Found problems: 85335
2017 Denmark MO - Mohr Contest, 5
In a chess tournament, each pair of players play one game. A lost game yields 0 points, a won game yields 1 point and a tied game yields $\frac12$ point. After the tournament, it turns out that in each group of three players, at least one got $1 \frac12$ points in the games against the two others. What is the largest number of players that may have participated?
1969 AMC 12/AHSME, 29
If $x=t^{(1/(t-1))}$ and $x=t^{(t/(t-1))}$, $t>0$, $t\not=1$, a relation between $x$ and $y$ is
$\textbf{(A)}\ y^x=x^{1/y}\qquad
\textbf{(B)}\ y^{1/x}=x^{y} \qquad
\textbf{(C)}\ y^x=x^{y}\qquad
\textbf{(D)}\ x^x=y^y\\
\textbf{(E)}\ \text{none of these}$
2018 Bulgaria JBMO TST, 1
For real numbers $a$ and $b$, define
$$f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}.$$
Find the smallest possible value of the expression $$f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b).$$
2021 China Second Round A2, 2
Find the maximum value of $M$, if you choose $10$ different real numbers randomly in $[1,M]$, there must be $3$ numbers $a<b<c$, satisfy $ax^2+bx+c=0$ has no real root.
2023 239 Open Mathematical Olympiad, 1
Each cell of an $100\times 100$ board is divided into two triangles by drawing some diagonal. What is the smallest number of colors in which it is always possible to paint these triangles so that any two triangles having a common side or vertex have different colors?
2014 AMC 10, 22
Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?
[asy]
scale(200);
draw(scale(.5)*((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle));
path p = arc((.25,-.5),.25,0,180)--arc((-.25,-.5),.25,0,180);
draw(p);
p=rotate(90)*p; draw(p);
p=rotate(90)*p; draw(p);
p=rotate(90)*p; draw(p);
draw(scale((sqrt(5)-1)/4)*unitcircle);
[/asy]
$\text{(A) } \dfrac{1+\sqrt2}4 \quad \text{(B) } \dfrac{\sqrt5-1}2 \quad \text{(C) } \dfrac{\sqrt3+1}4 \quad \text{(D) } \dfrac{2\sqrt3}5 \quad \text{(E) } \dfrac{\sqrt5}3$
2017-IMOC, A6
Show that for all positive reals $a,b,c$ with $a+b+c=3$,
$$\sum_{\text{cyc}}\sqrt{a+3b+\frac2c}\ge3\sqrt6.$$
Kvant 2023, M2749
We have $n{}$ coins, one of which is fake, which differs in weight from the real ones and a two-pan scale which works correctly if the weights on the pans are different, but can show any outcome if the weights on the pans are equal. For what $n{}$ can we determine which coin is fake and whether it is lighter or heavier than the real coins, in at most $k{}$ weightings?
[i]Proposed by A. Zaslavsky[/i]
2022 CCA Math Bonanza, TB4
Let $f(x)$ be a function such that $f(1) = 1234$, $f(2)=1800$, and $f(x) = f(x-1) + 2f(x-2)-1$ for all integers $x$. Evaluate the number of divisors of
\[\sum_{i=1}^{2022}f(i)\]
[i]2022 CCA Math Bonanza Tiebreaker Round #4[/i]
2025 Kyiv City MO Round 2, Problem 4
Point \( A_1 \) inside the acute-angled triangle \( ABC \) is such that
\[
\angle ACB = 2\angle A_1BC \quad \text{and} \quad \angle ABC = 2\angle A_1CB.
\]
Point \( A_2 \) is chosen so that points \( A \) and \( A_2 \) lie on opposite sides of line \( BC \), \( AA_2 \perp BC \), and the perpendicular bisector of \( AA_2 \) is tangent to the circumcircle of \( \triangle ABC \). Define points \( B_1, B_2, C_1, C_2 \) analogously. Prove that the circumcircles of \( \triangle AA_1A_2 \), \( \triangle BB_1B_2 \), and \( \triangle CC_1C_2 \) intersect at exactly two common points.
[i]Proposed by Vadym Solomka[/i]
2013 Romania Team Selection Test, 2
Let $K$ be a convex quadrangle and let $l$ be a line through the point of intersection of the diagonals of $K$. Show that the length of the segment of intersection $l\cap K$ does not exceed the length of (at least) one of the diagonals of $K$.
2016 NIMO Problems, 6
Let $S$ be the sum of all positive integers that can be expressed in the form $2^a \cdot 3^b \cdot 5^c$, where $a$, $b$, $c$ are positive integers that satisfy $a+b+c=10$. Find the remainder when $S$ is divided by $1001$.
[i]Proposed by Michael Ren[/i]
2005 Kyiv Mathematical Festival, 3
Two players by turn paint the vertices of triangles on the given picture each with his colour. At the end, each of small triangles is painted by the colour of the majority of its vertices. The winner is one who gets at least 6 triangles of his colour. If both players get at most 5, then it is a draw. Does any of them have winning strategy? If yes, then who wins?
\[ \begin{picture}(40,50) \put(2,2){\put(0,0){\line(6,0){42}} \put(7,14){\line(6,0){28}} \put(14,28){\line(6,0){14}} \put(0,0){\line(1,2){21}} \put(14,0){\line(1,2){14}} \put(28,0){\line(1,2){7}} \put(14,28){\line(1,2){7}} \put(14,0){\line( \minus{} 1,2){7}} \put(28,0){\line( \minus{} 1,2){14}} \put(42,0){\line( \minus{} 1,2){21}} \put(0,0){\circle*{3}} \put(14,0){\circle*{3}} \put(28,0){\circle*{3}} \put(42,0){\circle*{3}} \put(7,14){\circle*{3}} \put(21,14){\circle*{3}} \put(35,14){\circle*{3}} \put(14,28){\circle*{3}} \put(28,28){\circle*{3}} \put(21,42){\circle*{3}}} \end{picture}\]
2020/2021 Tournament of Towns, P4
There are two round tables with $n{}$ dwarves sitting at each table. Each dwarf has only two friends: his neighbours to the left and to the right. A good wizard wants to seat the dwarves at one round table so that each two neighbours are friends. His magic allows him to make any $2n$ pairs of dwarves into pairs of friends (the dwarves in a pair may be from the same or from different tables). However, he knows that an evil sorcerer will break $n{}$ of those new friendships. For which $n{}$ is the good wizard able to achieve his goal no matter what the evil sorcerer does?
[i]Mikhail Svyatlovskiy[/i]
2018 AMC 8, 5
What is the value of $1+3+5+\cdots+2017+2019-2-4-6-\cdots-2016-2018$?
$\textbf{(A) }-1010\qquad\textbf{(B) }-1009\qquad\textbf{(C) }1008\qquad\textbf{(D) }1009\qquad \textbf{(E) }1010$
2019 Teodor Topan, 1
Solve in the natural numbers the equation $ \log_{6n-19} (n!+1) =2. $
[i]Dragoș Crișan[/i]
2020 Serbian Mathematical Olympiad, Problem 3
We are given a triangle $ABC$. Points $D$ and $E$ on the line $AB$ are such that $AD=AC$ and $BE=BC$, with the arrangment of points $D - A - B - E$. The circumscribed circles of the triangles $DBC$ and $EAC$ meet again at the point $X\neq C$, and the circumscribed circles of the triangles $DEC$ and $ABC$ meet again at the point $Y\neq C$. Find the measure of $\angle ACB$ given the condition $DY+EY=2XY$.
2017 Peru MO (ONEM), 4
Let $A, B, C, D$ be points in a line $l$ in this order where $AB = BC$ and $AC = CD$. Let $w$ be a circle that passes in the points $B$ and $D$, a line that passes by $A$ intersects $w$ in the points $P$ and $Q$(the point $Q$ is in the segment $AP$). Let $M$ be the midpoint of $PD$ and $R$ is the symmetric of $Q$ by the line $l$, suppose that the segments $PR$ and $MB$ intersect in the point $N$. Prove that the quadrilateral $PMNC$ is cyclic
2013-2014 SDML (High School), 5
How many ways are there to make two $3$-digit numbers $m$ and $n$ such that $n=3m$ and each of six digits $1$, $2$, $3$, $6$, $7$, $8$ are used exactly once?
2015 IFYM, Sozopol, 2
On the VI-th International Festival of Young Mathematicians in Sozopol $n$ teams were participating, each of which was with $k$ participants ($n>k>1$). The organizers of the competition separated the $nk$ participants into $n$ groups, each with $k$ people, in such way that no two teammates are in the same group. Prove that there can be found $n$ participants no two of which are in the same team or group.
2024 Bulgaria National Olympiad, 3
Find all functions $f:\mathbb {R}^{+} \rightarrow \mathbb{R}^{+}$, such that $$f(af(b)+a)(f(bf(a))+a)=1$$ for any positive reals $a, b$.
2016 Sharygin Geometry Olympiad, 1
An altitude $AH$ of triangle $ABC$ bisects a median $BM$. Prove that the medians of triangle $ABM$ are sidelengths of a right-angled triangle.
by Yu.Blinkov
2009 Purple Comet Problems, 22
The diagram shows a parabola, a line perpendicular to the parabola's axis of symmetry, and three similar isosceles triangles each with a base on the line and vertex on the parabola. The two smaller triangles are congruent and each have one base vertex on the parabola and one base vertex shared with the larger triangle. The ratio of the height of the larger triangle to the height of the smaller triangles is $\tfrac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers, and $a$ and $c$ are relatively prime. Find $a + b + c$.
[asy]
size(200);
real f(real x) {return 1.2*exp(2/3*log(16-x^2));}
path Q=graph(f,-3.99999,3.99999);
path [] P={(-4,0)--(-2,0)--(-3,f(-3))--cycle,(-2,0)--(2,0)--(0,f(0))--cycle,(4,0)--(2,0)--(3,f(3))--cycle};
for(int k=0;k<3;++k)
{
fill(P[k],grey); draw(P[k]);
}
draw((-6,0)--(6,0),linewidth(1)); draw(Q,linewidth(1));[/asy]
2016 IFYM, Sozopol, 4
Prove that for each $n\geq 3$ the equation: $x^n+y^n+z^n+u^n=v^{n-1}$ has infinitely many solutions in natural numbers.
2011 QEDMO 10th, 1
Find all functions $f: R\to R$ with the property that $xf (y) + yf (x) = (x + y) f (xy)$ for all $x, y \in R$.