Found problems: 85335
Kvant 2024, M2824
There are $15$ boys and $15$ girls in the class. The first girl is friends with $4$ boys, the second with $5$, the third with $6$, . . . , the $11$th with $14$, and each of the other four girls is friends with all the boys. It turned out that there are exactly $3 \cdot 2^{25}$ ways to split the entire class into pairs, so that each pair has a boy and a girl who are friends. Prove that any of the friends of the first girl are friends with all the other girls too.
[i]G.M.Sharafetdinova[/i]
2004 AMC 12/AHSME, 17
Let $ f$ be a function with the following properties:
(i) $f(1) \equal{} 1$, and
(ii) $ f(2n) \equal{} n\times f(n)$, for any positive integer $ n$.
What is the value of $ f(2^{100})$?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2^{99} \qquad \textbf{(C)}\ 2^{100} \qquad \textbf{(D)}\ 2^{4950} \qquad \textbf{(E)}\ 2^{9999}$
2007 ITest, 27
The face diagonal of a cube has length $4$. Find the value of $n$ given that $n\sqrt2$ is the $\textit{volume}$ of the cube.
2003 Abels Math Contest (Norwegian MO), 2b
Let $a_1,a_2,...,a_n$ be $n$ different positive integers where $n\ge 1$. Show that $$\sum_{i=1}^n a_i^3 \ge \left(\sum_{i=1}^n a_i\right)^2$$
2009 Tournament Of Towns, 1
A rectangle is dissected into several smaller rectangles. Is it possible that for each pair of these rectangles, the line segment connecting their centers intersects some third rectangle?
2023 IFYM, Sozopol, 2
Given a triangle $ABC$, a line in its plane is called a [i]cool[/i] if it divides the triangle into two parts with equal areas and perimeters.
a) Does there exist a triangle $ABC$ with at least seven [i]cool[/i] lines?
b) Prove that all [i]cool[/i] lines intersect at a point $X$. If $\angle AXB = 126^\circ$, prove that $(8\sin^2 \angle ACB - 5)^2$ is an integer.
2010 German National Olympiad, 6
Let $A,B,C,D,E,F,G$ and $H$ be eight pairwise distinct points on the surface of a sphere. The quadruples $(A,B,C,D), (A,B,F,E),(B,C,G,F),(C,D,H,G)$ and $(D,A,E,H)$ of points are coplanar.
Prove that the quadruple $(E,F,G,H)$ is coplanar aswell.
2018 India IMO Training Camp, 1
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.
[i]Proposed by Jeck Lim, Singapore[/i]
2004 All-Russian Olympiad Regional Round, 8.7
A set of five-digit numbers $\{N_1,... ,N_k\}$ is such that any five-digit a number whose digits are all in ascending order is the same in at least one digit with at least one of the numbers $N_1$,$...$ ,$N_k$. Find the smallest possible value of $k$.
1969 Miklós Schweitzer, 12
Let $ A$ and $ B$ be nonsingular matrices of order $ p$, and let $ \xi$ and $ \eta$ be independent random vectors of dimension $ p$. Show that if $ \xi,\eta$ and $ \xi A\plus{} \eta B$ have the same distribution, if their first and second moments exist, and if their covariance matrix is the identity matrix, then these random vectors are normally distributed.
[i]B. Gyires[/i]
2022 LMT Fall, 1
Let $x$ be the positive integer satisfying $5^2 +28^2 +39^2 = 24^2 +35^2 + x^2$. Find $x$.
2021 Malaysia IMONST 1, 17
Determine the sum of all positive integers $n$ that satisfy the following condition:
when $6n + 1$ is written in base $10$, all its digits are equal.
2016 India IMO Training Camp, 1
Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $A_1,B_1$ and $C_1$ be respectively the midpoints of the arcs $BAC,CBA$ and $ACB$ of $\Gamma$. Show that the inradius of triangle $A_1B_1C_1$ is not less than the inradius of triangle $ABC$.
2021 HMNT, 2
Joey wrote a system of equations on a blackboard, where each of the equations was of the form $a + b = c$ or $a \cdot b = c$ for some variables or integers $a, b, c$. Then Sean came to the board and erased all of the plus signs and multiplication signs, so that the board reads:
$x\,\,\,\, z = 15$
$x\,\,\,\, y = 12$
$x\,\,\,\, x = 36$
If $x, y, z$ are integer solutions to the original system, find the sum of all possible values of $100x+10y+z$.
2008 Germany Team Selection Test, 1
Let $ a_1, a_2, \ldots, a_{100}$ be nonnegative real numbers such that $ a^2_1 \plus{} a^2_2 \plus{} \ldots \plus{} a^2_{100} \equal{} 1.$ Prove that
\[ a^2_1 \cdot a_2 \plus{} a^2_2 \cdot a_3 \plus{} \ldots \plus{} a^2_{100} \cdot a_1 < \frac {12}{25}.
\]
[i]Author: Marcin Kuzma, Poland[/i]
2007 Tuymaada Olympiad, 3
Several knights are arranged on an infinite chessboard. No square is attacked by more than one knight (in particular, a square occupied by a knight can be attacked by one knight but not by two). Sasha outlined a $ 14\times 16$ rectangle. What maximum number of knights can this rectangle contain?
2017 Rioplatense Mathematical Olympiad, Level 3, 3
Show that there are infinitely many pairs of positive integers $(m,n)$, with $m<n$, such that
$m$ divides $n^{2016}+n^{2015}+\dots+n^2+n+1$ and $n$ divides $m^{2016}+m^{2015} +\dots+m^2+m+1$.
2017 VTRMC, 2
Evaluate $ \int _ { 0 } ^ { a } d x / ( 1 + \cos x + \sin x ) $ for $ - \pi / 2 < a < \pi $. Use your answer to show that $ \int _ { 0 } ^ { \pi / 2 } d x / ( 1 + \cos x + \sin x ) = \ln 2 $.
1966 Swedish Mathematical Competition, 5
Let $f(r)$ be the number of lattice points inside the circle radius $r$, center the origin.
Show that $\lim_{r\to \infty} \frac{f(r)}{r^2}$ exists and find it. If the limit is $k$, put $g(r) = f(r) - kr^2$.
Is it true that $\lim_{r\to \infty} \frac{g(r)}{r^h} = 0$ for any $h < 2$?
1983 AIME Problems, 9
Find the minimum value of
\[\frac{9x^2 \sin^2 x + 4}{x \sin x}\]
for $0 < x < \pi$.
2017 CMIMC Geometry, 6
Cyclic quadrilateral $ABCD$ satisfies $\angle ABD = 70^\circ$, $\angle ADB=50^\circ$, and $BC=CD$. Suppose $AB$ intersects $CD$ at point $P$, while $AD$ intersects $BC$ at point $Q$. Compute $\angle APQ-\angle AQP$.
1994 Mexico National Olympiad, 1
The sequence $1, 2, 4, 5, 7, 9 ,10, 12, 14, 16, 17, ... $ is formed as follows. First we take one odd number, then two even numbers, then three odd numbers, then four even numbers, and so on. Find the number in the sequence which is closest to $1994$.
2021 China Second Round Olympiad, Problem 4
When the expression $$(xy-5x+3y-15)^n$$ for some positive integer $n$ is expanded and like terms are combined, the expansion contains at least 2021 distinct terms. Compute the minimum possible value of $n$.
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 4)[/i]
2005 MOP Homework, 6
Consider the three disjoint arcs of a circle determined by three points of the circle. We construct a circle around each of the midpoint of every arc which goes the end points of the arc. Prove that the three circles pass through a common point.
2016 Belarus Team Selection Test, 1
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.