Found problems: 85335
2004 Germany Team Selection Test, 2
Find all functions $f: \Bbb{R}_{0}^{+}\rightarrow \Bbb{R}_{0}^{+}$ with the following properties:
(a) We have $f\left( xf\left( y\right) \right) \cdot f\left( y\right) =f\left( x+y\right)$ for all $x$ and $y$.
(b) We have $f\left(2\right) = 0$.
(c) For every $x$ with $0 < x < 2$, the value $f\left(x\right)$ doesn't equal $0$.
[b]NOTE.[/b] We denote by $\Bbb{R}_{0}^{+}$ the set of all non-negative real numbers.
2015 Latvia Baltic Way TST, 14
Let $S(a)$ denote the sum of the digits of the number $a$. Given a natural $R$ can one find a natural $n$ such that $\frac{S (n^2)}{S (n)}= R$?
2008 Hong Kong TST, 2
Let $ a$, $ b$, $ c$ be the three sides of a triangle. Determine all possible values of \[ \frac{a^2\plus{}b^2\plus{}c^2}{ab\plus{}bc\plus{}ca}\]
2012 National Olympiad First Round, 20
For each permutation $(a_1,a_2,\dots,a_{11})$ of the numbers $1,2,3,4,5,6,7,8,9,10,11$, we can determine at least $k$ of $a_i$s when we get $(a_1+a_3, a_2+a_4,a_3+a_5,\dots,a_8+a_{10},a_9+a_{11})$. $k$ can be at most ?
$ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ \text{None}$
2015 Online Math Open Problems, 8
The two numbers $0$ and $1$ are initially written in a row on a chalkboard. Every minute thereafter, Denys writes the number $a+b$ between all pairs of consecutive numbers $a$, $b$ on the board. How many odd numbers will be on the board after $10$ such operations?
[i]Proposed by Michael Kural[/i]
MathLinks Contest 2nd, 5.3
Let $n \ge 3$ and $\sigma \in S_n$ a permutation of the first $n$ positive integers. Prove that the numbers $\sigma (1), 2\sigma (2), 3\sigma(3), ... , n\sigma (n)$ cannot form an arithmetic, nor a geometric progression.
2007 Harvard-MIT Mathematics Tournament, 2
Two reals $x$ and $y$ are such that $x-y=4$ and $x^3-y^3=28$. Compute $xy$.
2020 Federal Competition For Advanced Students, P2, 1
Let $ABCD$ be a convex cyclic quadrilateral with the diagonal intersection $S$. Let further be $P$ the circumcenter of the triangle $ABS$ and $Q$ the circumcenter of the triangle $BCS$. The parallel to $AD$ through $P$ and the parallel to $CD$ through $Q$ intersect at point $R$. Prove that $R$ is on $BD$.
(Karl Czakler)
1968 Dutch Mathematical Olympiad, 1
On the base $AB$ of the isosceles triangle $ABC$, lies the point $P$ such that $AP : PB = 1 : 2$. Determine the minimum of $\angle ACP$.
1988 Swedish Mathematical Competition, 6
The sequence $(a_n)$ is defined by $a_1 = 1$ and $a_{n+1} = \sqrt{a_n^2 +\frac{1}{a_n}}$ for $n \ge 1$.
Prove that there exists $a$ such that $\frac{1}{2} \le \frac{a_n}{n^a} \le 2$ for $n \ge 1$.
2020 Iran MO (2nd Round), P4
Let $\omega_1$ and $\omega_2$ be two circles that intersect at point $A$ and $B$. Define point $X$ on $\omega_1$ and point $Y$ on $\omega_2$ such that the line $XY$ is tangent to both circles and is closer to $B$. Define points $C$ and $D$ the reflection of $B$ WRT $X$ and $Y$ respectively. Prove that the angle $\angle{CAD}$ is less than $90^{\circ}$
2024 Harvard-MIT Mathematics Tournament, 1
Inside an equilateral triangle of side length $6$, three congruent equilateral triangles of side length $x$ with sides parallel to the original equilateral triangle are arranged so that each has a vertex on a side of the larger triangle, and a vertex on another one of the three equilateral triangles, as shown below.
[img]https://cdn.artofproblemsolving.com/attachments/3/f/ff48c885154ce065c0d0420d1580769aa98eb1.png[/img]
A smaller equilateral triangle formed between the three congruent equilateral triangles has side length $1$. Compute $x$.
2025 Poland - Second Round, 3
Let $P$ be a point inside an acute triangle $ABC$ such that $\angle BPC=90^\circ$. We build triangles $AQB$ and $ARC$, outside of the triangle $ABC$, such that $\angle ABQ = \angle PBC$, $\angle QAB = \angle PAC$, $\angle RCA = \angle PCB$, and $\angle CAR = \angle BAP$. Prove that $P$, $Q$, $R$ are collinear.
2003 Czech-Polish-Slovak Match, 3
Numbers $p,q,r$ lies in the interval $(\frac{2}{5},\frac{5}{2})$ nad satisfy $pqr=1$. Prove that there exist two triangles of the same area, one with the sides $a,b,c$ and the other with the sides $pa,qb,rc$.
2017 Azerbaijan BMO TST, 1
Let $\triangle ABC$ be a acute triangle. Let $H$ the foot of the C-altitude in $AB$ such that $AH=3BH$, let $M$ and $N$ the midpoints of $AB$ and $AC$ and let $P$ be a point such that $NP=NC$ and $CP=CB$ and $B$, $P$ are located on different sides of the line $AC$. Prove that $\measuredangle APM=\measuredangle PBA$.
2017 Harvard-MIT Mathematics Tournament, 8
Does there exist an irrational number $\alpha > 1$ such that
\[\lfloor \alpha^n \rfloor \equiv 0 \pmod{2017}\]
for all integers $n \ge 1$?
2025 Sharygin Geometry Olympiad, 15
A point $C$ lies on the bisector of an acute angle with vertex $S$. Let $P$, $Q$ be the projections of $C$ to the sidelines of the angle. The circle centered at $C$ with radius $PQ$ meets the sidelines at points $A$ and $B$ such that $SA\ne SB$. Prove that the circle with center $A$ touching $SB$ and the circle with center $B$ touching $SA$ are tangent.
Proposed by: A.Zaslavsky
2013 JBMO Shortlist, 2
$\boxed{\text{A2}}$ Find the maximum value of $|\sqrt{x^2+4x+8}-\sqrt{x^2+8x+17}|$ where $x$ is a real number.
Math Hour Olympiad, Grades 8-10, 2014.1
Sherlock and Mycroft are playing Battleship on a $4\times4$ grid. Mycroft hides a single $3\times1$ cruiser somewhere on the board. Sherlock can pick squares on the grid and fire upon them. What is the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser?
1974 IMO Shortlist, 1
Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then they give the cards back. After a number $(\geq 2)$ of games we find out that A has $20$ points, $B$ has $10$ points and $C$ has $9$ points. We also know that in the last game B had the card with the biggest number. Who had in the first game the card with the second value (this means the middle card concerning its value).
Russian TST 2016, P3
The diagonals of a cyclic quadrilateral $ABCD$ intersect at $P$, and there exist a circle $\Gamma$ tangent to the extensions of $AB,BC,AD,DC$ at $X,Y,Z,T$ respectively. Circle $\Omega$ passes through points $A,B$, and is externally tangent to circle $\Gamma$ at $S$. Prove that $SP\perp ST$.
2010 China Team Selection Test, 2
Let $ABCD$ be a convex quadrilateral. Assume line $AB$ and $CD$ intersect at $E$, and $B$ lies between $A$ and $E$. Assume line $AD$ and $BC$ intersect at $F$, and $D$ lies between $A$ and $F$. Assume the circumcircles of $\triangle BEC$ and $\triangle CFD$ intersect at $C$ and $P$. Prove that $\angle BAP=\angle CAD$ if and only if $BD\parallel EF$.
2020 GQMO, 2
Geoff has an infinite stock of sweets, which come in $n$ flavours. He arbitrarily distributes some of the sweets amongst $n$ children (a child can get sweets of any subset of all flavours, including the empty set). Call a distribution $k-\textit{nice}$ if every group of $k$ children together has sweets in at least $k$ flavours. Find all subsets $S$ of $\{ 1, 2, \dots, n \}$ such that if a distribution of sweets is $s$-nice for all $s \in S$, then it is $s$-nice for all $s \in \{ 1, 2, \dots, n \}$.
[i]Proposed by Kyle Hess, USA[/i]
2008 Tournament Of Towns, 1
Each of ten boxes contains a di fferent number of pencils. No two pencils in the same box are of the same colour. Prove that one can choose one pencil from each box so that no two are of the same colour.
1986 National High School Mathematics League, 8
$f(x)=|1-2x|,x\in[0,1]$. Then the number of solutions to $f(f(f(x)))=\frac{1}{2}x$ is________.