Found problems: 85335
1960 IMO Shortlist, 7
An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given.
a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P$;
b) Calculate the distance of $p$ from either base;
c) Determine under what conditions such points $P$ actually exist. Discuss various cases that might arise.
2000 Moldova National Olympiad, Problem 5
Several crocodiles, dragons and snakes were left on an island. Animals were eating each other according to the following rules. Every day at the breakfast, each snake ate one dragon; at the lunch, each dragon ate one crocodile; and at the dinner, each crocodile ate one snake. On the Saturday after the dinner, only one crocodile and no snakes and dragons remained on the island. How many crocodiles, dragons and snakes were there on the Monday in the same week before the breakfast?
2024 Korea - Final Round, P1
Let $a, b, c, d$ be odd positive integers and pairwise coprime. For a positive integer $n$, let $$f(n) = \left[\frac{n}{a}
\right]+\left[\frac{n}{b}\right]+\left[\frac{n}{c}\right]+\left[\frac{n}{d}\right]$$ Prove that $$\sum_{n=1}^{abcd}(-1)^{f(n)}=1$$
2016 District Olympiad, 3
Let $ \alpha ,\beta $ be real numbers. Find the greatest value of the expression
$$ |\alpha x +\beta y| +|\alpha x-\beta y| $$
in each of the following cases:
[b]a)[/b] $ x,y\in \mathbb{R} $ and $ |x|,|y|\le 1 $
[b]b)[/b] $ x,y\in \mathbb{C} $ and $ |x|,|y|\le 1 $
2010 Miklós Schweitzer, 5
Given the vectors $ v_ {1}, \dots, v_ {n} $ and $ w_ {1}, \dots, w_ {n} $ in the plane with the following properties:
for every $ 1 \leq i \leq n $ ,$ \left | v_{i} -w_{i} \right | \leq 1, $ and for every $ 1 \leq i <j \leq n $ ,$ \left | v_{i} -v_{j} \right | \ge 3 $ and $ v_{i} -w_ {i} \ne v_ {j} -w_ {j} $. Prove that for sets $ V = \left \{v_ {1}, \dots, v_{n } \right \} $ and $ W = \left \{w_ {1}, \dots, w_ {n} \right \}$, the set of $ V + (V \cup W) $ must have at least $ cn^{3/2} $ elements ,for some universal constant $ c>0 $ .
2024 PErA, P5
Find all functions $f\colon \mathbb{R}^+ \to \mathbb{R}^+$ such that
\[
f(xf(x)+y^2) = x^2+yf(y)
\]
for any positive reals $x,y$.
2018 Indonesia MO, 7
Suppose there are three empty buckets and $n \ge 3$ marbles. Ani and Budi play a game. For the first turn, Ani distributes all the marbles into the buckets so that each bucket has at least one marble. Then Budi and Ani alternate turns, starting with Budi. On a turn, the current player may choose a bucket and take 1, 2, or 3 marbles from it. The player that takes the last marble wins. Find all $n$ such that Ani has a winning strategy, including what Ani's first move (distributing the marbles) should be for these $n$.
Russian TST 2016, P3
A simple graph has $N{}$ vertices and less than $3(N-1)/2$ edges. Prove that its vertices can be divided into two non-empty groups so that each vertex has at most one neighbor in the group it doesn't belong to.
1992 Mexico National Olympiad, 5
$x, y, z$ are positive reals with sum $3$. Show that $$6 < \sqrt{2x+3} + \sqrt{2y+3} + \sqrt{2z+3}\le 3\sqrt5$$
2009 Today's Calculation Of Integral, 439
Find the volume of the solid defined by the inequality $ x^2 \plus{} y^2 \plus{} \ln (1 \plus{} z^2)\leq \ln 2$.
Note that you may not directively use double integral here for Japanese high school students who don't study it.
1935 Moscow Mathematical Olympiad, 002
Given the lengths of two sides of a triangle and that of the bisector of the angle between these sides, construct the triangle.
2013 CHMMC (Fall), 3
Bill plays a game in which he rolls two fair standard six-sided dice with sides labeled one through six. He wins if the number on one of the dice is three times the number on the other die. If Bill plays this game three times, compute the probability that he wins at least once.
2009 Croatia Team Selection Test, 2
Every natural number is coloured in one of the $ k$ colors. Prove that there exist four distinct natural numbers $ a, b, c, d$, all coloured in the same colour, such that $ ad \equal{} bc$, $ \displaystyle \frac b a$ is power of 2 and $ \displaystyle \frac c a$ is power of 3.
2014 Iran Team Selection Test, 2
Point $D$ is an arbitary point on side $BC$ of triangle $ABC$. $I$,$I_1$ and$I_2$ are the incenters of triangles $ABC$,$ABD$ and $ACD$ respectively. $M\not=A$ and $N\not=A$ are the intersections of circumcircle of triangle $ABC$ and circumcircles of triangles $IAI_1$ and $IAI_2$ respectively. Prove that regardless of point $D$, line $MN$ goes through a fixed point.
1958 February Putnam, A1
If $a_0 , a_1 ,\ldots, a_n$ are real number satisfying
$$ \frac{a_0 }{1} + \frac{a_1 }{2} + \ldots + \frac{a_n }{n+1}=0,$$
show that the equation $a_n x^n + \ldots +a_1 x+a_0 =0$ has at least one real root.
2004 Postal Coaching, 3
Let $a,b,c,d,$ be real and $ad-bc = 1$. Show that $Q = a^2 + b^2 + c^2 + d^2 + ac +bd$ $\not= 0, 1, -1$
1998 Finnish National High School Mathematics Competition, 3
Consider the geometric sequence $1/2, \ 1 / 4, \ 1 / 8,...$
Can one choose a subsequence, finite or infinite, for which the ratio of consecutive terms is not $1$ and whose sum is $1/5?$
1968 Polish MO Finals, 6
Consider a set of $n > 3$ points in the plane, no three of which are collinear, and a natural number $k < n$. Prove the following statements:
(a) If $k \le \frac{n}{2}$, then each point can be connected with at least k other points by segments so that no three segments form a triangle.
(b) If $k \ge \frac{n}{2}$, and each point is connected with at least k other points by segments, then some three segments form a triangle.
2019 PUMaC Team Round, 9
Find the integer $\sqrt[5]{55^5 + 3183^5 + 28969^5 + 85282^5}$.
II Soros Olympiad 1995 - 96 (Russia), 11.7
Three edges of a parallelepiped lie on three intersecting diagonals of the lateral faces of a triangular prism. Find the ratio of the volumes of the parallelepiped and the prism.
2001 Junior Balkan MO, 3
Let $ABC$ be an equilateral triangle and $D$, $E$ points on the sides $[AB]$ and $[AC]$ respectively. If $DF$, $EF$ (with $F\in AE$, $G\in AD$) are the interior angle bisectors of the angles of the triangle $ADE$, prove that the sum of the areas of the triangles $DEF$ and $DEG$ is at most equal with the area of the triangle $ABC$. When does the equality hold?
[i]Greece[/i]
2024 Dutch IMO TST, 1
Let $ABC$ be a triangle with orthocenter $H$ and circumcircle $\Gamma$. Let $D$ be the reflection of $A$ in $B$ and let $E$ the reflection of $A$ in $C$. Let $M$ be the midpoint of segment $DE$. Show that the tangent to $\Gamma$ in $A$ is perpendicular to $HM$.
2021 USMCA, 11
Let $f_1 (x) = x^2 - 3$ and $f_n (x) = f_1(f_{n-1} (x))$ for $n \ge 2$. Let $m_n$ be the smallest positive root of $f_n$, and $M_n$ be the largest positive root of $f_n$. If $x$ is the least number such that $M_n \le m_n \cdot x$ for all $n \ge 1$, compute $x^2$.
2013 India IMO Training Camp, 1
For a prime $p$, a natural number $n$ and an integer $a$, we let $S_n(a,p)$ denote the exponent of $p$ in the prime factorisation of $a^{p^n} - 1$. For example, $S_1(4,3) = 2$ and $S_2(6,2) = 0$. Find all pairs $(n,p)$ such that $S_n(2013,p) = 100$.
1962 AMC 12/AHSME, 6
A square and an equilateral triangle have equal perimeters. The area of the triangle is $ 9 \sqrt{3}$ square inches. Expressed in inches the diagonal of the square is:
$ \textbf{(A)}\ \frac{9}{2} \qquad
\textbf{(B)}\ 2 \sqrt{5} \qquad
\textbf{(C)}\ 4 \sqrt{2} \qquad
\textbf{(D)}\ \frac{9 \sqrt{2}}{2} \qquad
\textbf{(E)}\ \text{none of these}$