Found problems: 85335
2010 Indonesia TST, 4
Let $n$ be a positive integer with $n = p^{2010}q^{2010}$ for two odd primes $p$ and $q$. Show that there exist exactly $\sqrt[2010]{n}$ positive integers $x \le n$ such that $p^{2010}|x^p - 1$ and $q^{2010}|x^q - 1$.
II Soros Olympiad 1995 - 96 (Russia), 11.5
The space is filled in the usual way with unit cubes. (Each cube is adjacent to $6$ others that have a common face with it.) On three edges of one of the cubes emerging from one vertex, points are marked at a distance of $1/19$, $1/9$ and $1/7$ from it, respectively. A plane is drawn through these points. Let's consider the many different polygons formed when this plane intersects with the cubes filling the space. How many different (unequal) polygons are there in this set?
2006 Switzerland Team Selection Test, 3
Let $n$ be natural number. Each of the numbers $\in\{1,2,\ldots ,n\}$ is coloured in black or white. When we choose a number, we flip it's colour and the colour of all the numbers which have at least one common divider with the chosen number. At the beginning all the numbers were coloured white. For which $n$ are all the numbers black after a finite number of changes?
Denmark (Mohr) - geometry, 1997.2
Two squares, both with side length $1$, are arranged so that one has one vertex in the center of the other. Determine the area of the gray area.
[img]https://1.bp.blogspot.com/-xt3pe0rp1SI/XzcGLgEw1EI/AAAAAAAAMYM/vFKxvvVuLvAJ5FO_yX315X3Fg_iFaK2fACLcBGAsYHQ/s0/1997%2BMohr%2Bp2.png[/img]
2023 Junior Balkan Team Selection Tests - Romania, P4
Let $M \geq 1$ be a real number. Determine all natural numbers $n$ for which there exist distinct natural numbers $a$, $b$, $c > M$, such that
$n = (a,b) \cdot (b,c) + (b,c) \cdot (c,a) + (c,a) \cdot (a,b)$
(where $(x,y)$ denotes the greatest common divisor of natural numbers $x$ and $y$).
2000 Kazakhstan National Olympiad, 1
Two guys are playing the game "Sea Battle-2000". On the board $ 1 \times 200 $, they take turns placing the letter "$ S $" or "$ O $" on the empty squares of the board. The winner is the one who gets the word "$ SOS $" first. Prove that the second player wins when played correctly.
2009 Thailand Mathematical Olympiad, 8
Let $a, b, c$ be side lengths of a triangle, and define $s =\frac{a+b+c}{2}$. Prove that
$$\frac{2a(2a-s)}{b + c}+\frac{2b(2b - s)}{c + a}+\frac{2c(2c - s)}{a + b}\ge s.$$
MathLinks Contest 7th, 3.1
Let $ p$ be a prime and let $ d \in \left\{0,\ 1,\ \ldots,\ p\right\}$. Prove that
\[ \sum_{k \equal{} 0}^{p \minus{} 1}{\binom{2k}{k \plus{} d}}\equiv r \pmod{p},
\]where $ r \equiv p\minus{}d \pmod 3$, $ r\in\{\minus{}1,0,1\}$.
2022 USAMO, 5
A function $f: \mathbb{R}\to \mathbb{R}$ is [i]essentially increasing[/i] if $f(s)\leq f(t)$ holds whenever $s\leq t$ are real numbers such that $f(s)\neq 0$ and $f(t)\neq 0$.
Find the smallest integer $k$ such that for any 2022 real numbers $x_1,x_2,\ldots , x_{2022},$ there exist $k$ essentially increasing functions $f_1,\ldots, f_k$ such that \[f_1(n) + f_2(n) + \cdots + f_k(n) = x_n\qquad \text{for every } n= 1,2,\ldots 2022.\]
2011 Vietnam National Olympiad, 3
Let $AB$ be a diameter of a circle $(O)$ and let $P$ be any point on the tangent drawn at $B$ to $(O).$ Define $AP\cap (O)=C\neq A,$ and let $D$ be the point diametrically opposite to $C.$ If $DP$ meets $(O)$ second time in $E,$ then,
[b](i)[/b] Prove that $AE, BC, PO$ concur at $M.$
[b](ii)[/b] If $R$ is the radius of $(O),$ find $P$ such that the area of $\triangle AMB$ is maximum, and calculate the area in terms of $R.$
1991 Poland - Second Round, 5
$ P_1, P_2, \ldots, P_n $ are different two-element subsets of $ \{1,2,\ldots,n\} $. The sets $ P_i $, $ P_j $ for $ i\neq j $ have a common element if and only if the set $ \{i,j\} $ is one of the sets $ P_1, P_2, \ldots, P_n $. Prove that each of the numbers $ 1,2,\ldots,n $ is a common element of exactly two sets from $ P_1, P_2, \ldots, P_n $.
1977 Spain Mathematical Olympiad, 1
Given the determinant of order $n$
$$\begin{vmatrix}
8 & 3 & 3 & \dots & 3 \\
3 & 8 & 3 & \dots & 3 \\
3 & 3 & 8 & \dots & 3 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
3 & 3 & 3 & \dots & 8
\end{vmatrix}$$
Calculate its value and determine for which values of $n$ this value is a multiple of $10$.
2020-21 KVS IOQM India, 23
Find the largest positive integer $N$ such that the number of integers In the set ${1,2,3,...,N}$ which are divisible by $3$ is equal to the number of integers which are divisible by $5$ or $7$ (or both),
2025 Kyiv City MO Round 1, Problem 4
Distinct real numbers \( a, b, c \) satisfy the following condition:
\[
\frac{a - b}{a^3b^3} + \frac{b - c}{b^3c^3} + \frac{c - a}{c^3a^3} = 0.
\]
What are the possible values of the expression
\[
\frac{a^4 + b^4 + c^4}{a^2b^2 + b^2c^2 + c^2a^2}?
\]
[i]Proposed by Vadym Solomka[/i]
2016 CMIMC, 1
For how many distinct ordered triples $(a,b,c)$ of boolean variables does the expression $a \lor (b \land c)$ evaluate to true?
1968 IMO Shortlist, 2
Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.
2021 ELMO Problems, 1
In $\triangle ABC$, points $P$ and $Q$ lie on sides $AB$ and $AC$, respectively, such that the circumcircle of $\triangle APQ$ is tangent to $BC$ at $D$. Let $E$ lie on side $BC$ such that $BD = EC$. Line $DP$ intersects the circumcircle of $\triangle CDQ$ again at $X$, and line $DQ$ intersects the circumcircle of $\triangle BDP$ again at $Y$. Prove that $D$, $E$, $X$, and $Y$ are concyclic.
2009 Federal Competition For Advanced Students, P2, 1
If $x,y,K,m \in N$, let us define:
$a_m= \underset{k \, twos}{2^{2^{,,,{^{2}}}}}$, $A_{km} (x)= \underset{k \, twos}{ 2^{2^{,,,^{x^{a_m}}}}}$, $B_k(y)= \underset{m \, fours}{4^{4^{4^{,,,^{4^y}}}}}$,
Determine all pairs $(x,y)$ of non-negative integers, dependent on $k>0$, such that $A_{km} (x)=B_k(y)$
1984 All Soviet Union Mathematical Olympiad, 391
The white fields of $3x3$ chess-board are filled with either $+1$ or $-1$. For every field, let us calculate the product of neighbouring numbers. Then let us change all the numbers by the respective products. Prove that we shall obtain only $+1$'s, having repeated this operation finite number of times.
2011 Junior Balkan Team Selection Tests - Romania, 4
Let $m$ be a positive integer. Determine the smallest positive integer $n$ for which there exist real numbers $x_1, x_2,...,x_n \in (-1, 1)$ such that $|x_1| + |x_2| +...+ |x_n| = m + |x_1 + x_2 + ... + x_n|$.
1992 Brazil National Olympiad, 2
Show that there is a positive integer n such that the first 1992 digits of $n^{1992}$ are 1.
2022 Israel TST, 2
Define a [b]ring[/b] in the plane to be the set of points at a distance of at least $r$ and at most $R$ from a specific point $O$, where $r<R$ are positive real numbers. Rings are determined by the three parameters $(O, R, r)$. The area of a ring is labeled $S$. A point in the plane for which both its coordinates are integers is called an integer point.
[b]a)[/b] For each positive integer $n$, show that there exists a ring not containing any integer point, for which $S>3n$ and $R<2^{2^n}$.
[b]b)[/b] Show that each ring satisfying $100\cdot R<S^2$ contains an integer point.
2011 Math Prize For Girls Problems, 16
Let $N$ be the number of ordered pairs of integers $(x, y)$ such that
\[
4x^2 + 9y^2 \le 1000000000.
\]
Let $a$ be the first digit of $N$ (from the left) and let $b$ be the second digit of $N$. What is the value of $10a + b$ ?
1973 Czech and Slovak Olympiad III A, 1
Consider a triangle such that \[\sin^2\alpha+\sin^2\beta+\sin^2\gamma=2.\] Show that the triangle is right.
2000 India Regional Mathematical Olympiad, 2
Solve the equation $y^3 = x^3 + 8x^2 - 6x +8$, for positive integers $x$ and $y$.