Found problems: 85335
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 $y$.
2012 Online Math Open Problems, 49
Find the magnitude of the product of all complex numbers $c$ such that the recurrence defined by $x_1 = 1$, $x_2 = c^2 - 4c + 7$, and $x_{n+1} = (c^2 - 2c)^2 x_n x_{n-1} + 2x_n - x_{n-1}$ also satisfies $x_{1006} = 2011$.
[i]Author: Alex Zhu[/i]
Kvant 2024, M2805
Find the largest positive integer $n$, such that there exists a finite set $A$ of $n$ reals, such that for any two distinct elements of $A$, there exists another element from $A$, so that the arithmetic mean of two of these three elements equals the third one.
1996 IMC, 3
The linear operator $A$ on a finite-dimensional vector space $V$ is called an involution if
$A^{2}=I$, where $I$ is the identity operator. Let $\dim V=n$.
i) Prove that for every involution $A$ on $V$, there exists a basis of $V$ consisting of eigenvectors
of $A$.
ii) Find the maximal number of distinct pairwise commuting involutions on $V$.