Found problems: 85335
Let $O, I, H$ be the circumcenter, the incenter, and the orthocenter of $\triangle ABC$. The lines $AI$ and $AH$ intersect the circumcircle of $\triangle ABC$ for the second time at $D$ and $E$, respectively. Prove that if $OI \parallel BC$, then the circumcenter of $\triangle OIH$ lies on $DE$.
(Fedir Yudin)
[b]a)[/b] Let $ h:\mathbb{R}\longrightarrow\mathbb{R} $ he a function that admits a primitive $ H $ such that the function $ h/H $ is constant. Prove that there is a real number $ \gamma $ such that $ h(x)=\gamma\cdot\exp \left( x\cdot\frac{h}{H} (x) \right) , $ for any real number $ x. $
[b]b)[/b] Find the functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ that admit the primitives $ F,G, $ respectively, that satisfy $ f=\frac{G+g}{2},g=\frac{F+f}{2} $ and $ f(0)=g(0)=0. $
Let $n$ be a positive integer. For a fi nite set $M$ of positive integers and each $i \in \{0,1,..., n-1\}$, we denote $s_i$ the number of non-empty subsets of $M$ whose sum of elements gives remainder $i$ after division by $n$. We say that $M$ is "$n$-balanced" if $s_0 = s_1 =....= s_{n-1}$. Prove that for every odd number $n$ there exists a non-empty $n$-balanced subset of $\{0,1,..., n\}$.
For example if $n = 5$ and $M = \{1,3,4\}$, we have $s_0 = s_1 = s_2 = 1, s_3 = s_4 = 2$ so $M$ is not $5$-balanced.(Czech Republic)
Let $a$, $b$ and $c$ be positive real numbers such that $a \geq b \geq c$. Prove the inequality:
$\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \leq \frac{b}{a}+\frac{c}{b}+\frac{a}{c}$
Determine all pairs $(a, b)$ of positive integers for which the equation $x^3 - 17x^2 + ax - b^2 = 0$ has three integer roots (not necessarily different).
Let $S$ be the set of the points $(x_1, x_2, . . . , x_{2012})$ in $2012$-dimensional space such that $|x_1|+|x_2|+...+|x_{2012}| \le 1$. Let $T$ be the set of points in $2012$-dimensional space such that $\max^{2012}_{i=1}|x_i| = 2$. Let $p$ be a randomly chosen point on $T$. What is the probability that the closest point in $S$ to $p$ is a vertex of $S$?
Let $ABCD$ be an inscribed quadrilateral in a circle $c(O,R)$ (of circle $O$ and radius $R$). With centers the vertices $A,B,C,D$, we consider the circles $C_{A},C_{B},C_{C},C_{D}$ respectively, that do not intersect to each other . Circle $C_{A}$ intersects the sides of the quadrilateral at points $A_{1} , A_{2}$ , circle $C_{B}$ intersects the sides of the quadrilateral at points $B_{1} , B_{2}$ , circle $C_{C}$ at points $C_{1} , C_{2}$ and circle $C_{D}$ at points $C_{1} , C_{2}$ . Prove that the quadrilateral defined by lines $A_{1}A_{2} , B_{1}B_{2} , C_{1}C_{2} , D_{1}D_{2}$ is cyclic.
Given $P$ a real polynomial with degree greater than $ 1$.
Find all pairs $(f,Q)$ with function $f : R \to R$ and the real polynomial $Q$ satisfying the following two conditions:
i) for all $x, y \in R$, we have $f(P(x) + f(y)) = y + Q(f(x))$.
ii) there exists $x_0 \in R$ such that $f(P(x_0)) = Q(f(x_0))$.
$S,T$ are two trees without vertices of degree 2. To each edge is associated a positive number which is called length of this edge. Distance between two arbitrary vertices $v,w$ in this graph is defined by sum of length of all edges in the path between $v$ and $w$. Let $f$ be a bijective function from leaves of $S$ to leaves of $T$, such that for each two leaves $u,v$ of $S$, distance of $u,v$ in $S$ is equal to distance of $f(u), f(v)$ in $T$. Prove that there is a bijective function $g$ from vertices of $S$ to vertices of $T$ such that for each two vertices $u,v$ of $S$, distance of $u,v$ in $S$ is equal to distance of $g(u)$ and $g(v)$ in $T$.
Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$.
[i]Proposed by Juhan Aru, Estonia[/i]
An unfair coin has the property that when flipped four times, it has the same nonzero probability
of turning up $2$ heads and $2$ tails (in any order) as $3$ heads and $1$ tail (in any order). What is the probability of getting a head in any one flip?
A box contains red, green, blue, and white balls, 111 balls in all. If you take out 100 balls without looking, then there will always be 4 balls of different colors among them. What is the smallest number of balls you must take out without looking to guarantee that among them there will always be balls of at least 3 different colors?
(a) Prove that $(x+y+z)^2 \le 3(x^2 +y^2 +z^2)$ for any real numbers $x,y,z$.
(b) If positive numbers $a,b,c$ satisfy $a+b+c \ge abc$, prove that $a^2 +b^2 +c^2 \ge \sqrt3 abc$
(a) Find all positive integers $n$ for which the equation $(a^a)^n = b^b$ has a solution
in positive integers $a,b$ greater than $1$.
(b) Find all positive integers $a, b$ satisfying $(a^a)^5=b^b$
Given are two circles $\omega_1,\omega_2$ which intersect at points $X,Y$. Let $P$ be an arbitrary point on $\omega_1$. Suppose that the lines $PX,PY$ meet $\omega_2$ again at points $A,B$ respectively. Prove that the circumcircles of all triangles $PAB$ have the same radius.
Chords $AB$ and $CD$ of a circle are perpendicular and intersect at a point $P$. If $AP = 6, BP = 12$, and $CD = 22$, find the area of the circle.
How many $1 < n \le 2018$ such that the set $$\{0, 1, 1+2,...,1+2+3+...+i,..., 1+2+...+n-1\}$$ is a permutation of $\{0, 1, 2, 3, 4,...,; n -1\}$ when reduced modulo $n$?
Let $\xi_{(k_1, k_2)}, k_1, k_2 \in\mathbb N$ be random variables uniformly bounded. Let $c_l, l\in\mathbb N$ be a positive real strictly increasing infinite sequence such that $c_{l+1}/ c_l$ is bounded. Let $d_l=\log \left(c_{l+1}/c_l\right), l\in\mathbb N$ and suppose that $D_n=\sum_{l=1}^n d_l\uparrow \infty$ when $n\to\infty$
Suppose there exist $C>0$ and $\varepsilon>0$ such that
$$\left| \mathbb E \left\{ \xi_{(k_1,k_2)}\xi_{(l_1,l_2)}\right\}\right| \leq C\prod_{i=1}^2 \left\{ \log_+\log_+\left( \frac{c_{\max\{ k_i, l_i\}}}{c_{\min\{ k_i, l_i\}}}\right)\right\}^{-(1+\varepsilon)}$$
for each $(k_1, k_2), (l_1,l_2)\in\mathbb N^2$ ($\log_+$ is the positive part of the natural logarithm). Show that
$$\lim_{\substack{n_1\to\infty \\ n_2\to\infty}} \frac{1}{D_{n_1}D_{n_2}}\sum_{k_1=1}^{n_1} \sum_{k_2=1}^{n_2} d_{k_1}d_{k_2}\xi_{(k_1,k_2)}=0$$
almost surely.
(translated by j___d)
Find the set of all real values of $a$ for which the real polynomial equation $P(x)=x^2-2ax+b=0$ has real roots, given that $P(0)\cdot P(1)\cdot P(2)\neq 0$ and $P(0),P(1),P(2)$ form a geometric progression.
Suppose that $x$, $y$, and $z$ are complex numbers such that $xy = -80-320i$, $yz = 60$, and $zx = -96+24i$, where $i = \sqrt{-1}$. Then there are real numbers $a$ and $b$ such that $x+y+z = a+bi$. Find $a^2 + b^2$.
A closed container in the shape of a rectangular parallelepiped contains $1$ liter of water. If the container rests horizontally on three different sides, the water level is $2$ cm, $4$ cm and $5$ cm. Calculate the volume of the parallelepiped.
A mink is standing in the center of a field shaped like a regular polygon. The field is surrounded by a fence, and the mink can only exit through the vertices of the polygon. A dog is standing on one of the vertices, and can move along the fence. The mink wants to escape the field, while the dog tries to prevent it. Each of them moves with constant velocity. For what ratio of velocities could the mink escape if:
a. The field is a regular triangle?
b. The field is a square?
Let $\vartriangle ABC$ be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in $A$ to the circumcircle intersects line $BC$ in $D$. Prove that $BD$ is not an integer.
The diagonals of a cyclic quadrilateral $ABCD$ intersect at $E$. Let $F$ and $G$ be the midpoints of $AB$ and $CD$ respectively. Prove that the lines through $E,F$ and $G$ perpendicular to $AD,BD$ and $AC$, respectively, intersect in a single point.
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.
[i]Proposed by Jeck Lim, Singapore[/i]