Found problems: 85335
1994 IMO Shortlist, 5
Let $ f(x) \equal{} \frac{x^2\plus{}1}{2x}$ for $ x \neq 0.$ Define $ f^{(0)}(x) \equal{} x$ and $ f^{(n)}(x) \equal{} f(f^{(n\minus{}1)}(x))$ for all positive integers $ n$ and $ x \neq 0.$ Prove that for all non-negative integers $ n$ and $ x \neq \{\minus{}1,0,1\}$
\[ \frac{f^{(n)}(x)}{f^{(n\plus{}1)}(x)} \equal{} 1 \plus{} \frac{1}{f \left( \left( \frac{x\plus{}1}{x\minus{}1} \right)^{2n} \right)}.\]
1978 Bulgaria National Olympiad, Problem 2
$k_1$ denotes one of the arcs formed by intersection of the circumference $k$ and the chord $AB$. $C$ is the middle point of $k_1$. On the half line (ray) $PC$ is drawn the segment $PM$. Find the locus formed from the point $M$ when $P$ is moving on $k_1$.
[i]G. Ganchev[/i]
2015 Indonesia MO Shortlist, G8
$ABC$ is an acute triangle with $AB> AC$. $\Gamma_B$ is a circle that passes through $A,B$ and is tangent to $AC$ on $A$. Define similar for $ \Gamma_C$. Let $D$ be the intersection $\Gamma_B$ and $\Gamma_C$ and $M$ be the midpoint of $BC$. $AM$ cuts $\Gamma_C$ at $E$. Let $O$ be the center of the circumscibed circle of the triangle $ABC$. Prove that the circumscibed circle of the triangle $ODE$ is tangent to $\Gamma_B$.
2020 Baltic Way, 4
Find all functions $f:\mathbb{R} \to \mathbb{R}$ so that
\[f(f(x)+x+y) = f(x+y) + y f(y)\]
for all real numbers $x, y$.
2019 Sharygin Geometry Olympiad, 23
In the plane, let $a$, $b$ be two closed broken lines (possibly self-intersecting), and $K$, $L$, $M$, $N$ be four points. The vertices of $a$, $b$ and the points $K$ $L$, $M$, $N$ are in general position (i.e. no three of these points are collinear, and no three segments between them concur at an interior point). Each of segments $KL$ and $MN$ meets $a$ at an even number of points, and each of segments $LM$ and $NK$ meets $a$ at an odd number of points. Conversely, each of segments $KL$ and $MN$ meets $b$ at an odd number of points, and each of segments $LM$ and $NK$ meets $b$ at an even number of points. Prove that $a$ and $b$ intersect.
2018 MIG, 19
Rectangle $ABCD$, with integer side lengths, has equal area and perimeter. What is the positive difference between the two possible areas of $ABCD$?
$\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 5\qquad\textbf{(E) } 6$
2021 Grand Duchy of Lithuania, 3
Let $ABCD$ be a convex quadrilateral satisfying $\angle ADB + \angle ACB = \angle CAB + \angle DBA = 30^o$, $AD = BC$. Prove that there exists a right-angled triangle with side lengths $AC$, $BD$, $CD$.
2018 Brazil National Olympiad, 6
Let $S(n)$ be the sum of digits of $n$. Determine all the pairs $(a, b)$ of positive integers, such that the expression
$S(an + b) - S(n)$ has a finite number of values, where $n$ is varying in the positive integers.
1950 AMC 12/AHSME, 50
A privateer discovers a merchantman $10$ miles to leeward at 11:45 a.m. and with a good breeze bears down upon her at $11$ mph, while the merchantman can only make $8$ mph in her attempt to escape. After a two hour chase, the top sail of the privateer is carried away; she can now make only $17$ miles while the merchantman makes $15$. The privateer will overtake the merchantman at:
$\textbf{(A)}\ 3\text{:}45\text{ p.m.} \qquad
\textbf{(B)}\ 3\text{:}30\text{ p.m.} \qquad
\textbf{(C)}\ 5\text{:}00\text{ p.m.} \qquad
\textbf{(D)}\ 2\text{:}45\text{ p.m.} \qquad
\textbf{(E)}\ 5\text{:}30\text{ p.m.}$
2007 All-Russian Olympiad, 8
Given a matrix $\{a_{ij}\}_{i,j=0}^{9}$, $a_{ij}=10i+j+1$. Andrei is going to cover its entries by $50$ rectangles $1\times 2$ (each such rectangle contains two adjacent entries) so that the sum of $50$ products in these rectangles is minimal possible. Help him.
[i]A. Badzyan[/i]
1999 All-Russian Olympiad Regional Round, 10.5
Are there $10$ different integers such that all the sums made up of $9$ of them are perfect squares?
2014 Contests, 3
Let $\Gamma_1$ be a circle and $P$ a point outside of $\Gamma_1$. The tangents from $P$ to $\Gamma_1$ touch the circle at $A$ and $B$. Let $M$ be the midpoint of $PA$ and $\Gamma_2$ the circle through $P$, $A$ and $B$. Line $BM$ cuts $\Gamma_2$ at $C$, line $CA$ cuts $\Gamma_1$ at $D$, segment $DB$ cuts $\Gamma_2$ at $E$ and line $PE$ cuts $\Gamma_1$ at $F$, with $E$ in segment $PF$. Prove lines $AF$, $BP$, and $CE$ are concurrent.
2013 Estonia Team Selection Test, 1
Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\
a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$
2005 Today's Calculation Of Integral, 82
Let $0<a<b$.Prove the following inequaliy.
\[\frac{1}{b-a}\int_a^b \left(\ln \frac{b}{x}\right)^2 dx<2\]
2012 Sharygin Geometry Olympiad, 3
In triangle $ABC$, the bisector $CL$ was drawn. The incircles of triangles $CAL$ and $CBL$ touch $AB$ at points $M$ and $N$ respectively. Points $M$ and $N$ are marked on the picture, and then the whole picture except the points $A, L, M$, and $N$ is erased. Restore the triangle using a compass and a ruler.
(V.Protasov)
2010 Stanford Mathematics Tournament, 6
A triangle has side lengths $7, 9,$ and $12$. What is the area of the triangle?
2002 Estonia Team Selection Test, 6
Place a pebble at each [i]non-positive[/i] integer point on the real line, and let $n$ be a fixed positive integer. At each step we choose some n consecutive integer points, remove one of the pebbles located at these points and rearrange all others arbitrarily within these points (placing at most one pebble at each point).
Determine whether there exists a positive integer $n$ such that for any given $N > 0$ we can place a pebble at a point with coordinate greater than $N$ in a finite number of steps described above.
2009 All-Russian Olympiad Regional Round, 11.6
Point $D$ on side $BC$ of acute triangle ABC is such that $AB=AD$. The circumcircle of triangle $ABD$ intersects side $AC$ at points $A$ and $K$. Line $DK$ intersects the perpendicular drawn from $B$ on $AC$, at the point $L$. Prove that $CL= BC$
2006 Korea National Olympiad, 1
Given that for reals $a_1,\cdots, a_{2004},$ equation $x^{2006}-2006x^{2005}+a_{2004}x^{2004}+\cdots +a_2x^2+a_1x+1=0$ has $2006$ positive real solution, find the maximum possible value of $a_1.$
2019 Kosovo National Mathematical Olympiad, 4
Find all real numbers $x,y,z$ such that satisfied the following equalities at same time:
$\sqrt{x^3-y}=z-1 \wedge \sqrt{y^3-z}=x-1\wedge \sqrt{z^3-x}=y-1$
2020 Estonia Team Selection Test, 1
The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).
2004 Junior Balkan Team Selection Tests - Romania, 3
A finite set of positive integers is called [i]isolated [/i]if the sum of the numbers in any given proper subset is co-prime with the sum of the elements of the set.
a) Prove that the set $A=\{4,9,16,25,36,49\}$ is isolated;
b) Determine the composite numbers $n$ for which there exist the positive integers $a,b$ such that the set
\[ A=\{(a+b)^2, (a+2b)^2,\ldots, (a+nb)^2\}\] is isolated.
2022 Belarusian National Olympiad, 11.4
On plane circles $\omega_1, \omega_2, \omega_3$ with centers $O_1,O_2,O_3$ are given such that $\omega_1$ is externally tangent $\omega_2$ and $\omega_3$ at points $P, Q$ respectively. On $\omega_1$ point $C$ is chosen arbitrarily. Line $CP$ intersects $\omega_2$ at $B$, line $CQ$ intersects $\omega_3$ at $A$. Point $O$ is the circumcenter of $ABC$.
Prove that the locus of points $O$ (when $C$ changes) is a circle, the center of which lies on the circumcircle of $O_1O_2O_3$
2022 Assara - South Russian Girl's MO, 5
Find all pairs of prime numbers $p, q$ such that the number $pq + p - 6$ is also prime.
1964 IMO Shortlist, 6
In tetrahedron $ABCD$, vertex $D$ is connected with $D_0$, the centrod if $\triangle ABC$. Line parallel to $DD_0$ are drawn through $A,B$ and $C$. These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_2, B_1,$ and $C_1$, respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$. Is the result if point $D_o$ is selected anywhere within $\triangle ABC$?