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 Online Math Open Problems, 1
Tags:
Leonhard has five cards. Each card has a nonnegative integer written on it, and any two cards show relatively prime numbers. Compute the smallest possible value of the sum of the numbers on Leonhard's cards. Note: Two integers are relatively prime if no positive integer other than $1$ divides both numbers. [i]Proposed by ABCDE and Tristan Shin

1954 Moscow Mathematical Olympiad, 286
Tags: number theory , combinatorics , sum , 10-digit , subset
Consider the set of all $10$-digit numbers expressible with the help of figures $1$ and $2$ only. Divide it into two subsets so that the sum of any two numbers of the same subset is a number which is written with not less than two $3$’s.

1961 Putnam, B5
Tags: factorial , inequalities
Let $k$ be a positive integer, and $n$ a positive integer greater than $2$. Define $$f_{1}(n)=n,\;\; f_{2}(n)=n^{f_{1}(n)},\;\ldots\;, f_{j+1}(n)=n^{f_{j}(n)}.$$ Prove either part of the inequality $$f_{k}(n) < n!! \cdots ! < f_{k+1}(n),$$ where the middle term has $k$ factorial symbols.

2023 BMT, Tie 1
Tags: combinatorics
Mataio has a weighted die numbered $1$ to $6$, where the probability of rolling a side $n$ for $1 \le n \le 6$ is inversely proportional to the value of $n$. If Mataio rolls the die twice, what is the probability that the sum of the two rolls is $7$?

2008 All-Russian Olympiad, 3
Tags: euler , circumcircle , vector , geometry
In a scalene triangle $ ABC, H$ and $ M$ are the orthocenter an centroid respectively. Consider the triangle formed by the lines through $ A,B$ and $ C$ perpendicular to $ AM,BM$ and $ CM$ respectively. Prove that the centroid of this triangle lies on the line $ MH$.

2024 Bosnia and Herzegovina Junior BMO TST, 1.
Tags: algebra
Let $a$,$b$,$c$ be real numbers different from 0 for which $ab$ + $bc$+ $ca$ = 0 holds a) Prove that ($a$+$b$)($b$+$c$)($c$+$a$)≠ 0 b) Let $X$ = $a$ + $b$ + $c$ and $Y$ = $\frac{1}{a+b}$ + $\frac{1}{b+c}$ + $\frac{1}{c+a}$. Prove that numbers $X$ and $Y$ are both positive or both negative.

2002 China Team Selection Test, 3
Tags: number theory
For positive integers $a,b,c$ let $ \alpha, \beta, \gamma$ be pairwise distinct positive integers such that \[ \begin{cases}{c} \displaystyle a &= \alpha + \beta + \gamma, \\ b &= \alpha \cdot \beta + \beta \cdot \gamma + \gamma \cdot \alpha, \\ c^2 &= \alpha\beta\gamma. \end{cases} \] Also, let $ \lambda$ be a real number that satisfies the condition \[\lambda^4 -2a\lambda^2 + 8c\lambda + a^2 - 4b = 0.\] Prove that $\lambda$ is an integer if and only if $\alpha, \beta, \gamma$ are all perfect squares.

2015 Bosnia And Herzegovina - Regional Olympiad, 2
Tags: number theory , prime
Find all triplets $(p,a,b)$ of positive integers such that $$p=b\sqrt{\frac{a-8b}{a+8b}}$$ is prime

2000 All-Russian Olympiad Regional Round, 8.3
Tags: combinatorics , combinatorial geometry , geometry
What is the smallest number of sides that an polygon can have (not necessarily convex), which can be cut into parallelograms?

2014 China Team Selection Test, 6
Tags: combinatorics
Let $n\ge 2$ be a positive integer. Fill up a $n\times n$ table with the numbers $1,2,...,n^2$ exactly once each. Two cells are termed adjacent if they have a common edge. It is known that for any two adjacent cells, the numbers they contain differ by at most $n$. Show that there exist a $2\times 2$ square of adjacent cells such that the diagonally opposite pairs sum to the same number.

2004 Brazil Team Selection Test, Problem 2
Tags: number theory , combinatorics
Show that there exist infinitely many pairs of positive integers $(m,n)$ such that $\binom m{n-1}=\binom{m-1}n$.

2013 Macedonia National Olympiad, 3
Tags: ratio , circumcircle , trigonometry , geometry
Acute angle triangle is given such that $ BC $ is the longest side. Let $ E $ and $ G $ be the intersection points from the altitude from $ A $ to $ BC $ with the circumscribed circle of triangle $ ABC $ and $ BC $ respectively. Let the center $ O $ of this circle is positioned on the perpendicular line from $ A $ to $ BE $. Let $ EM $ be perpendicular to $ AC $ and $ EF $ be perpendicular to $ AB $. Prove that the area of $ FBEG $ is greater than the area of $ MFE $.

2020 ELMO Problems, P5
Tags: polynomial , number theory , algebra
Let $m$ and $n$ be positive integers. Find the smallest positive integer $s$ for which there exists an $m \times n$ rectangular array of positive integers such that [list] [*]each row contains $n$ distinct consecutive integers in some order, [*]each column contains $m$ distinct consecutive integers in some order, and [*]each entry is less than or equal to $s$. [/list] [i]Proposed by Ankan Bhattacharya.[/i]

2022 Korea National Olympiad, 2
Tags: geometry , circumcircle , circumcenter , parallel
In a scalene triangle $ABC$, let the angle bisector of $A$ meets side $BC$ at $D$. Let $E, F$ be the circumcenter of the triangles $ABD$ and $ADC$, respectively. Suppose that the circumcircles of the triangles $BDE$ and $DCF$ intersect at $P(\neq D)$, and denote by $O, X, Y$ the circumcenters of the triangles $ABC, BDE, DCF$, respectively. Prove that $OP$ and $XY$ are parallel.

Kyiv City MO Juniors 2003+ geometry, 2010.9.4
Tags: geometry , circumcircle , bisects segment
In an acute-angled triangle $ABC$, the point $O$ is the center of the circumcircle, $CH$ is the height of the triangle, and the point $T$ is the foot of the perpendicular dropped from the vertex $C$ on the line $AO$. Prove that the line $TH$ passes through the midpoint of the side $BC$ .

2018 Philippine MO, 2
Tags: number theory
Suppose $a_1, a_2, \ldots$ is a sequence of integers, and $d$ is some integer. For all natural numbers $n$, \begin{align*}\text{(i)} |a_n| \text{ is prime;} && \text{(ii)} a_{n+2} = a_{n+1} + a_n + d. \end{align*} Show that the sequence is constant.

1992 India National Olympiad, 7
Tags: combinatorics , counting , chessboard , arithmetic progression
Let $n\geq 3$ be an integer. Find the number of ways in which one can place the numbers $1, 2, 3, \ldots, n^2$ in the $n^2$ squares of a $n \times n$ chesboard, one on each, such that the numbers in each row and in each column are in arithmetic progression.

2010 Contests, 3
Tags:
Find the number of $4$-digit numbers (in base $10$) having non-zero digits and which are divisible by $4$ but not by $8$.

2012 Sharygin Geometry Olympiad, 24
Tags: combinatorial geometry , geometry
Given are $n$ $(n > 2)$ points on the plane such that no three of them are collinear. In how many ways this set of points can be divided into two non-empty subsets with non-intersecting convex envelops?

2010 Austria Beginners' Competition, 1
Tags: perfect square , number theory , difference of squares
Prove that $2010$ cannot be represented as the difference between two square numbers. (B. Schmidt, Graz University of Technology)

2022 Korea Junior Math Olympiad, 8
Tags: number theory , rational number , diophantine equation , pythagorean triple , discriminant
Find all pairs $(x, y)$ of rational numbers such that $$xy^2=x^2+2x-3$$

2010 Tuymaada Olympiad, 1
Tags: combinatorics
Misha and Sahsa play a game on a $100\times 100$ chessboard. First, Sasha places $50$ kings on the board, and Misha places a rook, and then they move in turns, as following (Sasha begins): At his move, Sasha moves each of the kings one square in any direction, and Misha can move the rook on the horizontal or vertical any number of squares. The kings cannot be captured or stepped over. Sasha's purpose is to capture the rook, and Misha's is to avoid capture. Is there a winning strategy available for Sasha?

2022 IFYM, Sozopol, 5
Tags: algebra , inequality , sum
Prove that $\sum_{n=1}^{2022^{2022}} \frac{1}{\sqrt{n^3+2n^2+n}}<\frac{19}{10}$.

Indonesia Regional MO OSP SMA - geometry, 2002.4
Tags: geometry , circumcircle , equilateral
Given an equilateral triangle $ABC$ and a point $P$ so that the distances $P$ to $A$ and to $C$ are not farther than the distances $P$ to $B$. Prove that $PB = PA + PC$ if and only if $P$ lies on the circumcircle of $\vartriangle ABC$.

2013 Saint Petersburg Mathematical Olympiad, 3
Tags: geometry
$ABC$ is triangle. $l_1$- line passes through $A$ and parallel to $BC$, $l_2$ - line passes through $C$ and parallel to $AB$. Bisector of $\angle B$ intersect $l_1$ and $l_2$ at $X,Y$. $XY=AC$. What value can take $\angle A- \angle C$ ?