Found problems: 85335
2011 Math Prize for Girls Olympiad, 3
Let $n$ be a positive integer such that $n + 1$ is divisible by 24. Prove that the sum of all the positive divisors of $n$ is divisible by 24.
2015 Romania National Olympiad, 1
Show that among the square roots of the first $ 2015 $ natural numbers, we cannot choose an arithmetic sequence composed of $ 45 $ elements.
2016 IFYM, Sozopol, 7
Let $S$ be a set of integers which has the following properties:
1) There exists $x,y\in S$ such that $(x,y)=(x-2,y-2)=1$;
2) For $\forall$ $x,y\in S, x^2-y\in S$.
Prove that $S\equiv \mathbb{Z}$ .
2018 Hanoi Open Mathematics Competitions, 9
How many ways of choosing four edges in a cube such that any two among those four choosen edges have no common point.
2006 All-Russian Olympiad, 2
Show that there exist four integers $a$, $b$, $c$, $d$ whose absolute values are all $>1000000$ and which satisfy $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=\frac{1}{abcd}$.
2013 Stanford Mathematics Tournament, 11
Sara has an ice cream cone with every meal. The cone has a height of $2\sqrt2$ inches and the base of the cone has a diameter of $2$ inches. Ice cream protrudes from the top of the cone in a perfect hempisphere. Find the surface area of the ice cream cone, ice cream included, in square inches.
2007 AMC 10, 3
An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. It is filled with water to a height of 40 cm. A brick with a rectangular base that measures 40 cm by 20 cm and a height of 10 cm is placed in the aquarium. By how many centimeters does the water rise?
$ \textbf{(A)}\ 0.5 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 1.5 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 2.5$
2014 Contests, 1
A sequence $a_0,a_1,a_2,\cdots$ satisfies the conditions $a_0 = 0$ , $a_{n-1}^2 - a_{n-1} = a_n^2 + a_n$
1) determine the two possible values of $a_1$ . then determine all possible values of $a_2$ .
2)for each $n$, prove that $a_{n+1}=a_n+1$ or $a_{n+1} = -a_n$
3)Describe the possible values of $a_{1435}$
4)Prove that the values that you got in (3) are correct
2010 National Olympiad First Round, 5
Let $ABCD$ be a convex quadrilateral such that $|AB|=10$, $|CD|=3\sqrt 6$, $m(\widehat{ABD})=60^\circ$, $m(\widehat{BDC})=45^\circ$, and $|BD|=13+3\sqrt 3$. What is $|AC|$ ?
$ \textbf{(A)}\ 20
\qquad\textbf{(B)}\ 18
\qquad\textbf{(C)}\ 16
\qquad\textbf{(D)}\ 14
\qquad\textbf{(E)}\ 12
$
2015 ASDAN Math Tournament, 11
In the following diagram, each circle has radius $6$ and each circle passes through the center of the other two circles. Compute the area of the white center region and express your answer in terms of $\pi$.
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2002 BAMO, 4
For $n \ge 1$, let $a_n$ be the largest odd divisor of $n$, and let $b_n = a_1+a_2+...+a_n$.
Prove that $b_n \ge \frac{ n^2 + 2}{3}$, and determine for which $n$ equality holds.
For example, $a_1 = 1, a_2 = 1, a_3 = 3, a_4 = 1, a_5 = 5, a_6 = 3$, thus $b_6 = 1 + 1 + 3 + 1 + 5 + 3 = 14 \ge \frac{ 6^2 + 2}{3}= 12\frac23$
.
2022 China Girls Math Olympiad, 2
Let $n$ be a positive integer. There are $3n$ women's volleyball teams in the tournament, with no more than one match between every two teams (there are no ties in volleyball). We know that there are $3n^2$ games played in this tournament.
Proof: There exists a team with at least $\frac{n}{4}$ win and $\frac{n}{4}$ loss
2011 Indonesia TST, 4
Let $a, b$, and $c$ be positive integers such that $gcd(a, b) = 1$. Sequence $\{u_k\}$, is given such that $u_0 = 0$, $u_1 = 1$, and u$_{k+2} = au_{k+1} + bu_k$ for all $k \ge 0$. Let $m$ be the least positive integer such that $c | u_m$ and $n$ be an arbitrary positive integer such that $c | u_n$. Show that $m | n$.
[hide=PS.] There was a typo in the last line, as it didn't define what n does. Wording comes from [b]tst-2011-1.pdf[/b] from [url=https://sites.google.com/site/imoidn/idntst/2011tst]here[/url]. Correction was made according to #2[/hide]
KoMaL A Problems 2023/2024, A. 869
Let $A$ and $B$ be given real numbers. Let the sum of real numbers $0\le x_1\le x_2\le\ldots \le x_n$ be $A$, and let the sum of real numbers $0\le y_1\le y_2\le \ldots\le y_n$ be $B$. Find the largest possible value of
\[\sum_{i=1}^n (x_i-y_i)^2.\]
[i]Proposed by Péter Csikvári, Budapest[/i]
2012 Junior Balkan Team Selection Tests - Moldova, 4
How many solutions does the system have:
$ \{\begin{matrix}&(3x+2y) *(\frac{3}{x}+\frac{1}{y})=2\\ & x^2+y^2\leq 2012\\ \end{matrix} $
where $ x,y $ are non-zero integers
2009 All-Russian Olympiad, 2
Consider the sequence of numbers $(a_n)$ ($n = 1, 2, \ldots$) defined as follows: $ a_1\in (1, 2)$, $ a_{k + 1} = a_k + \frac{k}{a_k}$ ($k = 1, 2, \ldots$). Prove that there exists at most one pair of distinct positive integers $(i, j)$ such that $a_i + a_j$ is an integer.
MOAA Accuracy Rounds, 2023.4
A two-digit number $\overline{ab}$ is [i]self-loving[/i] if $a$ and $b$ are relatively prime positive integers and $\overline{ab}$ is divisible by $a+b$. How many self-loving numbers are there?
[i]Proposed by Anthony Yang and Andy Xu[/i]
2025 Bundeswettbewerb Mathematik, 1
Fridolin the frog jumps on the number line: He starts at $0$, then jumps in some order on each of the numbers $1,2,\dots,9$ exactly once and finally returns with his last jump to $0$. Can the total distance he travelled with these $10$ jumps be a) $20$, b) $25$?
2010 India IMO Training Camp, 12
Prove that there are infinitely many positive integers $m$ for which there exists consecutive odd positive integers $p_m<q_m$ such that $p_m^2+p_mq_m+q_m^2$ and $p_m^2+m\cdot p_mq_m+q_m^2$ are both perfect squares. If $m_1, m_2$ are two positive integers satisfying this condition, then we have $p_{m_1}\neq p_{m_2}$
1985 Traian Lălescu, 1.4
Let $ ABC $ a right triangle in $ A. $ Let $ D $ a point on the segment $ AC, $ and $ E,F $ the projections of $ A $ upon the lines $ BD, $ respectively, $ BC. $ Show that the quadrilateral $ CDEF $ is concyclic.
2004 Peru MO (ONEM), 3
Let $x,y,z$ be positive real numbers, less than $\pi$, such that:
$$\cos x + \cos y + \cos z = 0$$
$$\cos 2x + \cos 2 y + \cos 2z = 0$$
$$\cos 3x + \cos 3y + \cos 3z = 0$$
Find all the values that $\sin x + \sin y + \sin z$ can take.
2020 CHMMC Winter (2020-21), 4
Select a random real number $m$ from the interval $(\frac{1}{6}, 1)$. A track is in the shape of an equilateral triangle of side length $50$ feet. Ch, Hm, and Mc are all initially standing at one of the vertices of the track. At the time $t = 0$, the three people simultaneously begin walking around the track in clockwise direction. Ch, Hm, and Mc walk at constant rates of $2, 3$, and $4$ feet per second, respectively. Let $T$ be the set of all positive real numbers $t_0$ satisfying the following criterion:
[i]If we choose a random number $t_1$ from the interval $[0, t_0]$, the probability that the three people are on the same side of the track at the time $t = t_1$ is precisely $m$.[/i]
The probability that $|T| = 17$ (i.e., $T$ has precisely $17$ elements) equals $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2014 USAJMO, 6
Let $ABC$ be a triangle with incenter $I$, incircle $\gamma$ and circumcircle $\Gamma$. Let $M,N,P$ be the midpoints of sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ and let $E,F$ be the tangency points of $\gamma$ with $\overline{CA}$ and $\overline{AB}$, respectively. Let $U,V$ be the intersections of line $EF$ with line $MN$ and line $MP$, respectively, and let $X$ be the midpoint of arc $BAC$ of $\Gamma$.
(a) Prove that $I$ lies on ray $CV$.
(b) Prove that line $XI$ bisects $\overline{UV}$.
2013 Dutch IMO TST, 5
Let $ABCDEF$ be a cyclic hexagon satisfying $AB\perp BD$ and $BC=EF$.Let $P$ be the intersection of lines $BC$ and $AD$ and let $Q$ be the intersection of lines $EF$ and $AD$.Assume that $P$ and $Q$ are on the same side of $D$ and $A$ is on the opposite side.Let $S$ be the midpoint of $AD$.Let $K$ and $L$ be the incentres of $\triangle BPS$ and $\triangle EQS$ respectively.Prove that $\angle KDL=90^0$.
2018 CMIMC Number Theory, 10
Let $a_1 < a_2 < \cdots < a_k$ denote the sequence of all positive integers between $1$ and $91$ which are relatively prime to $91$, and set $\omega = e^{2\pi i/91}$. Define \[S = \prod_{1\leq q < p\leq k}\left(\omega^{a_p} - \omega^{a_q}\right).\] Given that $S$ is a positive integer, compute the number of positive divisors of $S$.