Found problems: 85335
In a trick tournament $2k$ people sign up. All possible matches are played with the condition that in each match, each of the four players knows his partner and does not know any of his two opponents. Determine the maximum number of matches that can be in such a tournament.
Suppose that $ a$ and $ b$ are digits, not both nine and not both zero, and the repeating decimal $ 0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$
Prove that, for all integer $a>1$, the prime divisors of $5a^4-5a^2+1$ have the form $20k\pm1,k\in\mathbb{Z}$.
[i]Proposed by Géza Kós.[/i]
Let $a_1, a_2,\cdots ,a_n$ be positive integers and let $a$ be an integer greater than $1$ and divisible
by the product $a_1a_2\cdots a_n$. Prove that $a^{n+1} + a-1$ is not divisible by the product
$(a + a_1 - 1)(a + a_2 - 1) \cdots (a + a_n - 1)$.
At a table tennis tournament, each one of the $ n\ge 2 $ participants play with all the others exactly once. Show that, at the end of the tournament, one and only one of these propositions will be true:
$ \text{(1)} $ The players can be labeled with the numbers $ 1,2,...,n, $ such that $ 1 $ won $ 2, 2 $ won $ 3,...,n-1 $ won $ n $ and $ n $ won $ 1. $
$ \text{(2)} $ The players can be partitioned in two nonempty subsets $ A,B, $ such that whichever one from $ A $ won all that are in $ B. $
Several circles are drawn on the plane and all points of their meeting or touching are marked. May be that each circle contains exactly four marked points and exactly four marked points lie on each circle?
Given an acute triangle $ABC$ with $AB>AC$ that has circumcenter $O$. Line $BO$ and $CO$ meet the bisector of $\angle BAC$ at $P$ and $Q$, respectively. Moreover, line $BQ$ and $CP$ meet at $R$. Show that $AR$ is perpendicular to $BC$.
[i]Proposer: Soewono and Fajar Yuliawan[/i]
$(YUG 1)$ Suppose that positive real numbers $x_1, x_2, x_3$ satisfy
$x_1x_2x_3 > 1, x_1 + x_2 + x_3 <\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}$
Prove that:
$(a)$ None of $x_1, x_2, x_3$ equals $1$.
$(b)$ Exactly one of these numbers is less than $1.$
Let $ABC$ be a triangle with $\angle A<90^o, AB \ne AC$. Denote $H$ the orthocenter of triangle $ABC$, $N$ the midpoint of segment $[AH]$, $M$ the midpoint of segment $[BC]$ and $D$ the intersection point of the angle bisector of $\angle BAC$ with the segment $[MN]$. Prove that $<ADH=90^o$
If $x_{1},x_{2},...,x_{n}(n>2)$ are positive real numbers with $x_{1}+x_{2}+...+x_{n}=1$. Prove that $x_{1}^{2}x_{2}+x_{2}^{2}x_{3}+...+x_{n}^{2}x_{1}\leq\frac{4}{27}$.
Let $T_n$ be the number of non-congruent triangles with positive area and integer side lengths summing to $n$. Prove that $T_{2022}=T_{2019}$.
[i]Proposed by Constantinos Papachristoforou[/i]
For every positive integer $n$, form the number $n/s(n)$, where $s(n)$ is the sum of digits of $n$ in base 10. Determine the minimum value of $n/s(n)$ in each of the following cases:
(i) $10 \leq n \leq 99$
(ii) $100 \leq n \leq 999$
(iii) $1000 \leq n \leq 9999$
(iv) $10000 \leq n \leq 99999$
Compare the number of distinct prime divisors of $200^2 \cdot 201^2 \cdot ... \cdot 900^2$ and $(200^2 -1)(201^2 -1)\cdot ... \cdot (900^2 -1) .$
The product of three consecutive positive integers is $ 8$ times their sum. What is the sum of their squares?
$ \textbf{(A)}\ 50 \qquad
\textbf{(B)}\ 77 \qquad
\textbf{(C)}\ 110 \qquad
\textbf{(D)}\ 149 \qquad
\textbf{(E)}\ 194$
How many primes $p$ are there such that the number of positive divisors of $p^2+23$ is equal to $14$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \text{None of above}
$
Let two circles $C_1,C_2$ with different radius be externally tangent at a point $T$.
Let $A$ be on $C_1$ and $B$ be on $C_2$, with $A,B \ne T$ such that $\angle ATB = 90^o$.
(a) Prove that all such lines $AB$ are concurrent.
(b) Find the locus of the midpoints of all such segments $AB$.
Let $S$ be the smallest set of positive integers such that
a) $2$ is in $S,$
b) $n$ is in $S$ whenever $n^2$ is in $S,$ and
c) $(n+5)^2$ is in $S$ whenever $n$ is in $S.$
Which positive integers are not in $S?$
(The set $S$ is ``smallest" in the sense that $S$ is contained in any other such set.)
There are $2019$ students sitting around circular table. Initially each of them have one candy. Teacher is allowed to pick one student, who has at least one can candy, and this student can decide, whether he gives his candy to his neighbour on the right or on the left. Prove that no matter what students teacher picks during the process, students can always ensure that any point of time no student has more than $2$ candies.
Find the real values of $p$ for which the equation
\[\sqrt{2p+ 1 - x^2} +\sqrt{3x + p + 4} = \sqrt{x^2 + 9x+ 3p + 9}\]
in $x$ has exactly two real distinct roots.($\sqrt t $ means the positive square root of $t$).
In triangle $ABC,$ $AB = 13,$ $BC = 14,$ and $CA = 15.$ A circle of radius $r$ passes through point $A$ and is tangent to line $BC$ at $C.$ If $r = m/n,$ where $m$ and $n$ are relatively prime positive integers, find $100m + n.$
[i]Proposed by Michael Tang[/i]
Consider the sequence $a_{1}, a_{2}, a_{3},\ldots$ defined by $a_{1}=2024^{2024}$ and for each positive integer $n$, $$a_{n+1}=\left|a_{n}-\sqrt{2}\right|.$$ Prove that there exists an integer $k$ such that $a_{k+2}=a_k$.
[i]Here [/i]$\left|x\right|$[i] denotes the absolute value of [/i]$x$.
A mathematical frog jumps along the number line. The frog starts at $1$, and jumps according to the following rule: if the frog is at integer $n$, then it can jump either to $n+1$ or to $n + 2^{m_n+1}$ where $2^{m_n}$ is the largest power of $2$ that is a factor of $n.$ Show that if $k \geq 2$ is a positive integer and $i$ is a nonnegative integer, then the minimum number of jumps needed to reach $2^ik$ is greater than the minimum number of jumps needed to reach $2^i.$
All edges of a regular right pyramid are equal to $1$, and all vertices lie on the side surface of a (infinite) right circular cylinder of radius $R$. Find all possible values of $R$.
Every square of a $3\times3$ board is assigned a sign $+$ or $-$. In every move, one square is selected and the signs are changed in the selected square and all the neighboring squares (two squares are neighboring if they have a common side). Is it true that, no matter how the signs were initially distributed, one can obtain a table in which all signs are $-$ after finitely many moves?
Find all integers that can be written in the form $\frac{(x+y+z)^2}{xyz}$, where $x,y,z$ are positive integers.