_3*...*x_{100}$. Prove that $x_{100}>0.99$.

2018 India PRMO, 17
Tags: geometry , equal angles
Triangles $ABC$ and $DEF$ are such that $\angle A = \angle D, AB = DE = 17, BC = EF = 10$ and $AC - DF = 12$. What is $AC + DF$?

2025 Bangladesh Mathematical Olympiad, P4
Tags: prime numbers , modular arithmetic , Diophantine equation , number theory , Bounding
Find all prime numbers $p, q$ such that$$p(p+1)(p^2+1) = q^2(q^2+q+1) + 2025.$$ [i]Proposed by Md. Fuad Al Alam[/i]

2019 Purple Comet Problems, 24
Tags: geometry
A $12$-sided polygon is inscribed in a circle with radius $r$. The polygon has six sides of length $6\sqrt3$ that alternate with six sides of length $2$. Find $r^2$.

2006 Iran Team Selection Test, 4
Tags: algebra , polynomial , number theory , divisibility tests , number theory proposed
Let $n$ be a fixed natural number. Find all $n$ tuples of natural pairwise distinct and coprime numbers like $a_1,a_2,\ldots,a_n$ such that for $1\leq i\leq n$ we have \[ a_1+a_2+\ldots+a_n|a_1^i+a_2^i+\ldots+a_n^i \]

2010 LMT, 10
Tags:
A two digit prime number is such that the sum of its digits is $13.$ Determine the integer.

2000 Vietnam National Olympiad, 2
Tags: geometry , circumcircle , geometric transformation , geometry proposed
Two circles $ (O_1)$ and $ (O_2)$ with respective centers $ O_1$, $ O_2$ are given on a plane. Let $ M_1$, $ M_2$ be points on $ (O_1)$, $ (O_2)$ respectively, and let the lines $ O_1M_1$ and $ O_2M_2$ meet at $ Q$. Starting simultaneously from these positions, the points $ M_1$ and $ M_2$ move clockwise on their own circles with the same angular velocity. (a) Determine the locus of the midpoint of $ M_1M_2$. (b) Prove that the circumcircle of $ \triangle M_1QM_2$ passes through a fixed point.

2006 Kyiv Mathematical Festival, 4
Tags: geometry , circumcircle , incenter , trigonometry , trig identities , Law of Sines , geometry proposed
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.