Found problems: 85335
2016 AMC 10, 2
For what value of $x$ does $10^{x}\cdot 100^{2x}=1000^{5}$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$
2007 Nicolae Coculescu, 3
Show that for any three numbers $ a,b,c\in (1,\infty ) , $ the following inequality is true:
$$ \log_{ab} c +\log_{bc} a +\log_{ca} b\ge log_{a^2bc} bc +log_{b^2ca} ca +log_{c^2ab} ab$$
[i]Costel Anghel[/i]
2013 Bosnia Herzegovina Team Selection Test, 5
Let $x_1,x_2,\ldots,x_n$ be nonnegative real numbers of sum equal to $1$.
Let $F_n=x_1^{2}+x_2^{2}+\cdots +x_n^{2}-2(x_1x_2+x_2x_3+\cdots +x_nx_1)$.
Find:
a) $\min F_3$;
b) $\min F_4$;
c) $\min F_5$.
2019 South East Mathematical Olympiad, 2
$ABCD$ is a parallelogram with $\angle BAD \neq 90$. Circle centered at $A$ radius $BA$ denoted as $\omega _1$ intersects the extended side of $AB,CB$ at points $E,F$ respectively. Suppose the circle centered at $D$ with radius $DA$, denoted as $\omega _2$, intersects $AD,CD$ at points $M,N$ respectively. Suppose $EN,FM$ intersects at $G$, $AG$ intersects $ME$ at point $T$. $MF$ intersects $\omega _1$ at $Q \neq F$, and $EN$ intersects $\omega _2$ at $P \neq N$. Prove that $G,P,T,Q$ concyclic.
2015 Germany Team Selection Test, 3
Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively.
Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes.
[i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]
2005 Italy TST, 1
A stage course is attended by $n \ge 4$ students. The day before the final exam, each group of three students conspire against another student to throw him/her out of the exam. Prove that there is a student against whom there are at least $\sqrt[3]{(n-1)(n- 2)} $conspirators.
1996 China National Olympiad, 2
Find the smallest positive integer $ K$ such that every $ K$-element subset of $ \{1,2,...,50 \}$ contains two distinct elements $ a,b$ such that $ a\plus{}b$ divides $ ab$.
1987 Balkan MO, 1
Let $a$ be a real number and let $f : \mathbb{R}\rightarrow \mathbb{R}$ be a function satisfying $f(0)=\frac{1}{2}$ and
\[f(x+y)=f(x)f(a-y)+f(y)f(a-x), \quad \forall x,y \in \mathbb{R}.\]
Prove that $f$ is constant.
2021/2022 Tournament of Towns, P2
On a blank paper there were drawn two perpendicular axes $x$ and $y$ with the same scale. The graph of a function $y=f(x)$ was drawn in this coordinate system. Then the $y$ axis and all the scale marks on the $x$ axis were erased. Provide a way how to draw again the $y$ axis using pencil, ruler and compass:
(a) $f(x)= 3^x$;
(b) $f(x)= \log_a x$, where $a>1$ is an unknown number.
2011 Saudi Arabia Pre-TST, 1.4
Let $ABC$ be a triangle with $AB=AC$ and $\angle BAC = 40^o$. Points $S$ and $T$ lie on the sides $AB$ and $BC$, such that $\angle BAT = \angle BCS = 10^o$. Lines $AT$ and $CS$ meet at $P$. Prove that $BT = 2PT$.
2007 QEDMO 4th, 1
Find all primes $p,$ $q,$ $r$ satisfying $p^{2}+2q^{2}=r^{2}.$
1997 China Team Selection Test, 3
There are 1997 pieces of medicine. Three bottles $A, B, C$ can contain at most 1997, 97, 19 pieces of medicine respectively. At first, all 1997 pieces are placed in bottle $A$, and the three bottles are closed. Each piece of medicine can be split into 100 part. When a bottle is opened, all pieces of medicine in that bottle lose a part each. A man wishes to consume all the medicine. However, he can only open each of the bottles at most once each day, consume one piece of medicine, move some pieces between the bottles, and close them. At least how many parts will be lost by the time he finishes consuming all the medicine?
2013 Balkan MO Shortlist, G5
Let $ABC$ be an acute triangle with $AB < AC < BC$ inscribed in a circle $(c)$ and let $E$ be an arbitrary point on its altitude $CD$. The circle $(c_1)$ with diameter $EC$, intersects the circle $(c)$ at point $K$ (different than $C$), the line $AC$ at point $L$ and the line $BC$ at point $M$. Finally the line $KE$ intersects $AB$ at point $N$. Prove that the quadrilateral $DLMN$ is cyclic.
2017 Princeton University Math Competition, A5/B7
Rectangle $HOMF$ has $HO=11$ and $OM=5$. Triangle $ABC$ has orthocenter $H$ and circumcenter $O$. $M$ is the midpoint of $BC$ and altitude $AF$ meets $BC$ at $F$. Find the length of $BC$.
2005 Baltic Way, 19
Is it possible to find $2005$ different positive square numbers such that their sum is also a square number ?
2014 Contests, 1
A natural number $k$ is such that $k^2 < 2014 < (k +1)^2$. What is the largest prime factor of $k$?
2022 CIIM, 5
Define in the plane the sequence of vectors $v_1, v_2, \ldots$ with initial values $v_1 = (1, 0)$, $v_2 = (-1/\sqrt{2}, 1/\sqrt{2})$ and satisfying the relationship $$v_n=\frac{v_{n-1}+v_{n-2}}{\lVert v_{n-1}+v_{n-2}\rVert},$$ for $n \geq 3$. Show that the sequence is convergent and determine its limit.
[b]Note:[/b] The expression $\lVert v \rVert$ denotes the length of the vector $v$.
2014 ELMO Shortlist, 3
We say a finite set $S$ of points in the plane is [i]very[/i] if for every point $X$ in $S$, there exists an inversion with center $X$ mapping every point in $S$ other than $X$ to another point in $S$ (possibly the same point).
(a) Fix an integer $n$. Prove that if $n \ge 2$, then any line segment $\overline{AB}$ contains a unique very set $S$ of size $n$ such that $A, B \in S$.
(b) Find the largest possible size of a very set not contained in any line.
(Here, an [i]inversion[/i] with center $O$ and radius $r$ sends every point $P$ other than $O$ to the point $P'$ along ray $OP$ such that $OP\cdot OP' = r^2$.)
[i]Proposed by Sammy Luo[/i]
2013 Math Prize For Girls Problems, 4
The MathMatters competition consists of 10 players $P_1$, $P_2$, $\dots$, $P_{10}$ competing in a ladder-style tournament. Player $P_{10}$ plays a game with $P_9$: the loser is ranked 10th, while the winner plays $P_8$. The loser of that game is ranked 9th, while the winner plays $P_7$. They keep repeating this process until someone plays $P_1$: the loser of that final game is ranked 2nd, while the winner is ranked 1st. How many different rankings of the players are possible?
2023 Indonesia TST, G
Given circle $\Omega_1$ and $\Omega_2$ interesting at $P$ and $Q$. $X$ and $Y$ on line $PQ$ such that $X, P, Q, Y$ in that order. Point $A$ and $B$ on $\Omega_1$ and $\Omega_2$ respectively such that the intersections of $\Omega_1$ with $AX$ and $AY$, intersections of $\Omega_2$ with $BX$ and $BY$ are all in one line. $l$. Prove that $AB, l$ and perpendicular bisector of $PQ$ are concurrent.
2009 Postal Coaching, 3
Let $n \ge 3$ be a positive integer. Find all nonconstant real polynomials $f_1(x), f_2(x), ..., f_n(x)$ such that $f_k(x)f_{k+1}(x) = f_{k+1}(f_{k+2}(x))$, $1 \le k \le n$ for all real x. [All suffixes are taken modulo $n$.]
1994 IMO, 4
Find all ordered pairs $ (m,n)$ where $ m$ and $ n$ are positive integers such that $ \frac {n^3 \plus{} 1}{mn \minus{} 1}$ is an integer.
2008 AMC 10, 18
A right triangle has perimeter $ 32$ and area $ 20$. What is the length of its hypotenuse?
$ \textbf{(A)}\ \frac{57}{4} \qquad
\textbf{(B)}\ \frac{59}{4} \qquad
\textbf{(C)}\ \frac{61}{4} \qquad
\textbf{(D)}\ \frac{63}{4} \qquad
\textbf{(E)}\ \frac{65}{4}$
1992 Romania Team Selection Test, 2
Let $ a_1, a_2, ..., a_k $ be distinct positive integers such that the $2^k$ sums $\displaystyle\sum\limits_{i=1}^{k}{\epsilon_i a_i}$, $\epsilon_i\in\left\{0,1\right\}$ are distinct.
a) Show that $ \dfrac{1}{a_1}+\dfrac{1}{a_2}+...+\dfrac{1}{a_k}\le2(1-2^{-k}) $;
b) Find the sequences $(a_1,a_2,...,a_k)$ for which the equality holds.
[i]Șerban Buzețeanu[/i]
2022 Indonesia TST, A
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying
\[ f(a^2) - f(b^2) \leq (f(a)+b)(a-f(b)) \] for all $a,b \in \mathbb{R}$.