Found problems: 85335
1973 Poland - Second Round, 1
Prove that if positive numbers $ x, y, z $ satisfy the inequality
$$
\frac{x^2+y^2-z^2}{2xy} + \frac{y^2+z^2-x^2}{2yz} + \frac{z^2+x^2-y ^2}{2xz} > 1,$$
then they are the lengths of the sides of a certain triangle.
2014 Contests, 2
Let $M$ be the set of palindromic integers of the form $5n+4$ where $n\ge 0$ is an integer.
[list=a]
[*]If we write the elements of $M$ in increasing order, what is the $50^{\text{th}}$ number?
[*]Among all numbers in $M$ with nonzero digits which sum up to $2014$ which is the largest and smallest one?[/list]
1984 IMO Longlists, 38
Determine all continuous functions $f: \mathbb R \to \mathbb R$ such that
\[f(x + y)f(x - y) = (f(x)f(y))^2, \quad \forall(x, y) \in\mathbb{R}^2.\]
2005 Taiwan National Olympiad, 2
In triangle $ABC$, $D$ is the midpoint of side $AB$. $E$ and $F$ are points arbitrarily chosen on segments $AC$ and $BC$, respectively. Show that $[DEF] < [ADE] + [BDF]$.
1965 German National Olympiad, 1
For a given positive real parameter $p$, solve the equation $\sqrt{p+x}+\sqrt{p-x }= x$.
2010 Contests, 1
Find all functions $ f : R \to R$ that satisfies $$xf(y) - yf(x)= f\left(\frac{y}{x}\right)$$ for all $x, y \in R$.
2010 China Team Selection Test, 3
Let $A$ be a finite set, and $A_1,A_2,\cdots, A_n$ are subsets of $A$ with the following conditions:
(1) $|A_1|=|A_2|=\cdots=|A_n|=k$, and $k>\frac{|A|}{2}$;
(2) for any $a,b\in A$, there exist $A_r,A_s,A_t\,(1\leq r<s<t\leq n)$ such that
$a,b\in A_r\cap A_s\cap A_t$;
(3) for any integer $i,j\, (1\leq i<j\leq n)$, $|A_i\cap A_j|\leq 3$.
Find all possible value(s) of $n$ when $k$ attains maximum among all possible systems $(A_1,A_2,\cdots, A_n,A)$.
2009 Middle European Mathematical Olympiad, 4
Determine all integers $ k\ge 2$ such that for all pairs $ (m$, $ n)$ of different positive integers not greater than $ k$, the number $ n^{n\minus{}1}\minus{}m^{m\minus{}1}$ is not divisible by $ k$.
2020 MIG, 18
When $171$ is written as the sum of $19$ consecutive integers, the median of those numbers is $M$. When $171$ is written as the sum of $18$ consecutive integers, the median of those numbers is $N$. Find $|M - N|$.
$\textbf{(A) }{-}1\qquad\textbf{(B) }{-}0.5\qquad\textbf{(C) }0\qquad\textbf{(D) }0.5\qquad\textbf{(E) }1$
2005 China Team Selection Test, 1
Prove that for any $n$ ($n \geq 2$) pairwise distinct fractions in the interval $(0,1)$, the sum of their denominators is no less than $\frac{1}{3} n^{\frac{3}{2}}$.
2015 Romania Team Selection Test, 2
Let $ABC$ be a triangle, and let $r$ denote its inradius. Let $R_A$ denote the radius of the circle internally tangent at $A$ to the circle $ABC$ and tangent to the line $BC$; the radii $R_B$ and $R_C$ are defined similarly. Show that $\frac{1}{R_A} + \frac{1}{R_B} + \frac{1}{R_C}\leq\frac{2}{r}$.
2015 Postal Coaching, Problem 2
Find all pairs of cubic equations $x^3+ax^2+bx+c=0$ and $x^3+bx^2+ax+c=0$ where $a, b$ are positive integers, $c\neq 0$ is an integer, such that each equation has three integer roots and exactly one of these three roots is common to both the equations.
2001 Estonia National Olympiad, 1
Solve the system of equations $$\begin{cases} \sin x = y \\ \sin y = x \end{cases}$$
1956 Moscow Mathematical Olympiad, 344
* Let $A, B, C$ be three nodes of a graph paper. Prove that if $\vartriangle ABC$ is an acute one, then there is at least one more node either inside $\vartriangle ABC$ or on one of its sides.
2017 Federal Competition For Advanced Students, 3
Anna and Berta play a game in which they take turns in removing marbles from a table. Anna takes the first turn. At the beginning of a turn there are n ≥ 1 marbles on the table, then the player whose turn is removes k marbles, where k ≥ 1 either is an even number with $k \le \frac{n}{2}$ or an odd number with $ \frac{n}{2}\le k \le n$. A player wins the game if she removes the last marble from the table.
Determine the smallest number $N\ge100000$ which Berta has wining strategy.
[i]proposed by Gerhard Woeginger[/i]
2023 Kyiv City MO Round 1, Problem 3
Let $I$ be the incenter of triangle $ABC$ with $AB < AC$. Point $X$ is chosen on the external bisector of $\angle ABC$ such that $IC = IX$. Let the tangent to the circumscribed circle of $\triangle BXC$ at point $X$ intersect the line $AB$ at point $Y$. Prove that $AC = AY$.
[i]Proposed by Oleksiy Masalitin[/i]
2019 USA EGMO Team Selection Test, 4
For every pair $(m, n)$ of positive integers, a positive real number $a_{m, n}$ is given. Assume that
\[a_{m+1, n+1} = \frac{a_{m, n+1} a_{m+1, n} + 1}{a_{m, n}}\]
for all positive integers $m$ and $n$. Suppose further that $a_{m, n}$ is an integer whenever $\min(m, n) \le 2$. Prove that $a_{m, n}$ is an integer for all positive integers $m$ and $n$.
1999 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 1
Compute:
$ \frac{777^2 \minus{} 66^2}{777\plus{}66}$
2010 Brazil Team Selection Test, 3
Let $a$, $b$, $c$ be positive real numbers such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = a+b+c$. Prove that:
\[\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}.\]
[i]Proposed by Juhan Aru, Estonia[/i]
2002 Junior Balkan MO, 2
Two circles with centers $O_{1}$ and $O_{2}$ meet at two points $A$ and $B$ such that the centers of the circles are on opposite sides of the line $AB$. The lines $BO_{1}$ and $BO_{2}$ meet their respective circles again at $B_{1}$ and $B_{2}$. Let $M$ be the midpoint of $B_{1}B_{2}$. Let $M_{1}$, $M_{2}$ be points on the circles of centers $O_{1}$ and $O_{2}$ respectively, such that $\angle AO_{1}M_{1}= \angle AO_{2}M_{2}$, and $B_{1}$ lies on the minor arc $AM_{1}$ while $B$ lies on the minor arc $AM_{2}$. Show that $\angle MM_{1}B = \angle MM_{2}B$.
[i]Ciprus[/i]
1997 USAMO, 4
To [i]clip[/i] a convex $n$-gon means to choose a pair of consecutive sides $AB, BC$ and to replace them by the three segments $AM, MN$, and $NC$, where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC$. In other words, one cuts off the triangle $MBN$ to obtain a convex $(n+1)$-gon. A regular hexagon ${\cal P}_6$ of area 1 is clipped to obtain a heptagon ${\cal P}_7$. Then ${\cal P}_7$ is clipped (in one of the seven possible ways) to obtain an octagon ${\cal P}_8$, and so on. Prove that no matter how the clippings are done, the area of ${\cal P}_n$ is greater than $\frac 13$, for all $n \geq 6$.
2012 Today's Calculation Of Integral, 788
For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions:
(1) Find $f'(x)$.
(2) Sketch the graph of $y=f(x)$.
(3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.
1998 National Olympiad First Round, 18
Let $ p_{1} <p_{2} <\ldots <p_{24}$ be the prime numbers on the interval $ \left[3,100\right]$. Find the smallest value of $ a\ge 0$ such that $ \sum _{i\equal{}1}^{24}p_{i}^{99!} \equiv a\, \, \left(mod\, 100\right)$.
$\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 48 \qquad\textbf{(D)}\ 50 \qquad\textbf{(E)}\ 99$
2009 Estonia Team Selection Test, 4
Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$
2002 APMO, 3
Let $ABC$ be an equilateral triangle. Let $P$ be a point on the side $AC$ and $Q$ be a point on the side $AB$ so that both triangles $ABP$ and $ACQ$ are acute. Let $R$ be the orthocentre of triangle $ABP$ and $S$ be the orthocentre of triangle $ACQ$. Let $T$ be the point common to the segments $BP$ and $CQ$. Find all possible values of $\angle CBP$ and $\angle BCQ$ such that the triangle $TRS$ is equilateral.