Found problems: 85335
2023 Sharygin Geometry Olympiad, 23
An ellipse $\Gamma_1$ with foci at the midpoints of sides $AB$ and $AC$ of a triangle $ABC$ passes through $A$, and an ellipse $\Gamma_2$ with foci at the midpoints of $AC$ and $BC$ passes through $C$. Prove that the common points of these ellipses and the orthocenter of triangle $ABC$ are collinear.
2017 China Western Mathematical Olympiad, 5
Let $a_1,a_2,\cdots ,a_9$ be $9$ positive integers (not necessarily distinct) satisfying: for all $1\le i<j<k\le 9$, there exists $l (1\le l\le 9)$ distinct from $i,j$ and $j$ such that $a_i+a_j+a_k+a_l=100$. Find the number of $9$-tuples $(a_1,a_2,\cdots ,a_9)$ satisfying the above conditions.
1949-56 Chisinau City MO, 39
Solve the equation: $\log_{x} 2 \cdot \log_{2x} 2 = \log_{4x} 2$.
VMEO III 2006 Shortlist, N1
$f(n)$ denotes the largest integer $k$ such that that $2^k|n$.
$2006$ integers $a_i$ are such that $a_1<a_2<...<a_{2016}$.
Is it possible to find integers $k$ where $1 \le k\le 2006$ and $f(a_i-a_j)\ne k$ for every $1 \le j \le i \le 2006$ ?
2004 Bulgaria Team Selection Test, 1
Find all $k>0$ such that there exists a function $f : [0,1]\times[0,1] \to [0,1]$ satisfying the following conditions:
$f(f(x,y),z)=f(x,f(y,z))$;
$f(x,y) = f(y,x)$;
$f(x,1)=x$;
$f(zx,zy) = z^{k}f(x,y)$, for any $x,y,z \in [0,1]$
1995 VJIMC, Problem 4
Let $\{x_n\}_{n=1}^\infty$ be a sequence such that $x_1=25$, $x_n=\operatorname{arctan}(x_{n-1})$. Prove that this sequence has a limit and find it.
2006 AMC 10, 8
A parabola with equation $ y \equal{} x^2 \plus{} bx \plus{} c$ passes through the points $ (2,3)$ and $ (4,3)$. What is $ c$?
$ \textbf{(A) } 2 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 11$
1987 IMO Longlists, 73
Let $f(x)$ be a periodic function of period $T > 0$ defined over $\mathbb R$. Its first derivative is continuous on $\mathbb R$. Prove that there exist $x, y \in [0, T )$ such that $x \neq y$ and
\[f(x)f'(y)=f'(x)f(y).\]
1992 Tournament Of Towns, (333) 1
Prove that the product of all integers from $2^{1917} +1$ up to $2^{1991} -1$ is not the square of an integer.
(V. Senderov, Moscow)
2015 Online Math Open Problems, 18
Alex starts with a rooted tree with one vertex (the root). For a vertex $v$, let the size of the subtree of $v$ be $S(v)$. Alex plays a game that lasts nine turns. At each turn, he randomly selects a vertex in the tree, and adds a child vertex to that vertex. After nine turns, he has ten total vertices. Alex selects one of these vertices at random (call the vertex $v_1$). The expected value of $S(v_1)$ is of the form $\tfrac{m}{n}$ for relatively prime positive integers $m, n$. Find $m+n$.
[b]Note:[/b] In a rooted tree, the subtree of $v$ consists of its indirect or direct descendants (including $v$ itself).
[i]Proposed by Yang Liu[/i]