Found problems: 85335
2019 Caucasus Mathematical Olympiad, 7
On sides $BC$, $CA$, $AB$ of a triangle $ABC$ points $K$, $L$, $M$ are chosen, respectively, and a point $P$ is inside $ABC$ is chosen so that $PL\parallel BC$, $PM\parallel CA$, $PK\parallel AB$. Determine if it is possible that each of three trapezoids $AMPL$, $BKPM$, $CLPK$ has an inscribed circle.
2011 Gheorghe Vranceanu, 2
Let $ a\ge 3 $ and a polynom $ P. $ Show that:
$$ \max_{1\le k\le \text{grad} P} \left| a^{k-1}-P(k-1) \right| \ge 1 $$
1990 Federal Competition For Advanced Students, P2, 1
Determine the number of integers $ n$ with $ 1 \le n \le N\equal{}1990^{1990}$ such that $ n^2\minus{}1$ and $ N$ are coprime.
2014 Contests, 4
Find all functions $f$ defined on all real numbers and taking real values such that \[f(f(y)) + f(x - y) = f(xf(y) - x),\] for all real numbers $x, y.$
2022 Harvard-MIT Mathematics Tournament, 9
Consider permutations $(a_0, a_1, . . . , a_{2022})$ of $(0, 1, . . . , 2022)$ such that
$\bullet$ $a_{2022} = 625$,
$\bullet$ for each $0 \le i \le 2022$, $a_i \ge \frac{625i}{2022}$ ,
$\bullet$ for each $0 \le i \le 2022$, $\{a_i, . . . , a_{2022}\}$ is a set of consecutive integers (in some order).
The number of such permutations can be written as $\frac{a!}{b!c!}$ for positive integers $a, b, c$, where $b > c$ and $a$ is minimal. Compute $100a + 10b + c$.
2025 USA IMO Team Selection Test, 4
Let $ABC$ be a triangle, and let $X$, $Y$, and $Z$ be collinear points such that $AY=AZ$, $BZ=BX$, and $CX=CY$. Points $X'$, $Y'$, and $Z'$ are the reflections of $X$, $Y$, and $Z$ over $BC$, $CA$, and $AB$, respectively. Prove that if $X'Y'Z'$ is a nondegenerate triangle, then its circumcenter lies on the circumcircle of $ABC$.
[i]Michael Ren[/i]
2023 JBMO TST - Turkey, 3
Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$f(x+f(x))=f(-x)$ and for all $x \leq y$ it satisfies $f(x) \leq f(y)$
2000 Belarusian National Olympiad, 7
(a) Find all positive integers $n$ for which the equation $(a^a)^n = b^b$ has a solution
in positive integers $a,b$ greater than $1$.
(b) Find all positive integers $a, b$ satisfying $(a^a)^5=b^b$
2023 MIG, 4
Which operation makes the following expression true: $(4 \underline{~~~~} 1) \times (3 \underline{~~~~} 2 - 1) = 2$?
$\textbf{(A) } +\qquad\textbf{(B) } -\qquad\textbf{(C) } \times\qquad\textbf{(D) } \div\qquad\textbf{(E) } \text{There is no such operation}$
KoMaL A Problems 2024/2025, A. 898
Let $n$ be a given positive integer. Ana and Bob play the following game: Ana chooses a polynomial $p$ of degree $n$ with integer coefficients. In each round, Bob can choose a finite set $S$ of positive integers, and Ana responds with a list containing the values of the polynomial $p$ evaluated at the elements of $S$ with multiplicity (sorted in increasing order). Determine, in terms of $n$, the smallest positive integer $k$ such that Bob can always determine the polynomial $p$ in at most $k$ rounds.
[i]Proposed by: Andrei Chirita, Cambridge[/i]
2018 Tajikistan Team Selection Test, 5
Problem 5. Consider the integer number n>2. Let a_1,a_2,…,a_n and b_1,b_2,…,b_n be two permutations of 0,1,2,…,n-1. Prove that there exist some i≠j such that:
n|a_i b_i-a_j b_j
[color=#00f]Moved to HSO. ~ oVlad[/color]
2023 Balkan MO, 1
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
\[xf(x+f(y))=(y-x)f(f(x)).\]
[i]Proposed by Nikola Velov, Macedonia[/i]
1979 Brazil National Olympiad, 5
[list=i]
[*] ABCD is a square with side 1. M is the midpoint of AB, and N is the midpoint of BC. The lines CM and DN meet at I. Find the area of the triangle CIN.
[*] The midpoints of the sides AB, BC, CD, DA of the parallelogram ABCD are M, N, P, Q respectively. Each midpoint is joined to the two vertices not on its side. Show that the area outside the resulting 8-pointed star is $\frac{2}{5}$ the area of the parallelogram.
[*] ABC is a triangle with CA = CB and centroid G. Show that the area of AGB is $\frac{1}{3}$ of the area of ABC.
[*] Is (ii) true for all convex quadrilaterals ABCD?
[/list]
2010 Today's Calculation Of Integral, 664
For a positive integer $n$, let $I_n=\int_{-\pi}^{\pi} \left(\frac{\pi}{2}-|x|\right)\cos nx\ dx$.
Find $I_1+I_2+I_3+I_4$.
[i]1992 University of Fukui entrance exam/Medicine[/i]
2010 SEEMOUS, Problem 2
Inside a square consider circles such that the sum of their circumferences is twice the perimeter of the square.
a) Find the minimum number of circles having this property.
b) Prove that there exist infinitely many lines which intersect at least 3 of these circles.
2018 German National Olympiad, 2
We are given a tetrahedron with two edges of length $a$ and the remaining four edges of length $b$ where $a$ and $b$ are positive real numbers. What is the range of possible values for the ratio $v=a/b$?
1981 Swedish Mathematical Competition, 3
Find all polynomials $p(x)$ of degree $5$ such that $p(x) + 1$ is divisible by $(x-1)^3$ and $p(x) - 1$ is divisible by $(x+1)^3$.
1982 All Soviet Union Mathematical Olympiad, 335
Three numbers $a,b,c$ belong to $[0,\pi /2]$ interval with $$\cos a = a, \sin(\cos b) = b, \cos(\sin c ) = c$$ Sort those numbers in increasing order.
2021 Stars of Mathematics, 1
For every integer $n\geq 3$, let $s_n$ be the sum of all primes (strictly) less than $n$. Are there infinitely many integers $n\geq 3$ such that $s_n$ is coprime to $n$?
[i]Russian Competition[/i]
2009 International Zhautykov Olympiad, 1
On the plane, a Cartesian coordinate system is chosen. Given points $ A_1,A_2,A_3,A_4$ on the parabola $ y \equal{} x^2$, and points $ B_1,B_2,B_3,B_4$ on the parabola $ y \equal{} 2009x^2$. Points $ A_1,A_2,A_3,A_4$ are concyclic, and points $ A_i$ and $ B_i$ have equal abscissas for each $ i \equal{} 1,2,3,4$.
Prove that points $ B_1,B_2,B_3,B_4$ are also concyclic.
2004 Estonia National Olympiad, 2
Draw a line passing through a point $M$ on the angle bisector of the angle $\angle AOB$, that intersects $OA$ and $OB$ at points $K$ and $L$ respectively. Prove that the valus of the sum $\frac{1}{|OK|}+\frac{1}{|OL|}$ does not depend on the choice of the straight line passing through $M$, i.e. is defined by the size of the angle AOB and the selection of the point $M$ only.
2023 CMIMC Combo/CS, 7
Max has a light bulb and a defective switch. The light bulb is initially off, and on the $n$th time the switch is flipped, the light bulb has a $\tfrac 1{2(n+1)^2}$ chance of changing its state (i.e. on $\to$ off or off $\to$ on). If Max flips the switch 100 times, find the probability the light is on at the end.
[i]Proposed by Connor Gordon[/i]
2002 USAMTS Problems, 2
Find four distinct positive integers, $a$, $b$, $c$, and $d$, such that each of the four sums $a+b+c$, $a+b+d$,$a+c+d$, and $b+c+d$ is the square of an integer. Show that infinitely many quadruples $(a,b,c,d)$ with this property can be created.
2023 Canadian Mathematical Olympiad Qualification, 3
Let circles $\Gamma_1$ and $\Gamma_2$ have radii $r_1$ and $r_2$, respectively. Assume that $r_1 < r_2$. Let $T$ be an intersection point of $\Gamma_1$ and $\Gamma_2$, and let $S$ be the intersection of the common external tangents of $\Gamma_1$ and $\Gamma_2$. If it is given that the tangents to $\Gamma_1$ and $ \Gamma_2$ at $T$ are perpendicular, determine the length of $ST$ in terms of $r_1$ and $r_2$.
2013 239 Open Mathematical Olympiad, 1
Consider all permutations of natural numbers from $1$ to $100$. A permutation is called $\emph{double}$ when it has the following property: If you write this permutation twice in a row, then delete $100$ numbers from them you get the remaining numbers $1, 2, 3, \ldots , 100$ in order. How many $\emph{double}$ permutations are there?