Found problems: 85335
2018 IMC, 4
Find all differentiable functions $f:(0,\infty) \to \mathbb{R}$ such that
$$f(b)-f(a)=(b-a)f’(\sqrt{ab}) \qquad \text{for all}\qquad a,b>0.$$
[i]Proposed by Orif Ibrogimov, National University of Uzbekistan[/i]
2016 Rioplatense Mathematical Olympiad, Level 3, 2
Determine all positive integers $n$ for which there are positive real numbers $x,y$ and $z$ such that $\sqrt x +\sqrt y +\sqrt z=1$ and $\sqrt{x+n} +\sqrt{y+n} +\sqrt{z+n}$ is an integer.
2015 Moldova Team Selection Test, 4
Let $n$ and $k$ be positive integers, and let be the sets $X=\{1,2,3,...,n\}$ and $Y=\{1,2,3,...,k\}$.
Let $P$ be the set of all the subsets of the set $X$. Find the number of functions $ f: P \to Y$ that satisfy $f(A \cap B)=\min(f(A),f(B))$ for all $A,B \in P$.
2014 Postal Coaching, 1
Suppose $p,q,r$ are three distinct primes such that $rp^3+p^2+p=2rq^2+q^2+q$. Find all possible values of $pqr$.
2001 JBMO ShortLists, 9
Consider a convex quadrilateral $ABCD$ with $AB=CD$ and $\angle BAC=30^{\circ}$. If $\angle ADC=150^{\circ}$, prove that $\angle BCA= \angle ACD$.
2023-24 IOQM India, 9
Find the number of triples $(a, b, c)$ of positive integers such that
(a) $a b$ is a prime;
(b) $b c$ is a product of two primes;
(c) $a b c$ is not divisible by square of any prime and
(d) $a b c \leq 30$.
2019 Belarus Team Selection Test, 7.3
Given a positive integer $n$, determine the maximal constant $C_n$ satisfying the following condition: for any partition of the set $\{1,2,\ldots,2n \}$ into two $n$-element subsets $A$ and $B$, there exist labellings $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ of $A$ and $B$, respectively, such that
$$
(a_1-b_1)^2+(a_2-b_2)^2+\ldots+(a_n-b_n)^2\ge C_n.
$$
[i](B. Serankou, M. Karpuk)[/i]
2006 Kyiv Mathematical Festival, 4
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.
2005 Purple Comet Problems, 10
What is the $1000$ th digit to the right of the decimal point in the decimal representation of $\tfrac{37}{5500}$?
1975 AMC 12/AHSME, 25
A woman, her brother, her son and her daughter are chess players (all relations by birth). The worst player's twin (who is one of the four players) and the best player are of opposite sex. The worst player and the best player are the same age. Who is the worst player?
$ \textbf{(A)}\ \text{the woman} \qquad\textbf{(B)}\ \text{her son} \qquad\textbf{(C)}\ \text{her brother} \qquad\textbf{(D)}\ \text{her daughter} \\ \qquad\textbf{(E)}\ \text{No solution is consistent with the given information} $
2011 International Zhautykov Olympiad, 1
Find the maximum number of sets which simultaneously satisfy the following conditions:
[b]i)[/b] any of the sets consists of $4$ elements,
[b]ii)[/b] any two different sets have exactly $2$ common elements,
[b]iii)[/b] no two elements are common to all the sets.
2001 Moldova National Olympiad, Problem 7
Let $ABCD$ and $AB’C’D’$ be equally oriented squares. Prove that the lines $BB_1,CC_1,DD_1$ are concurrent.
2017 Junior Balkan Team Selection Tests - Romania, 2
Let $A$ be a point outside the circle $\omega$ . The tangents from $A$ touch the circle at $B$ and $C$. Let $P$ be an arbitrary point on extension of $AC$ towards $C$, $Q$ the projection of $C$ onto $PB$ and $E$ the second intersection point of the circumcircle of $ABP$ with the circle $\omega$ . Prove that $\angle PEQ = 2\angle APB$
2004 Federal Competition For Advanced Students, P2, 2
Show that every set $ \{p_1,p_2,\dots,p_k\}$ of prime numbers fulfils the following: The sum of all unit fractions (that are fractions of the type $ \frac{1}{n}$), whose denominators are exactly the $ k$ given prime factors (but in arbitrary powers with exponents unequal zero), is an unit fraction again.
How big is this sum if $ \frac{1}{2004}$ is among this summands?
Show that for every set $ \{p_1,p_2,\dots,p_k\}$ containing $ k$ prime numbers ($ k>2$) is the sum smaller than $ \frac{1}{N}$ with $ N=2\cdot 3^{k-2}(k-2)!$
2009 Argentina Team Selection Test, 1
On a $ 50 \times 50$ board, the centers of several unit squares are colored black. Find the maximum number of centers that can be colored black in such a way that no three black points form a right-angled triangle.
1989 IMO Longlists, 11
Given the equation \[ y^4 \plus{} 4y^2x \minus{} 11y^2 \plus{} 4xy \minus{} 8y \plus{} 8x^2 \minus{} 40x \plus{} 52 \equal{} 0,\] find all real solutions.
2019 PUMaC Algebra B, 8
A [i]weak binary representation[/i] of a nonnegative integer $n$ is a representation $n=a_0+2\cdot a_1+2^2\cdot a_2+\dots$ such that $a_i\in\{0,1,2,3,4,5\}$. Determine the number of such representations for $513$.
2015 Canadian Mathematical Olympiad Qualification, 4
Given an acute-angled triangle $ABC$ whose altitudes from $B$ and $C$ intersect at $H$, let $P$ be any point on side $BC$ and $X, Y$ be points on $AB, AC$, respectively, such that $PB = PX$ and $PC = PY$. Prove that the points $A, H, X, Y$ lie on a common circle.
2011 Regional Olympiad of Mexico Center Zone, 4
Show that if a $6n$-digit number is divisible by $7$, then the number that results from moving the ones digit to the beginning of the number is also a multiple of $7$.
Revenge EL(S)MO 2024, 5
Inscribe three mutually tangent pink disks of radii $450$, $450$, and $720$ in an uncolored circle $\Omega$ of radius $1200$. In one move, Elmo selects an uncolored region inside $\Omega$ and draws in it the largest possible pink disk. Can Elmo ever draw a disk with a radius that is a perfect square of a rational?
Proposed by [i]Ritwin Narra[/i]
2011 NIMO Problems, 3
Billy and Bobby are located at points $A$ and $B$, respectively. They each walk directly toward the other point at a constant rate; once the opposite point is reached, they immediately turn around and walk back at the same rate. The first time they meet, they are located 3 units from point $A$; the second time they meet, they are located 10 units from point $B$. Find all possible values for the distance between $A$ and $B$.
[i]Proposed by Isabella Grabski[/i]
2021 MIG, 19
Aprameya graphs the equation $2x = y + 4$ on the coordinate plane. It turns out that there is a unique point with a positive integer coordinate and a negative integer coordinate lying on Aprameya's graph. What is the sum of the coordinates of this point?
$\textbf{(A) }{-}3\qquad\textbf{(B) }{-}1\qquad\textbf{(C) }0\qquad\textbf{(D) }1\qquad\textbf{(E) }2$
2023 Moldova Team Selection Test, 12
The sequence $\left(a_n \right)$ is defined by $a_1=1, \ a_2=2$ and
$$a_{n+2} = 2a_{n+1}-pa_n, \ \forall n \ge 1,$$ for some prime $p.$ Find all $p$ for which there exists $m$ such that $a_m=-3.$
1963 Dutch Mathematical Olympiad, 5
You want to color the side faces of a cube in such a way that each face is colored evenly. Six colors are available:
[i]red, white, blue, yellow, purple, orange[/i]. Two cube colors are called the same if one arises from the other by a rotation of the cube.
(a) How many different cube colorings are there, using six colors?
(b) How many different cube colorings are there, using exactly five colors?
1986 Miklós Schweitzer, 3
(a) Prove that for every natural number $k$, there are positive integers $a_1<a_2<\ldots <a_k$ such that $a_i-a_j$ divides $a_i$ for all $1\leq i, j\leq k, i\neq j$.
(b) Show that there is an absolute constant $C>0$ such that $a_1>k^{Ck}$ for every sequence $a_1,\ldots, a_k$ of numbers that satisfy the above divisibility condition.
[A. Balogh, I. Z. Ruzsa]