Found problems: 85335
2024 Belarusian National Olympiad, 9.4
In some company, consisting of $n$ people, any two have at most $k \geq 2$ common friends. Lets call group of people working in the company unsocial if everyone in the group has at most one friend from the group.
Prove that there exists an unsocial group consisting of at least $\sqrt{\frac{2n}{k}}$ people
[i]M. Zorka[/i]
1973 Yugoslav Team Selection Test, Problem 2
A circle $k$ is drawn using a given disc (e.g. a coin). A point $A$ is chosen on $k$. Using just the given disc, determine the point $B$ on $k$ so that $AB$ is a diameter of $k$. (You are allowed to choose an arbitrary point in one of the drawn circles, and using the given disc it is possible to construct either of the two circles that passes through the points at a distance that is smaller than the radius of the circle.)
2013 Balkan MO Shortlist, N3
Determine all quadruplets ($x, y, z, t$) of positive integers, such that $12^x + 13^y - 14^z = 2013^t$.
2021 Winter Stars of Mathematics, 1
Let $a_1,a_2,a_3,a_4$ be positive real numbers satisfying \[\sum_{i<j}a_ia_j=1.\]Prove that \[\sum_{\text{sym}}\frac{a_1a_2}{1+a_3a_4}\geq\frac{6}{7}.\][i]* * *[/i]
2010 Iran MO (3rd Round), 4
a) prove that every discrete subgroup of $(\mathbb R^2,+)$ is in one of these forms:
i-$\{0\}$.
ii-$\{mv|m\in \mathbb Z\}$ for a vector $v$ in $\mathbb R^2$.
iii-$\{mv+nw|m,n\in \mathbb Z\}$ for tho linearly independent vectors $v$ and $w$ in $\mathbb R^2$.(lattice $L$)
b) prove that every finite group of symmetries that fixes the origin and the lattice $L$ is in one of these forms: $\mathcal C_i$ or $\mathcal D_i$ that $i=1,2,3,4,6$ ($\mathcal C_i$ is the cyclic group of order $i$ and $\mathcal D_i$ is the dyhedral group of order $i$).(20 points)
LMT Team Rounds 2010-20, A10 B18
Define a sequence $\{a_n\}_{n \geq 1}$ recursively by $a_1=1$, $a_2=2$, and for all integers $n \geq 2$, $a_{n+1}=(n+1)^{a_n}$. Determine the number of integers $k$ between $2$ and $2020$, inclusive, such that $k+1$ divides $a_k - 1$.
[i]Proposed by Taiki Aiba[/i]
2013 China Second Round Olympiad, 1
$AB$ is a chord of circle $\omega$, $P$ is a point on minor arc $AB$, $E,F$ are on segment $AB$ such that $AE=EF=FB$. $PE,PF$ meets $\omega$ at $C,D$ respectively. Prove that $EF\cdot CD=AC\cdot BD$.
1984 IMO Longlists, 58
Let $(a_n)_1^{\infty}$ be a sequence such that $a_n \le a_{n+m} \le a_n + a_m$ for all positive integers $n$ and $m$. Prove that $\frac{a_n}{n}$ has a limit as $n$ approaches infinity.
2020/2021 Tournament of Towns, P7
Let $p{}$ and $q{}$ be two coprime positive integers. A frog hops along the integer line so that on every hop it moves either $p{}$ units to the right or $q{}$ units to the left. Eventually, the frog returns to the initial point. Prove that for every positive integer $d{}$ with $d < p + q$ there are two numbers visited by the frog which differ just by $d{}$.
[i]Nikolay Belukhov[/i]
2022 CCA Math Bonanza, I7
Let
$$A = \{2, 4, \ldots, 1000\},$$
$$B = \{3, 6, \ldots, 999\},$$
$$C = \{5, 10, \ldots, 1000\},$$
$$D = \{7, 14, \ldots, 994\},$$
$$E = \{11, 22, \ldots, 990\},$$
$$\textrm{and } F = \{13, 26, \ldots, 988\}.$$
Find the number of elements in the set $(((((A\cup B)\cap C)\cup D)\cap E)\cup F)$.
[i]2022 CCA Math Bonanza Individual Round #7[/i]
2019 AMC 10, 12
What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$?
$\textbf{(A) } 11
\qquad\textbf{(B) } 14
\qquad\textbf{(C) } 22
\qquad\textbf{(D) } 23
\qquad\textbf{(E) } 27$
2002 Junior Balkan Team Selection Tests - Romania, 1
A square of side 1 is decomposed into 9 equal squares of sides 1/3 and the one in the center is painted in black. The remaining eight squares are analogously divided into nine squares each and the square in the center is painted in black. Prove that after 1000 steps the total area of black region exceeds 0.999[/b]
1988 Federal Competition For Advanced Students, P2, 3
Show that there is precisely one sequence $ a_1,a_2,...$ of integers which satisfies $ a_1\equal{}1, a_2>1,$ and $ a_{n\plus{}1}^3\plus{}1\equal{}a_n a_{n\plus{}2}$ for $ n \ge 1$.
2012 Kazakhstan National Olympiad, 1
For a positive reals $ x_{1},...,x_{n} $ prove inequlity:
$ \frac{1}{x_{1}+1}+...+\frac{1}{x_{n}+1}\le \frac{n}{1+\frac{n}{\frac{1}{x_{1}}+...+\frac{1}{x_{n}}}}$
1996 APMO, 2
Let $m$ and $n$ be positive integers such that $n \leq m$. Prove that
\[ 2^n n! \leq \frac{(m+n)!}{(m-n)!} \leq (m^2 + m)^n \]
2021 USMCA, 5
Let $A$ denote the set of all the positive integer divisors of $30.$ For each nonempty subset $s \subseteq A,$ define $p(s)$ to be the product of the elements in $s.$ Finally, let $B$ denote the set of all possible remainders when $p(s)$ is divided by $30.$ How many (distinct) elements are in $B?$
May Olympiad L1 - geometry, 2016.4
In a triangle $ABC$, let $D$ and $E$ point in the sides $BC$ and $AC$ respectively. The segments $AD$ and $BE$ intersects in $O$, let $r$ be line (parallel to $AB$) such that $r$ intersects $DE$ in your midpoint, show that the triangle $ABO$ and the quadrilateral $ODCE$ have the same area.
2007 Hanoi Open Mathematics Competitions, 9
A triangle is said to be the Heron triangle if it has
integer sides and integer area. In a Heron triangle, the sides a; b; c satisfy
the equation b=a(a-c). Prove that the triangle is isosceles.
2014 Romania National Olympiad, 2
Let $ a $ be an odd natural that is not a perfect square, and $ m,n\in\mathbb{N} . $ Then
[b]a)[/b] $ \left\{ m\left( a+\sqrt a \right) \right\}\neq\left\{ n\left( a-\sqrt a \right) \right\} $
[b]b)[/b] $ \left[ m\left( a+\sqrt a \right) \right]\neq\left[ n\left( a-\sqrt a \right) \right] $
Here, $ \{\},[] $ denotes the fractionary, respectively the integer part.
1974 IMO Shortlist, 7
Let $a_i, b_i$ be coprime positive integers for $i = 1, 2, \ldots , k$, and $m$ the least common multiple of $b_1, \ldots , b_k$. Prove that the greatest common divisor of $a_1 \frac{m}{b_1} , \ldots, a_k \frac{m}{b_k}$ equals the greatest common divisor of $a_1, \ldots , a_k.$
2009 Today's Calculation Of Integral, 410
Evaluate $ \int_0^{\frac{\pi}{4}} \frac{1}{\cos \theta}\sqrt{\frac{1\plus{}\sin \theta}{\cos \theta}}\ d\theta$.
2000 Irish Math Olympiad, 5
Consider all parabolas of the form $ y\equal{}x^2\plus{}2px\plus{}q$ for $ p,q \in \mathbb{R}$ which intersect the coordinate axes in three distinct points. For such $ p,q$, denote by $ C_{p,q}$ the circle through these three intersection points. Prove that all circles $ C_{p,q}$ have a point in common.
2015 Online Math Open Problems, 10
For any positive integer $n$, define a function $f$ by \[f(n)=2n+1-2^{\lfloor\log_2n\rfloor+1}.\] Let $f^m$ denote the function $f$ applied $m$ times.. Determine the number of integers $n$ between $1$ and $65535$ inclusive such that $f^n(n)=f^{2015}(2015).$
[i]Proposed by Yannick Yao[/i]
1979 IMO Longlists, 28
Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that: \[ a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} - (2-\sqrt2)^{n-1} \over\sqrt2}. \]
2012 China Western Mathematical Olympiad, 3
Let $n$ be a positive integer $\geq 2$ . Consider a $n$ by $n$ grid with all entries $1$. Define an operation on a square to be changing the signs of all squares adjacent to it but not the sign of its own. Find all $n$ such that it is possible after a finite sequence of operations to reach a $n$ by $n$ grid with all entries $-1$