Found problems: 85335
2018 Iran Team Selection Test, 6
$a_1,a_2,\ldots,a_n$ is a sequence of positive integers that has at least $\frac {2n}{3}+1$ distinct numbers and each positive integer has occurred at most three times in it. Prove that there exists a permutation $b_1,b_2,\ldots,b_n$ of $a_i $'s such that all the $n$ sums $b_i+b_{i+1}$ are distinct ($1\le i\le n $ , $b_{n+1}\equiv b_1 $)
[i]Proposed by Mohsen Jamali[/i]
2015 China Team Selection Test, 1
For a positive integer $n$, and a non empty subset $A$ of $\{1,2,...,2n\}$, call $A$ good if the set $\{u\pm v|u,v\in A\}$ does not contain the set $\{1,2,...,n\}$. Find the smallest real number $c$, such that for any positive integer $n$, and any good subset $A$ of $\{1,2,...,2n\}$, $|A|\leq cn$.
2011 Romania Team Selection Test, 2
Prove that the set $S=\{\lfloor n\pi\rfloor \mid n=0,1,2,3,\ldots\}$ contains arithmetic progressions of any finite length, but no infinite arithmetic progressions.
[i]Vasile Pop[/i]
2021 Bangladesh Mathematical Olympiad, Problem 6
Let $ABC$ be an acute-angled triangle. The external bisector of $\angle BAC$ meets the line $BC$ at point $N$. Let $M$ be the midpoint of $BC$. $P$ and $Q$ are two points on line $AN$ such that, $\angle PMN=\angle MQN=90^{\circ}$. If $PN=5$ and $BC=3$, then the length $QA$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are co-prime positive integers. What is the value of $(a+b)$?
2017 USA Team Selection Test, 1
You are cheating at a trivia contest. For each question, you can peek at each of the $n > 1$ other contestants' guesses before writing down your own. For each question, after all guesses are submitted, the emcee announces the correct answer. A correct guess is worth $0$ points. An incorrect guess is worth $-2$ points for other contestants, but only $-1$ point for you, since you hacked the scoring system. After announcing the correct answer, the emcee proceeds to read the next question. Show that if you are leading by $2^{n - 1}$ points at any time, then you can surely win first place.
[i]Linus Hamilton[/i]
2015 Latvia Baltic Way TST, 14
Let $S(a)$ denote the sum of the digits of the number $a$. Given a natural $R$ can one find a natural $n$ such that $\frac{S (n^2)}{S (n)}= R$?
1994 Poland - First Round, 2
Given a positive integer $n \geq 2$. Solve the following system of equations:
$
\begin{cases}
\ x_1|x_1| &= x_2|x_2| + (x_1-1)|x_1-1| \\
\ x_2|x_2| &= x_3|x_3| + (x_2-1)|x_2-1| \\
&\dots \\
\ x_n|x_n| &= x_1|x_1| + (x_n-1)|x_n-1|. \\
\end{cases}
$
2007 Moldova Team Selection Test, 4
Consider a convex polygon $A_{1}A_{2}\ldots A_{n}$ and a point $M$ inside it. The lines $A_{i}M$ intersect the perimeter of the polygon second time in the points $B_{i}$. The polygon is called balanced if all sides of the polygon contain exactly one of points $B_{i}$ (strictly inside). Find all balanced polygons.
[Note: The problem originally asked for which $n$ all convex polygons of $n$ sides are balanced. A misunderstanding made this version of the problem appear at the contest]
2012 Today's Calculation Of Integral, 789
Find the non-constant function $f(x)$ such that $f(x)=x^2-\int_0^1 (f(t)+x)^2dt.$
2007 Middle European Mathematical Olympiad, 3
Let $ k$ be a circle and $ k_{1},k_{2},k_{3},k_{4}$ four smaller circles with their centres $ O_{1},O_{2},O_{3},O_{4}$ respectively, on $ k$. For $ i \equal{} 1,2,3,4$ and $ k_{5}\equal{} k_{1}$ the circles $ k_{i}$ and $ k_{i\plus{}1}$ meet at $ A_{i}$ and $ B_{i}$ such that $ A_{i}$ lies on $ k$. The points $ O_{1},A_{1},O_{2},A_{2},O_{3},A_{3},O_{4},A_{4}$ lie in that order on $ k$ and are pairwise different.
Prove that $ B_{1}B_{2}B_{3}B_{4}$ is a rectangle.
2024 Assara - South Russian Girl's MO, 3
In the cells of the $4\times N$ table, integers are written, modulo no more than $2024$ (i.e. numbers from the set $\{-2024, -2023,\dots , -2, -1, 0, 1, 2, 3,\dots , 2024\}$) so that in each of the four lines there are no two equal numbers. At what maximum $N$ could it turn out that in each column the sum of the numbers is equal to $23$?
[i]G.M.Sharafetdinova[/i]
1999 Tournament Of Towns, 4
Every $24$ hours , the minute hand of an ordinary clock completes $24$ revolutions while the hour hand completes $2$. Every $24$ hours , the minute hand of an Italian clock completes $24$ revolutions while the hour hand completes only $1$ . The minute hand of each clock is longer than the hour hand, and "zero hour" is located at the top of the clock's face. How many positions of the two hands can occur on an Italian clock within a $24$-hour period that are possible on an ordinary one?
(Folklore)
2004 Paraguay Mathematical Olympiad, 4
In a square $ABCD$, $E$ is the midpoint of $BC$ and $F$ is the midpoint of $CD$. Prove that $AF$ and $AE$ divide the diagonal $BD$ in three equal segments.
2016 CMIMC, 2
The $\emph{Stooge sort}$ is a particularly inefficient recursive sorting algorithm defined as follows: given an array $A$ of size $n$, we swap the first and last elements if they are out of order; we then (if $n\ge3$) Stooge sort the first $\lceil\tfrac{2n}3\rceil$ elements, then the last $\lceil\tfrac{2n}3\rceil$, then the first $\lceil\tfrac{2n}3\rceil$ elements again. Given that this runs in $O(n^\alpha)$, where $\alpha$ is minimal, find the value of $(243/32)^\alpha$.
2023 MMATHS, 12
Let $ABC$ be a triangle with incenter $I.$ The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA,$ and $AB$ at points $D, E,$ and $F,$ respectively. Let $D'$ be the reflection of $D$ over $I.$ Let $P$ be a point on $\omega$ such that $\angle{ADP}=90^\circ.$ $\mathcal{H}$ is a hyperbola passing through $D', E, F, I,$ and $P.$ Given that $\angle{BAD}=45^\circ$ and $\angle{CAD}=30^\circ,$ the acute angle between the asymptotes of $\mathcal{H}$ can be expressed as $\left(\tfrac{m}{n}\right)^\circ,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
1994 Flanders Math Olympiad, 2
Determine all integer solutions (a,b,c) with $c\leq 94$ for which:
$(a+\sqrt c)^2+(b+\sqrt c)^2 = 60 + 20\sqrt c$
1998 Gauss, 1
The value of $\frac{1998- 998}{1000}$ is
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 1000 \qquad \textbf{(C)}\ 0.1 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 0.001$
2021 Alibaba Global Math Competition, 15
Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold with $n \ge 2$. Suppose $M$ is connected and $\text{Ric} \ge (n-1)g$, where $\text{Ric}$ is the Ricci tensor of $(M,g)$. Denote by $\text{d}g$ the Riemannian measure of $(M,g)$ and by $d(x,y)$ the geodesic distance between $x$ and $y$. Prove that
\[\int_{M \times M} \cos d(x,y) \text{d}g(x)\text{d}g(y) \ge 0.\]
Moreover, equality holds if and only if $(M,g)$ is isometric to the unit round sphere $S^n$.
2009 Federal Competition For Advanced Students, P2, 4
Let $ a$ be a positive integer. Consider the sequence $ (a_n)$ defined as $ a_0\equal{}a$
and $ a_n\equal{}a_{n\minus{}1}\plus{}40^{n!}$ for $ n > 0$. Prove that the sequence $ (a_n)$ has infinitely
many numbers divisible by $ 2009$.
I Soros Olympiad 1994-95 (Rus + Ukr), 10.5
A circle can be drawn around the quadrilateral $ABCD$. Let straight lines $AB$ and $CD$ intersect at point $M$, and straight lines $BC$ and $AD$ intersect at point $K$. (Points $B$ and $P$ lie on segments $AM$ and $AK$, respectively.) Let $P$ be the projection of point $M$ onto straight line $AK$, $L$ be the projection of point $K$ on the straight line $AM$. Prove that the straight line $LP$ divides the diagonal $BD$ in half.
2013 Stanford Mathematics Tournament, 10
Consider a sequence given by $a_n=a_{n-1}+3a_{n-2}+a_{n-3}$, where $a_0=a_1=a_2=1$. What is the remainder of $a_{2013}$ divided by $7$?
2021 Estonia Team Selection Test, 2
Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$.
Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.
1970 Poland - Second Round, 2
On the sides of the regular $ n $-gon, $ n + 1 $ points are taken dividing the perimeter into equal parts. At what position of the selected points is the area of the convex polygon with these $ n + 1 $ vertices
a) the largest,
b) the smallest?
1969 Vietnam National Olympiad, 4
Two circles centers $O$ and $O'$, radii $R$ and $R'$, meet at two points. A variable line $L$ meets the circles at $A, C, B, D$ in that order and $\frac{AC}{AD} = \frac{CB}{BD}$. The perpendiculars from $O$ and $O'$ to $L$ have feet $H$ and $H'$.
Find the locus of $H$ and $H'$.
If $OO'^2 < R^2 + R'^2$, find a point $P$ on $L$ such that $PO + PO'$ has the smallest possible value.
Show that this value does not depend on the position of $L$.
Comment on the case $OO'^2 > R^2 + R'^2$.
2021 CMIMC, 1.8
Let $ABC$ be a triangle with $AB < AC$ and $\omega$ be a circle through $A$ tangent to both the $B$-excircle and the $C$-excircle. Let $\omega$ intersect lines $AB, AC$ at $X,Y$ respectively and $X,Y$ lie outside of segments $AB, AC$. Let $O$ be the center of $\omega$ and let $OI_C, OI_B$ intersect line $BC$ at $J,K$ respectively. Suppose $KJ = 4$, $KO = 16$ and $OJ = 13$. Find $\frac{[KI_BI_C]}{[JI_BI_C]}$.
[i]Proposed by Grant Yu[/i]