Found problems: 85335
2021 Polish Junior MO Second Round, 2
Given is the square $ABCD$. Point $E$ lies on the diagonal $AC$, where $AE> EC$. On the side $AB$, a different point from $B$ has been selected for which $EF = DE$. Prove that $\angle DEF = 90^o$.
1998 AMC 12/AHSME, 15
A regular hexagon and an equilateral triangle have equal areas. What is the ratio of the length of a side of the triangle to the length of a side of the hexagon?
$ \textbf{(A)}\ \sqrt 3\qquad
\textbf{(B)}\ 2\qquad
\textbf{(C)}\ \sqrt 6\qquad
\textbf{(D)}\ 3\qquad
\textbf{(E)}\ 6$
2005 AMC 10, 19
On a certain math exam, $ 10 \%$ of the students got 70 points, $ 25 \%$ got 80 points, $ 20 \%$ got 85 points, $ 15 \%$ got 90 points, and the rest got 95 points. What is the difference between the mean and the median score on this exam?
$ \textbf{(A)}\ 0\qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ 2\qquad
\textbf{(D)}\ 4\qquad
\textbf{(E)}\ 5$
1999 Brazil Team Selection Test, Problem 5
(a) If $m, n$ are positive integers such that $2^n-1$ divides $m^2 + 9$, prove
that $n$ is a power of $2$;
(b) If $n$ is a power of $2$, prove that there exists a positive integer $m$ such
that $2^n-1$ divides $m^2 + 9$.
2010 Singapore MO Open, 5
A prime number $p$ and integers $x, y, z$ with $0 < x < y < z < p$ are given. Show that if the numbers $x^3, y^3, z^3$ give the same remainder when divided by $p$, then $x^2 + y^2 + z^2$ is divisible by $x + y + z.$
2025 China Team Selection Test, 19
Let $\left \{ x_n \right \} _{n\ge 1}$ and $\left \{ y_n \right \} _{n\ge 1}$ be two infinite sequences of integers. Prove that there exists an infinite sequence of integers $\left \{ z_n \right \} _{n\ge 1}$ such that for any positive integer \( n \), the following holds:
\[
\sum_{k|n} k \cdot z_k^{\frac{n}{k}} = \left( \sum_{k|n} k \cdot x_k^{\frac{n}{k}} \right) \cdot \left( \sum_{k|n} k \cdot y_k^{\frac{n}{k}} \right).
\]
2003 Irish Math Olympiad, 4
Eight players, Ann, Bob, Con, Dot, Eve, Fay, Guy and Hal compete in a chess tournament. No pair plays together more than once and there is no group of five people in which each one plays against all of the other four.
(a) Write down an arrangement for a tournament of $24$ games satisfying these conditions.
(b) Show that it is impossible to have a tournament of more than $24$ games satisfying these conditions.
2025 Canada Junior National Olympiad, 4
Determine all positive integers $a$, $b$, $c$, $p$, where $p$ and $p+2$ are odd primes and
\[2^ap^b=(p+2)^c-1.\]
2008 Sharygin Geometry Olympiad, 3
(D.Shnol) Two opposite angles of a convex quadrilateral with perpendicular diagonals are equal. Prove that a circle can be inscribed in this quadrilateral.
1996 AMC 8, 6
What is the smallest result that can be obtained from the following process?
*Choose three different numbers from the set $\{3,5,7,11,13,17\}$.
*Add two of these numbers.
*Multiply their sum by the third number.
$\text{(A)}\ 15 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 50 \qquad \text{(E)}\ 56$
II Soros Olympiad 1995 - 96 (Russia), 10.8
Is it possible to fill an $n \times n$ table with the numbers $-1$, $0$ and $1$ so that all $2n$ sums in each column and each row are different?
Solve the problem with
a) $n = 5$;
b) $n = 10$.
2004 Iran MO (3rd Round), 8
$\mathbb{P}$ is a n-gon with sides $l_1 ,...,l_n$ and vertices on a circle. Prove that no n-gon with this sides has area more than $\mathbb{P}$
2024-25 IOQM India, 25
A finite set $M$ of positive integers consists of distinct perfect squares and the number $92$. The average of the numbers in $M$ is $85$. If we remove $92$ from $M$, the average drops to $84$. If $N^2$ is the largest possible square in $M$, what is the value of $N$?
1977 Miklós Schweitzer, 3
Prove that if $ a,x,y$ are $ p$-adic integers different from $ 0$ and $ p | x, pa | xy$, then \[ \frac 1y \frac{(1\plus{}x)^y\minus{}1}{x} \equiv \frac{\log (1\plus{}x)}{x} \;\;\;\; ( \textrm{mod} \; a\ ) \\\\ .\]
[i]L. Redei[/i]
2012 Pre-Preparation Course Examination, 5
Suppose that for the linear transformation $T:V \longrightarrow V$ where $V$ is a vector space, there is no trivial subspace $W\subset V$ such that $T(W)\subseteq W$. Prove that for every polynomial $p(x)$, the transformation $p(T)$ is invertible or zero.
2004 All-Russian Olympiad Regional Round, 8.4
The cells of the $11 \times 111 \times11$ cube contain the numbers $ 1, 2, , . .. . . 1331$, once each number. Two worms are sent from one corner cube to the opposite corner. Each of them can crawl into a cube adjacent to the edge, while the first can crawl if the number in the adjacent cube differs by $8$, the second - if they differ by $ 9$. Is there such an arrangement of numbers that both worms can get to the opposite corner cube?
MathLinks Contest 7th, 5.3
If $ a\geq b\geq c\geq d > 0$ such that $ abcd\equal{}1$, then prove that \[ \frac 1{1\plus{}a} \plus{} \frac 1{1\plus{}b} \plus{} \frac 1{1\plus{}c} \geq \frac {3}{1\plus{}\sqrt[3]{abc}}.\]
2025 Caucasus Mathematical Olympiad, 6
A point $P$ is chosen inside a convex quadrilateral $ABCD$. Could it happen that$$PA = AB, \quad PB = BC, \quad PC = CD \quad \text{and} \quad PD = DA?$$
1993 ITAMO, 5
Prove the following inequality for any positive real numbers a,b,c not exceeding 1
$a^2b+b^2c+c^2a+1\ge a^2+b^2+c^2$
2018 Putnam, A5
Let $f: \mathbb{R} \to \mathbb{R}$ be an infinitely differentiable function satisfying $f(0) = 0$, $f(1) = 1$, and $f(x) \ge 0$ for all $x \in \mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$ such that $f^{(n)}(x) < 0$.
1994 Miklós Schweitzer, 8
Prove that a Hausdorff space X is countably compact iff for every open cover $\cal {U}$ there is a finite set $A \subset X$ such that $ \bigcup \{U \in {\cal U} : U \cap A \neq \emptyset \} = X$.
2018 Kyiv Mathematical Festival, 3
A circle is divided by $2018$ points into equal parts. Two players delete these points in turns. A player loses, if after his turn it is possible to draw a diameter of the circle such that there are no undeleted points on one side of it. Which player has a winning strategy?
2009 Princeton University Math Competition, 1
Find 100 times the area of a regular dodecagon inscribed in a unit circle. Round your answer to the nearest integer if necessary.
[asy]
defaultpen(linewidth(0.7)); real theta = 17; pen dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1);
draw(unitcircle,dg);
for(int i = 0; i < 12; ++i) {
draw(dir(30*i+theta)--dir(30*(i+1)+theta), db);
dot(dir(30*i+theta),Fill(rgb(0.8,0,0)));
} dot(dir(theta),Fill(dr)); dot((0,0),Fill(dr));
[/asy]
2011 QEDMO 9th, 6
Show that there are infinitely many pairs $(m, n)$ of natural numbers $m, n \ge 2$, for $m^m- 1$ is divisible by $n$ and $n^n- 1$ is divisible by $m$.
2000 Spain Mathematical Olympiad, 3
Two circles $C_1$ and $C_2$ with the respective radii $r_1$ and $r_2$ intersect in $A$ and $B.$ A variable line $r$ through $B$ meets $C_1$ and $C_2$ again at $P_r$ and $Q_r$ respectively. Prove that there exists a point $M,$ depending only on $C_1$ and $C_2,$ such that the perpendicular bisector of each segment $P_rQ_r$ passes through $M.$