Found problems: 85335
2001 IMO Shortlist, 8
Let $ABC$ be a triangle with $\angle BAC = 60^{\circ}$. Let $AP$ bisect $\angle BAC$ and let $BQ$ bisect $\angle ABC$, with $P$ on $BC$ and $Q$ on $AC$. If $AB + BP = AQ + QB$, what are the angles of the triangle?
2017 Pan-African Shortlist, G?
Let $ABC$ be a triangle with $H$ its orthocenter. The circle with diameter $[AC]$ cuts the circumcircle of triangle $ABH$ at $K$. Prove that the point of intersection of the lines $CK$ and $BH$ is the midpoint of the segment $[BH]$
2014 Singapore Senior Math Olympiad, 21
Let $n$ be an integer, and let $\triangle ABC$ be a right-angles triangle with right angle at $C$. It is given that $\sin A$ and $\sin B$ are the roots of the quadratic equation \[(5n+8)x^2-(7n-20)x+120=0.\] Find the value of $n$
2018 Malaysia National Olympiad, B3
Given $2018$ ones in a row: $$\underbrace{1\,\,\,1\,\,\,1\,\,\,1 \,\,\, ... \,\,\,1 \,\,\,1 \,\,\,1 \,\,\,1}_{2018 \,\,\, ones}$$
in which plus symbols $(+)$ are allowed to be inserted in between the ones. What is the maximum number of plus symbols $(+)$ that need to be inserted so that the resulting sum is 8102?
2010 Math Prize For Girls Problems, 2
Jane has two bags $X$ and $Y$. Bag $X$ contains 4 red marbles and 5 blue marbles (and nothing else). Bag $Y$ contains 7 red marbles and 6 blue marbles (and nothing else). Jane will choose one of her bags at random (each bag being equally likely). From her chosen bag, she will then select one of the marbles at random (each marble in that bag being equally likely). What is the probability that she will select a red marble?
1977 IMO Longlists, 34
Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.
1999 Singapore MO Open, 2
Call a natural number $n$ a [i]magic [/i] number if the number obtained by putting $n$ on the right of any natural number is divisible by $n$. Find the number of magic numbers less than $500$. Justify your answer
2011 Indonesia TST, 1
Let $Q^+$ denote the set of positive rationals. Determine all functions $f : Q^+ \to Q^+$ that satisfy both of these conditions:
(i) $f(x)$ is an integer if and only if $x$ is an integer;
(ii) $f(f(xf(y)) + x) = yf(x) + x$ for all $x, y \in Q^+$.
2012 Federal Competition For Advanced Students, Part 2, 3
We call an isosceles trapezoid $PQRS$ [i]interesting[/i], if it is inscribed in the unit square $ABCD$ in such a way, that on every side of the square lies exactly one vertex of the trapezoid and that the lines connecting the midpoints of two adjacent sides of the trapezoid are parallel to the sides of the square.
Find all interesting isosceles trapezoids and their areas.
2013 AIME Problems, 3
A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn down the $k$-th centimeter. Suppose it takes $T$ seconds for the candle to burn down completely. Then $\tfrac{T}{2}$ seconds after it is lit, the candle's height in centimeters will be $h$. Find $10h$.
1918 Eotvos Mathematical Competition, 2
Find three distinct natural numbers such that the sum of their reciprocals is an integer.
2009 Indonesia TST, 1
Let $ x_1,x_2,\dots,x_n$ be positive real numbers. Let $ m\equal{}\min\{x_1,x_2,\dots,x_n\}$, $ M\equal{}\max\{x_1,x_2,\dots,x_n\}$, $ A\equal{}\frac{1}{n}(x_1\plus{}x_2\plus{}\dots\plus{}x_n)$, and $ G\equal{}\sqrt[n]{x_1x_2 \dots x_n}$. Prove that \[ A\minus{}G \ge \frac{1}{n}(\sqrt{M}\minus{}\sqrt{m})^2.\]
2022 239 Open Mathematical Olympiad, 1
A piece is placed in the lower left-corner cell of the $15 \times 15$ board. It can move to the cells that are adjacent to the sides or the corners of its current cell. It must also alternate between horizontal and diagonal moves $($the first move must be diagonal$).$ What is the maximum number of moves it can make without stepping on the same cell twice$?$
1935 Moscow Mathematical Olympiad, 008
Prove that if the lengths of the sides of a triangle form an arithmetic progression, then the radius of the inscribed circle is one third of one of the heights of the triangle.
2021 Iran MO (3rd Round), 2
Given an acute triangle $ABC$, let $AD$ be an altitude and $H$ the orthocenter. Let $E$ denote the reflection of $H$ with respect to $A$. Point $X$ is chosen on the circumcircle of triangle $BDE$ such that $AC\| DX$ and point $Y$ is chosen on the circumcircle of triangle $CDE$ such that $DY\| AB$. Prove that the circumcircle of triangle $AXY$ is tangent to that of $ABC$.
2014 IPhOO, 3
Which of the following derived units is equivalent to units of velocity?
$ \textbf {(A) } \dfrac {\text {W}}{\text {N}} \qquad \textbf {(B) } \dfrac {\text {N}}{\text {W}} \qquad \textbf {(C) } \dfrac {\text {W}}{\text {N}^2} \qquad \textbf {(D) } \dfrac {\text {W}^2}{\text {N}} \qquad \textbf {(E) } \dfrac {\text {N}^2}{\text {W}^2} $
[i]Problem proposed by Ahaan Rungta[/i]
VMEO IV 2015, 12.3
Triangle $ABC$ is inscribed in circle $(O)$. $ P$ is a point on arc $BC$ that does not contain $ A$ such that $AP$ is the symmedian of triangle $ABC$. $E ,F$ are symmetric of $P$ wrt $CA, AB$ respectively . $K$ is symmetric of $A$ wrt $EF$. $L$ is the projection of $K$ on the line passing through $A$ and parallel to $BC$. Prove that $PA=PL$.
2009 All-Russian Olympiad Regional Round, 11.1
Square trinomial $f(x)$ is such that the polynomial (f(x))^5 - f(x) has exactly three real roots. Find the ordinate of the vertex of the graph of this trinomial.
1981 Tournament Of Towns, (008) 2
$M$ is a finite set of points in a plane. Point $O$ in the plane is called an “almost centre of symmetry” of set $M$ if it is possible to remove from $M$ one point in such a way that among the remaining members $O$ is the centre of symmetry in the usual sense. How many such “almost centres of symmetry” may a finite point set in a plane have? Indicate all such points.
(V Prasolov, Moscow)
2000 Belarus Team Selection Test, 2.2
Real numbers $a$, $b$, $c$ satisfy the equation $$2a^3-b^3+2c^3-6a^2b+3ab^2-3ac^2-3bc^2+6abc=0$$. If $a<b$, find which of the numbers $b$, $c$ is larger.
2008 National Chemistry Olympiad, 9
How many moles of oxygen gas are produced by the decomposition of $245$ g of potassium chlorate?
\[\ce{2KClO3(s)} \rightarrow \ce{2KCl(s)} + \ce{3O2(g)}\]
Given:
Molar Mass/ $\text{g} \cdot \text{mol}^{-1}$
$\ce{KClO3}$: $122.6$
$ \textbf{(A)}\hspace{.05in}1.50 \qquad\textbf{(B)}\hspace{.05in}2.00 \qquad\textbf{(C)}\hspace{.05in}2.50 \qquad\textbf{(D)}\hspace{.05in}3.00 \qquad $
2025 Taiwan TST Round 1, N
Find all positive integers $n$ such that there exist two permutations $a_0,a_1,\ldots,a_{n-1}$ and $b_0,b_1,\ldots,b_{n-1}$ of the set $\lbrace0,1,\ldots,n-1\rbrace$, satisfying the condition
$$ia_i\equiv b_i\pmod{n}$$
for all $0\le i\le n-1$.
[i]Proposed by Fysty[/i]
2023 Francophone Mathematical Olympiad, 3
Let $\Gamma$ and $\Gamma'$ be two circles with centres $O$ and $O'$, such that $O$ belongs to $\Gamma'$. Let $M$ be a point on $\Gamma'$, outside of $\Gamma$. The tangents to $\Gamma$ that go through $M$ touch $\Gamma$ in two points $A$ and $B$, and cross $\Gamma'$ again in two points $C$ and $D$. Finally, let $E$ be the crossing point of the lines $AB$ and $CD$. Prove that the circumcircles of the triangles $CEO'$ and $DEO'$ are tangent to $\Gamma'$.
1990 Baltic Way, 1
Numbers $1, 2, \dots , n$ are written around a circle in some order. What is the smallest possible sum of the absolute differences of adjacent numbers?
2011 Estonia Team Selection Test, 1
Two circles lie completely outside each other.Let $A$ be the point of intersection of internal common tangents of the circles and let $K$ be the projection of this point onto one of their external common tangents.The tangents,different from the common tangent,to the circles through point $K$ meet the circles at $M_1$ and $M_2$.Prove that the line $AK$ bisects angle $M_1 KM_2$.