Found problems: 85335
2012 AMC 12/AHSME, 7
Mary divides a circle into $12$ sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?
$ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12 $
2006 Taiwan National Olympiad, 1
Positive reals $a,b,c$ satisfy $abc=1$. Prove that
$\displaystyle 1+ \frac{3}{a+b+c} \ge \frac{6}{ab+bc+ca}$.
2014 Tournament of Towns., 4
Point L is marked on side BC of triangle ABC so that AL is twice as long as the median CM. Given that angle ALC is equal to 45 degrees prove that AL is perpendicular to CM.
2009 Postal Coaching, 2
Determine, with proof, all the integer solutions of the equation $x^3 + 2y^3 + 4z^3 - 6xyz = 1$.
2022-2023 OMMC FINAL ROUND, 6
Evan writes a random positive integer on a board: the integer $k$ has probability $2^{-k}$ of being written. He keeps writing integers in this way repeatedly until he writes an integer that he had written before. He then takes all the integers he has written besides his last, sorts them in the order he first drew them, and also sorts them in increasing order, forming two sequences. For example, if he wrote $5,8,2,3,6,10,2$ in that order then his two sequences would be $5,8,2,3,6,10$ and $2,3,5,6,8,10.$
Find the probability that for all $k \in \{ 1,4,34 \},$ that $k$ was written, and $k$ appears in the same position in both sequences.
1987 All Soviet Union Mathematical Olympiad, 454
Vertex $B$ of the $\angle ABC$ lies out the circle, and the $[BA)$ and $[BC)$ beams intersect it. Point $K$ belongs to the intersection of the $[BA)$ beam and the circumference. Chord $KP$ is orthogonal to the angle bisector of $\angle ABC$ . Line $(KP)$ intersects the beam $BC$ in the point $M$. Prove that the segment $[PM]$ is twice as long as the distance from the circle centre to the angle bisector of $\angle ABC$ .
1994 All-Russian Olympiad, 1
$a,b$ are natural numbers such that $\frac{a+1}{b}$ and $\frac{b+1}{a}$ are integers.Let $d=GCD(a;b)$.Prove that $d^2\le a+b$
2025 Kosovo National Mathematical Olympiad`, P3
Let $g_a$, $g_b$ and $g_c$ be the medians of a triangle $\triangle ABC$ erected from the vertices $A$, $B$ and $C$, respectively.
Similarly, let $g_x$, $g_y$ and $g_z$ be the medians of an another triangle $\triangle XYZ$. Show that if
$$g_a : g_b : g_c = g_x : g_y : g_z, $$
then the triangles $\triangle ABC$ and $\triangle XYZ$ are similar.
2021 Irish Math Olympiad, 6
A sequence whose first term is positive has the property that any given term is the area of an equilateral triangle whose perimeter is the preceding term. If the first three terms form an arithmetic progression, determine all possible values of the first term.
2024 PErA, P4
Let $ABC$ be a triangle, and let $E$, $F$ be the feet of the altitudes from $B$ and $C$ to sides $AC$ and $AB$, respectively. Let $P$ and $Q$ be the intersections of $EF$ with the tangents from $B$ and $C$ to $(ABC)$, respectively. If $M$ is the midpoint of $BC$, prove that $(PQM)$ is tangent to $BC$ at $M$.
2016 239 Open Mathematical Olympiad, 6
A graph is called $7-chip$ if it obtained by removing at most three edges that have no vertex in common from a complete graph with seven vertices. Consider a complete graph $G$ with $v$ vertices which each edge of its is colored blue or red. Prove that there is either a blue path with $100$ edges or a red $7-chip$.
2011 National Olympiad First Round, 17
Let $D$ be a point inside the equilateral triangle $\triangle ABC$ such that $|AD|=\sqrt{2}, |BD|=3, |CD|=\sqrt{5}$. $m(\widehat{ADB}) = ?$
$\textbf{(A)}\ 120^{\circ} \qquad\textbf{(B)}\ 105^{\circ} \qquad\textbf{(C)}\ 100^{\circ} \qquad\textbf{(D)}\ 95^{\circ} \qquad\textbf{(E)}\ 90^{\circ}$
2002 Junior Balkan Team Selection Tests - Romania, 3
Let $ABC$ be a triangle and $a = BC, b = CA$ and $c = AB$ be the lengths of its sides. Points $D$ and $E$ lie in the same halfplane determined by $BC$ as $A$. Suppose that $DB = c, CE = b$ and that the area of $DECB$ is maximal. Let $F$ be the midpoint of $DE$ and let $FB = x$. Prove that $FC = x$ and $4x^3 = (a^2+b^2 + c^2)x + abc$.
2011 JBMO Shortlist, 8
Decipher the equality $(\overline{LARN} -\overline{ACA}) : (\overline{CYP} +\overline{RUS}) = C^{Y^P} \cdot R^{U^S} $ where different symbols correspond to different digits and equal symbols correspond to equal digits. It is also supposed that all these digits are different from $0$.
2008 Regional Competition For Advanced Students, 4
For every positive integer $ n$ let
\[ a_n\equal{}\sum_{k\equal{}n}^{2n}\frac{(2k\plus{}1)^n}{k}\]
Show that there exists no $ n$, for which $ a_n$ is a non-negative integer.
2013 Czech-Polish-Slovak Junior Match, 1
Determine all pairs $(x, y)$ of integers for which satisfy the equality $\sqrt{x-\sqrt{y}}+ \sqrt{x+\sqrt{y}}= \sqrt{xy}$
2003 AIME Problems, 13
A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$
2021 AMC 10 Fall, 23
For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$
$\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11$
2000 Tournament Of Towns, 5
What is the largest number $N$ for which there exist $N$ consecutive positive integers such that the sum of the digits in the $k$-th integer is divisible by $k$ for $1 \le k \le N$ ?
(S Tokarev)
2023 Belarusian National Olympiad, 11.2
On a blackboard triangle $ABC$ is drawn. Vlad draws a random point $D$ inside it and then reflects $A,C,B$ across the midpoints of $CD, BD, AD$, gets $C_1, A_1, B_1$. When Vlad wasn't looking at the board, Dima deleted from it everything, except for $A_1,B_1,C_1$.
Can Vlad now using only chalk, ruler and compass draw the original point $D$?
2025 Euler Olympiad, Round 1, 4
Given any arc $AB$ on a circle and points $C$ and $D$ on segment $AB$, such that $$CD = DB = 2AC.$$ Find the ratio $\frac{CM}{MD}$, where $M$ is a point on arc $AB$, such that $\angle CMD$ is maximized.
[img]https://i.imgur.com/NfjRpgP.png[/img]
[i]
Proposed by Andria Gvaramia, Georgia [/i]
2008 Romania National Olympiad, 1
Let $ f : (0,\infty) \to \mathbb R$ be a continous function such that the sequences $ \{f(nx)\}_{n\geq 1}$ are nondecreasing for any real number $ x$. Prove that $ f$ is nondecreasing.
1997 Tournament Of Towns, (543) 4
A convex polygon $G$ is placed inside a convex polygon $ F$ so that their boundaries have no common points. A segment $s$ joining two points on the boundary of $F$ is called a support chord for $G$ if s contains a side or only a vertex of $G$. Prove that
(a) there exists a support chord for $G$ such that its midpoint lies on the boundary of $G$,
(b) there exist at least two such chords.
(P Pushkar)
2014 India Regional Mathematical Olympiad, 1
let $ABCD$ be a isosceles trapezium having an incircle with $AB$ parallel to $CD$.
let $CE$ be the perpendicular from $C$ on $AB$
prove that
$ CE^2 = AB. CD $
1985 AMC 12/AHSME, 18
Six bags of marbles contain $ 18$, $ 19$, $ 21$, $ 23$, $ 25$, and $ 34$ marbles, respectively. One bag contains chipped marbles only. The other $ 5$ bags contain no chipped marbles. Jane takes three of the bags and George takes two of the others. Only the bag of chipped marbles remains. If Jane gets twice as many marbles as George, how many chipped marbles are there?
$ \textbf{(A)}\ 18 \qquad \textbf{(B)}\ 19 \qquad \textbf{(C)}\ 21 \qquad \textbf{(D)}\ 23 \qquad \textbf{(E)}\ 25$