Found problems: 85335
2025 AIME, 5
Suppose $\triangle ABC$ has angles $\angle BAC = 84^\circ, \angle ABC=60^\circ,$ and $\angle ACB = 36^\circ.$ Let $D, E,$ and $F$ be the midpoints of sides $\overline{BC}, \overline{AC},$ and $\overline{AB},$ respectively. The circumcircle of $\triangle DEF$ intersects $\overline{BD}, \overline{AE},$ and $\overline{AF}$ at points $G, H,$ and $J,$ respectively. The points $G, D, E, H, J,$ and $F$ divide the circumcircle of $\triangle DEF$ into six minor arcs, as shown. Find $\overarc{DE}+2\cdot \overarc{HJ} + 3\cdot \overarc{FG},$ where the arcs are measured in degrees.
[asy]
import olympiad;
size(6cm);
defaultpen(fontsize(10pt));
pair B = (0, 0), A = (Cos(60), Sin(60)), C = (Cos(60)+Sin(60)/Tan(36), 0), D = midpoint(B--C), E = midpoint(A--C), F = midpoint(A--B);
guide circ = circumcircle(D, E, F);
pair G = intersectionpoint(B--D, circ), J = intersectionpoints(A--F, circ)[0], H = intersectionpoints(A--E, circ)[0];
draw(B--A--C--cycle);
draw(D--E--F--cycle);
draw(circ);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(H);
dot(J);
label("$A$", A, (0, .8));
label("$B$", B, (-.8, -.8));
label("$C$", C, (.8, -.8));
label("$D$", D, (0, -.8));
label("$E$", E, (.8, .2));
label("$F$", F, (-.8, .2));
label("$G$", G, (0, .8));
label("$H$", H, (-.2, -1));
label("$J$", J, (.2, -.8));
[/asy]
2011 Sharygin Geometry Olympiad, 7
Let a point $M$ not lying on coordinates axes be given. Points $Q$ and $P$ move along $Y$ - and $X$-axis respectively so that angle $P M Q$ is always right. Find the locus of points symmetric to $M$ wrt $P Q$.
2015 Sharygin Geometry Olympiad, 7
Let $ABC$ be an acute-angled, nonisosceles triangle. Altitudes $AA'$ and $BB' $meet at point $H$, and the medians of triangle $AHB$ meet at point $M$. Line $CM$ bisects segment $A'B'$. Find angle $C$.
(D. Krekov)
Russian TST 2017, P3
There are $n \geq 3$ islands in a city. Initially, the ferry company offers some routes between some pairs of islands so that it is impossible to divide the islands into two groups such that no two islands in different groups are connected by a ferry route.
After each year, the ferry company will close a ferry route between some two islands $X$ and $Y$. At the same time, in order to maintain its service, the company will open new routes according to the following rule: for any island which is connected to a ferry route to exactly one of $X$ and $Y$, a new route between this island and the other of $X$ and $Y$ is added.
Suppose at any moment, if we partition all islands into two nonempty groups in any way, then it is known that the ferry company will close a certain route connecting two islands from the two groups after some years. Prove that after some years there will be an island which is connected to all other islands by ferry routes.
2010 Serbia National Math Olympiad, 2
In an acute-angled triangle $ABC$, $M$ is the midpoint of side $BC$, and $D, E$ and $F$ the feet of the altitudes from $A, B$ and $C$, respectively. Let $H$ be the orthocenter of $\Delta ABC$, $S$ the midpoint of $AH$, and $G$ the intersection of $FE$ and $AH$. If $N$ is the intersection of the median $AM$ and the circumcircle of $\Delta BCH$, prove that $\angle HMA = \angle GNS$.
[i]Proposed by Marko Djikic[/i]
2022 HMIC, 3
For a nonnegative integer $n$, let $s(n)$ be the sum of the digits of the binary representation of $n$. Prove that
$$\sum_{n=1}^{2^{2022}-1} \frac{(-1)^{s(n)}}{n+2022}>0.$$
1993 Greece National Olympiad, 5
Let $P_0(x) = x^3 + 313x^2 - 77x - 8$. For integers $n \ge 1$, define $P_n(x) = P_{n - 1}(x - n)$. What is the coefficient of $x$ in $P_{20}(x)$?
2007 Romania Team Selection Test, 1
For $n\in\mathbb{N}$, $n\geq 2$, $a_{i}, b_{i}\in\mathbb{R}$, $1\leq i\leq n$, such that \[\sum_{i=1}^{n}a_{i}^{2}=\sum_{i=1}^{n}b_{i}^{2}=1, \sum_{i=1}^{n}a_{i}b_{i}=0. \] Prove that
\[\left(\sum_{i=1}^{n}a_{i}\right)^{2}+\left(\sum_{i=1}^{n}b_{i}\right)^{2}\leq n. \]
[i]Cezar Lupu & Tudorel Lupu[/i]
2012 Online Math Open Problems, 39
For positive integers $n,$ let $\nu_3 (n)$ denote the largest integer $k$ such that $3^k$ divides $n.$ Find the number of subsets $S$ (possibly containing 0 or 1 elements) of $\{1, 2, \ldots, 81\}$ such that for any distinct $a,b \in S$, $\nu_3 (a-b)$ is even.
[i]Author: Alex Zhu[/i]
[hide="Clarification"]We only need $\nu_3(a-b)$ to be even for $a>b$. [/hide]
1999 Korea Junior Math Olympiad, 2
Three integers are given. $A$ denotes the sum of the integers, $B$ denotes the sum of the square of the integers and $C$ denotes the sum of cubes of the integers(that is, if the three integers are $x, y, z$, then $A=x+y+z$, $B=x^2+y^2+z^2$, $C=x^3+y^3+z^3$). If $9A \geq B+60$ and $C \geq 360$, find $A, B, C$.
2014 Cezar Ivănescu, 1
[b]a)[/b] Let be three natural numbers, $ a>b\ge 3\le 3n, $ such that $ b^n|a^n-1. $ Prove that $ a^b>2^n. $
[b]b)[/b] Does there exist positive real numbers $ m $ which have the property that $ \log_8 (1+3\sqrt x) =\log_{27} (mx) $ if and only if $ 2^{x} +2^{1/x}\le 4? $
2005 Cono Sur Olympiad, 1
Let $a_n$ be the last digit of the sum of the digits of $20052005...2005$, where the $2005$ block occurs $n$ times. Find $a_1 +a_2 + \dots +a_{2005}$.
2017 Middle European Mathematical Olympiad, 1
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying
$$f(x^2 + f(x)f(y)) = xf(x + y)$$
for all real numbers $x$ and $y$.
2019 BAMO, A
Let $a$ and $b$ be positive whole numbers such that $\frac{4.5}{11}<\frac{a}{b}<\frac{5}{11}$.
Find the fraction $\frac{a}{b}$ for which the sum $a+b$ is as small as possible.
Justify your answer
2009 Today's Calculation Of Integral, 428
Let $ f(x)$ be a polynomial and $ C$ be a real number.
Find the $ f(x)$ and $ C$ such that $ \int_0^x f(y)dy\plus{}\int_0^1 (x\plus{}y)^2f(y)dy\equal{}x^2\plus{}C$.
2016 ASDAN Math Tournament, 6
A container is filled with a total of $51$ red and white balls and has at least $1$ red ball and $1$ white ball. The probability of picking up $3$ red balls and $1$ white ball, without replacement, is equivalent to the probability of picking up $1$ red ball and $2$ white balls, without replacement. Compute the original number of red balls in the container.
2017 BMT Spring, 1
You have $9$ colors of socks and $5$ socks of each type of color. Pick two socks randomly. What is the probability that they are the same color?
2017 CMIMC Team, 9
Circles $\omega_1$ and $\omega_2$ are externally tangent to each other. Circle $\Omega$ is placed such that $\omega_1$ is internally tangent to $\Omega$ at $X$ while $\omega_2$ is internally tangent to $\Omega$ at $Y$. Line $\ell$ is tangent to $\omega_1$ at $P$ and $\omega_2$ at $Q$ and furthermore intersects $\Omega$ at points $A$ and $B$ with $AP<AQ$. Suppose that $AP=2$, $PQ=4$, and $QB=3$. Compute the length of line segment $\overline{XY}$.
1986 Tournament Of Towns, (118) 6
Given the nonincreasing sequence of non-negative numbers in which $a_1 \ge a_2 \ge a_3 \ge ... \ge a_{2n-1}\ge 0$.
Prove that $a^2_1 -a^2_2 + a^2_3 - ... + a^2_{2n- l} \ge (a_1 - a_2 + a_3 - ... + a_{2n- l} )^2$ .
( L . Kurlyandchik , Leningrad )
2004 Denmark MO - Mohr Contest, 2
Show that if $a$ and $b$ are integer numbers, and $a^2 + b^2 + 9ab$ is divisible by $11$, then $a^2-b^2$ divisible by $11$.
2022 Durer Math Competition Finals, 11
In rectangle $ABCD$, diagonal $AC$ is met by the angle bisector from $B$ at $B'$ and the angle bisector from $D$ at $D'$. Diagonal $BD$ is met by the angle bisector from $A$ at $A'$ and the angle bisector from $C$ at $C'$. The area of quadrilateral $A'B'C'D'$ is $\frac{9}{16}$ the area of rectangle $ABCD$. What is the ratio of the longer side and shorter side of rectangle $ABCD$?
2008 South africa National Olympiad, 4
A pack of $2008$ cards, numbered from $1$ to $2008$, is shuffled in order to play a game in which each move has two steps:
(i) the top card is placed at the bottom;
(ii) the new top card is removed.
It turns out that the cards are removed in the order $1,2,\dots,2008$. Which card was at the top before the game started?
2022 VN Math Olympiad For High School Students, Problem 3
Let $ABC$ be a triangle with $\angle A,\angle B,\angle C <120^{\circ}$, $T$ is its [i]Fermat-Torricelli[/i] point.
Consider a point $P$ lying on the same plane with $\triangle ABC$. Prove that:
a)$$\dfrac{\overrightarrow {TA}}{TA}+\dfrac{\overrightarrow {TB}}{TB}+\dfrac{\overrightarrow {TC}}{TC}=\overrightarrow {0}.$$
b)$$PA + PB + PC \ge \frac{{\overrightarrow {PA} \overrightarrow {.TA} }}{{TA}} + \frac{{\overrightarrow {PB} .\overrightarrow {TB} }}{{TB}} + \frac{{\overrightarrow {PC} \overrightarrow {.TC} }}{{TC}}.$$
c)$$PA + PB + PC \ge TA + TB + TC$$and the equality occurs iff $P\equiv T$.
1995 AMC 12/AHSME, 10
The area of the triangle bounded by the lines $y = x, y = -x$ and $y = 6$ is
$
\mathbf{(A)}\; 12\qquad
\mathbf{(B)}\; 12\sqrt2\qquad
\mathbf{(C)}\; 24\qquad
\mathbf{(D)}\; 24\sqrt2\qquad
\mathbf{(E)}\; 36$
2019 Iranian Geometry Olympiad, 4
Quadrilateral $ABCD$ is given such that
$$\angle DAC = \angle CAB = 60^\circ,$$
and
$$AB = BD - AC.$$
Lines $AB$ and $CD$ intersect each other at point $E$. Prove that \[
\angle ADB = 2\angle BEC.
\]
[i]Proposed by Iman Maghsoudi[/i]