Found problems: 85335
2017 Sharygin Geometry Olympiad, P10
Points $K$ and $L$ on the sides $AB$ and $BC$ of parallelogram $ABCD$ are such that $\angle AKD = \angle CLD$. Prove that the circumcenter of triangle $BKL$ is equidistant from $A$ and $C$.
[i]Proposed by I.I.Bogdanov[/i]
1999 Bosnia and Herzegovina Team Selection Test, 2
Prove the inequality $$\frac{a^2}{b+c-a}+\frac{b^2}{a+c-b}+\frac{c^2}{a+b-c} \geq 3\sqrt{3}R$$ in triangle $ABC$ where $a$, $b$ and $c$ are sides of triangle and $R$ radius of circumcircle of $ABC$
1993 Flanders Math Olympiad, 3
For $a,b,c>0$ we have:
\[ -1 < \left(\dfrac{a-b}{a+b}\right)^{1993} + \left(\dfrac{b-c}{b+c}\right)^{1993} + \left(\dfrac{c-a}{c+a}\right)^{1993} < 1 \]
2018 IFYM, Sozopol, 1
Find all prime numbers $p$ and all positive integers $n$, such that
$n^8 - n^2 = p^5 + p^2$
2021 Princeton University Math Competition, A6 / B8
Let $f$ be a polynomial. We say that a complex number $p$ is a double attractor if there exists a polynomial $h(x)$ so that $f(x)-f(p) = h(x)(x-p)^2$ for all x \in R. Now, consider the polynomial $$f(x) = 12x^5 - 15x^4 - 40x^3 + 540x^2 - 2160x + 1,$$ and suppose that it’s double attractors are $a_1, a_2, ... , a_n$. If the sum $\sum^{n}_{i=1}|a_i|$ can be written as $\sqrt{a} +\sqrt{b}$, where $a, b$ are positive integers, find $a + b$.
1973 IMO, 3
Let $a_1, \ldots, a_n$ be $n$ positive numbers and $0 < q < 1.$ Determine $n$ positive numbers $b_1, \ldots, b_n$ so that:
[i]a.)[/i] $ a_{k} < b_{k}$ for all $k = 1, \ldots, n,$
[i]b.)[/i] $q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$ for all $k = 1, \ldots, n-1,$
[i]c.)[/i] $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$
1970 IMO Longlists, 46
Given a triangle $ABC$ and a plane $\pi$ having no common points with the triangle, find a point $M$ such that the triangle determined by the points of intersection of the lines $MA,MB,MC$ with $\pi$ is congruent to the triangle $ABC$.
2004 China Team Selection Test, 2
Let $p_1, p_2, \ldots, p_{25}$ are primes which don’t exceed 2004. Find the largest integer $T$ such that every positive integer $\leq T$ can be expressed as sums of distinct divisors of $(p_1\cdot p_2 \cdot \ldots \cdot p_{25})^{2004}.$
2022 Purple Comet Problems, 9
Let $a$ and $b$ be positive integers satisfying $3a < b$ and $a^2 + ab + b^2 = (b + 3)^2 + 27.$ Find the minimum possible value of $a + b.$
2024 HMNT, 33
A grid is called [i]groovy[/i] if each cell of the grid is labeled with the smallest positive integer that does not appear below it in the same column or to the left of it in the same row. Compute the sum of the entries of a groovy $14 \times 14$ grid whose bottom left entry is $1.$
1982 IMO Longlists, 1
[b](a)[/b] Prove that $\frac{1}{n+1} \cdot \binom{2n}{n}$ is an integer for $n \geq 0.$
[b](b)[/b] Given a positive integer $k$, determine the smallest integer $C_k$ with the property that $\frac{C_k}{n+k+1} \cdot \binom{2n}{n}$ is an integer for all $n \geq k.$
Geometry Mathley 2011-12, 5.1
Let $a, b$ be two lines intersecting each other at $O$. Point $M$ is not on either $a$ or $b$. A variable circle $(C)$ passes through $O,M$ intersecting $a, b$ at $A,B$ respectively, distinct from $O$. Prove that the midpoint of $AB$ is on a fixed line.
Hạ Vũ Anh
2022 CIIM, 6
Prove that $\tau ((n+1)!) \leq 2 \tau (n!)$ for all positive integers $n$.
2017 Mathematical Talent Reward Programme, SAQ: P 4
An irreducible polynomial is a not-constant polynomial that cannot be factored into product of two non-constant polynomials. Consider the following statements :-
[b]Statement 1 :[/b] $p(x)$ be any monic irreducible polynomial with integer coefficients and degree $\geq 4$. Then $p(n)$ is a prime for at least one natural number $n$
[b]Statement 2 :[/b] $n^2+1$ is prime for infinitely many values of natural number $n$
Show that if [b]Statement 1[/b] is true then [b]Statement 2[/b] is also true
STEMS 2024 Math Cat A, P1
Let $n$ be a positive integer and $S = \{ m \mid 2^n \le m < 2^{n+1} \}$. We call a pair of non-negative integers $(a, b)$ [i]fancy[/i] if $a + b$ is in $S$ and is a palindrome in binary. Find the number of [i]fancy[/i] pairs $(a, b)$.
2019 Denmark MO - Mohr Contest, 3
Seven positive integers are written on a piece of paper. No matter which five numbers one chooses, each of the remaining two numbers divides the sum of the five chosen numbers. How many distinct numbers can there be among the seven?
1992 Brazil National Olympiad, 3
Given positive real numbers $x_1, x_2, \ldots , x_n$ find the polygon $A_0A_1\ldots A_n$ with $A_iA_{i+1} = x_{i+1}$ and which has greatest area.
1990 Turkey Team Selection Test, 3
Let $n$ be an odd integer greater than $11$; $k\in \mathbb{N}$, $k \geq 6$, $n=2k-1$.
We define \[d(x,y) = \left | \{ i\in \{1,2,\dots, n \} \bigm | x_i \neq y_i \} \right |\] for $T=\{ (x_1, x_2, \dots, x_n) \bigm | x_i \in \{0,1\}, i=1,2,\dots, n \}$ and $x=(x_1,x_2,\dots, x_n), y=(y_1, y_2, \dots, y_n) \in T$.
Show that $n=23$ if $T$ has a subset $S$ satisfying
[list=i]
[*]$|S|=2^k$
[*]For each $x \in T$, there exists exacly one $y\in S$ such that $d(x,y)\leq 3$[/list]
2008 Indonesia MO, 1
Given triangle $ ABC$. Points $ D,E,F$ outside triangle $ ABC$ are chosen such that triangles $ ABD$, $ BCE$, and $ CAF$ are equilateral triangles. Prove that cicumcircles of these three triangles are concurrent.
2014 Balkan MO Shortlist, C2
Let $ M=\{1,2,...,2013\} $ and let $ \Gamma $ be a circle. For every nonempty subset $ B $ of the set $ M $, denote by $ S(B) $ sum of elements of the set $ B $, and define $ S(\varnothing)=0 $ ( $ \varnothing $ is the empty set ). Is it possible to join every subset $ B $ of $ M $ with some point $ A $ on the circle $ \Gamma $ so that following conditions are fulfilled:
$ 1 $. Different subsets are joined with different points;
$ 2 $. All joined points are vertices of a regular polygon;
$ 3 $. If $ A_1,A_2,...,A_k $ are some of the joined points, $ k>2 $ , such that $ A_1A_2...A_k $ is a regular $ k-gon $, then $ 2014 $ divides $ S(B_1)+S(B_2)+...+S(B_k) $ ?
2016 Croatia Team Selection Test, Problem 1
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for all real $x,y$:
$$ f(x^2) + xf(y) = f(x) f(x + f(y)) \, . $$
Russian TST 2019, P1
Suppose that $A$, $B$, $C$, and $D$ are distinct points, no three of which lie on a line, in the Euclidean plane. Show that if the squares of the lengths of the line segments $AB$, $AC$, $AD$, $BC$, $BD$, and $CD$ are rational numbers, then the quotient
\[\frac{\mathrm{area}(\triangle ABC)}{\mathrm{area}(\triangle ABD)}\]
is a rational number.
Azerbaijan Al-Khwarizmi IJMO TST 2025, 1
In isosceles triangle, the condition $AB=AC>BC$ is satisfied. Point $D$ is taken on the circumcircle of $ABC$ such that $\angle CAD=90^{\circ}$.A line parallel to $AC$ which passes from $D$ intersects $AB$ and $BC$ respectively at $E$ and $F$.Show that circumcircle of $ADE$ passes from circumcenter of $DFC$.
1968 Czech and Slovak Olympiad III A, 2
Show that for any integer $n$ the number \[a_n=\frac{\bigl(2+\sqrt3\bigr)^n-\bigl(2-\sqrt3\bigr)^n}{2\sqrt3}\] is also integer. Determine all integers $n$ such that $a_n$ is divisible by 3.
2016 Peru IMO TST, 3
Let $ABCD$ a convex quadrilateral such that $AD$ and $BC$ are not parallel. Let $M$ and $N$ the midpoints of $AD$ and $BC$ respectively. The segment $MN$ intersects $AC$ and $BD$ in $K$ and $L$ respectively, Show that at least one point of the intersections of the circumcircles of $AKM$ and $BNL$ is in the line $AB$.