Found problems: 85335
2023 Assara - South Russian Girl's MO, 1
A survey of participants was conducted at the Olympiad. $ 90\%$ of the participants liked the first round, $60\%$ of the participants liked the second round, $90\%$ of the participants liked the opening of the Olympiad. Each participant was known to enjoy at least two of these three events. Determine the percentage of participants who rated all three events positively.
2015 AMC 12/AHSME, 18
For every composite positive integer $n$, define $r(n)$ to be the sum of the factors in the prime factorization of $n$. For example, $r(50)=12$ because the prime factorization of $50$ is $ 2 \cdot 5^2 $, and $ 2 + 5 + 5 = 12 $. What is the range of the function $r$, $ \{ r(n) : n \ \text{is a composite positive integer} \} $?
[b](A)[/b] the set of positive integers
[b](B)[/b] the set of composite positive integers
[b](C)[/b] the set of even positive integers
[b](D)[/b] the set of integers greater than 3
[b](E)[/b] the set of integers greater than 4
2011 Middle European Mathematical Olympiad, 4
Let $n \geq 3$ be an integer. At a MEMO-like competition, there are $3n$ participants, there are n languages spoken, and each participant speaks exactly three different languages. Prove that at least $\left\lceil\frac{2n}{9}\right\rceil$ of the spoken languages can be chosen in such a way that no participant speaks more than two of the chosen languages.
[b]Note.[/b] $\lceil x\rceil$ is the smallest integer which is greater than or equal to $x$.
1994 Tournament Of Towns, (411) 2
The sequence of positive integers $a_1$, $a_2$,$...$ is such that for each $n = 1$,$2$, $...$ the quadratic equation $$a_{n+2}x^2 + a_{n+1}x+ a_n = 0$$ has a real root. Can the sequence consist of
(a) $10 $ terms,
(b) an infinite number of terms?
(A Shapovalov)
2020 AMC 8 -, 17
How many factors of $2020$ have more than $3$ factors? (As an example, $12$ has $6$ factors, namely $1$, $2$, $3$, $4$, $6$, and $12$.)
$\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 10$
2009 CentroAmerican, 5
Given an acute and scalene triangle $ ABC$, let $ H$ be its orthocenter, $ O$ its circumcenter, $ E$ and $ F$ the feet of the altitudes drawn from $ B$ and $ C$, respectively. Line $ AO$ intersects the circumcircle of the triangle again at point $ G$ and segments $ FE$ and $ BC$ at points $ X$ and $ Y$ respectively. Let $ Z$ be the point of intersection of line $ AH$ and the tangent line to the circumcircle at $ G$. Prove that $ HX$ is parallel to $ YZ$.
1995 Tournament Of Towns, (472) 6
A game is played on a $1 \times 1000$ board. There are n chips, all of which are initially in a box near the board. Two players move in turn. The first may choose $17$ chips or less, from either on or off the board. She then puts them into unoccupied cells on the board so that there is no more than one chip in each of the cells. The second player may take off the board any number of chips occupying consecutive cells and put them back in the box. The first player wins if she can put all n chips on the board so that they occupy consecutive cells.
(a) Show that she can win if $n = 98$.
(b) For what maximal value of $n$ can she win?
(A Shapovalov)
1954 Putnam, B6
Let $ x \in \mathbb{Q}^+$. Prove that there exits $\alpha_1,\alpha_2,...,\alpha_k \in \mathbb{N}$ and pairwe distinct such that
\[x= \sum_{i=1}^{k} \frac{1}{\alpha_i}\]
2008 Czech-Polish-Slovak Match, 1
Prove that there exists a positive integer $n$, such that the number $k^2+k+n$ does not have a prime divisor less than $2008$ for any integer $k$.
2016 Iran Team Selection Test, 3
Let $ABCD$ be a convex quadrilateral, and let $P$, $Q$, $R$, and $S$ be points on the sides $AB$, $BC$, $CD$, and $DA$, respectively. Let the line segment $PR$ and $QS$ meet at $O$. Suppose that each of the quadrilaterals $APOS$, $BQOP$, $CROQ$, and $DSOR$ has an incircle. Prove that the lines $AC$, $PQ$, and $RS$ are either concurrent or parallel to each other.
Russian TST 2021, P2
Let $ABC$ be a triangle with $AB < AC$, incenter $I$, and $A$ excenter $I_{A}$. The incircle meets $BC$ at $D$. Define $E = AD\cap BI_{A}$, $F = AD\cap CI_{A}$. Show that the circumcircle of $\triangle AID$ and $\triangle I_{A}EF$ are tangent to each other
2019 Mathematical Talent Reward Programme, MCQ: P 6
Find the limit $\lim \limits_{n \to \infty} \sin{n!}$
[list=1]
[*] 1
[*] 0
[*] $\frac{\pi}{4}$
[*] None of the above
[/list]
2006 Moldova Team Selection Test, 3
Positive real numbers $a,b,c$ satisfy the relation $abc=1$. Prove the inequality:
$\frac{a+3}{(a+1)^{2}}+\frac{b+3}{(b+1)^{2}}+\frac{c+3}{(c+1)^{2}}\geq3$.
2018 BMT Spring, 10
Let $a$,$b$,$c$ be the roots of the equation $x^{3} - 2018x +2018 = 0$. Let $q$ be the smallest positive integer for which there exists an integer $p, \, 0 < p \leq q$, such that $$\frac {a^{p+q} + b^{p+q} + c^{p+q}} {p+q} = \left(\frac {a^{p} + b^{p} + c^{p}} {p}\right)\left(\frac {a^{q} + b^{q} + c^{q}} {q}\right).$$ Find $p^{2} + q^{2}$.
2020 Iran MO (3rd Round), 2
For each $n$ find the number of ways one can put the numbers $\{1,2,3,...,n\}$ numbers on the circle, such that if for any $4$ numbers $a,b,c,d$ where $n|a+b-c-d$. The segments joining $a,b$ and $c,d$ do not meet inside the circle. (Two ways are said to be identical , if one can be obtained from rotaiting the other)
1997 Iran MO (3rd Round), 3
There are $30$ bags and there are $100$ similar coins in each bag (coins in each bag are similar, coins of different bags can be different). The weight of each coin is an one digit number in grams. We have a digital scale which can weigh at most $999$ grams in each weighing. Using this scale, we want to find the weight of coins of each bag.
[b](a)[/b] Show that this operation is possible by $10$ times of weighing, and
[b](b)[/b] It's not possible by $9$ times of weighing.
2009 Singapore MO Open, 3
for $k\in\mathbb{N}$ , define $A_n$ for $n=1,2,...$ by
$A_{n+1} = \frac{ nA_n+2(n+1)^{2k} }{n+2} , A_1=1$
Prove $A_n$ is integer for all $n\geq 1$, and $A_n$ is odd if and only if $n\equiv$1 or 2(mod 4)
2017 BMT Spring, 17
Triangle $ABC$ is drawn such that $\angle A = 80^o$, $\angle B = 60^o$, and $\angle C = 40^o$. Let the circumcenter of $\vartriangle ABC$ be $O$, and let $\omega$ be the circle with diameter $AO$. Circle $\omega$ intersects side $AC$ at point $P$. Let M be the midpoint of side $BC$, and let the intersection of $\omega$ and $PM$ be $K$. Find the measure of $\angle MOK$.
2011 District Olympiad, 1
Prove the rationality of the number $ \frac{1}{\pi }\int_{\sin\frac{\pi }{13}}^{\cos\frac{\pi }{13}} \sqrt{1-x^2} dx. $
PEN N Problems, 17
Suppose that $a$ and $b$ are distinct real numbers such that \[a-b, \; a^{2}-b^{2}, \; \cdots, \; a^{k}-b^{k}, \; \cdots\] are all integers. Show that $a$ and $b$ are integers.
PEN G Problems, 26
Prove that if $g \ge 2$ is an integer, then two series \[\sum_{n=0}^{\infty}\frac{1}{g^{n^{2}}}\;\; \text{and}\;\; \sum_{n=0}^{\infty}\frac{1}{g^{n!}}\] both converge to irrational numbers.
1959 Putnam, A4
If $f$ and $g$ are real-valued functions of one real variable, show that there exist $x$ and $y$ in $[0,1]$ such that $$|xy-f(x)-g(y)|\geq \frac{1}{4}.$$
2011 India IMO Training Camp, 1
Find all positive integer $n$ satisfying the conditions
$a)n^2=(a+1)^3-a^3$
$b)2n+119$ is a perfect square.
2012 Kurschak Competition, 2
Denote by $E(n)$ the number of $1$'s in the binary representation of a positive integer $n$. Call $n$ [i]interesting[/i] if $E(n)$ divides $n$. Prove that
(a) there cannot be five consecutive interesting numbers, and
(b) there are infinitely many positive integers $n$ such that $n$, $n+1$ and $n+2$ are each interesting.
2018 Hanoi Open Mathematics Competitions, 4
How many triangles are there for which the perimeters are equal to $30$ cm and the lengths of sides are integers in centimeters?
A. $16$ B. $17$ C. $18$ D. $19$ E. $20$