Found problems: 85335
2013 Balkan MO Shortlist, A6
Let $S$ be the set of positive real numbers. Find all functions $f\colon S^3 \to S$ such that, for all positive real numbers $x$, $y$, $z$ and $k$, the following three conditions are satisfied:
(a) $xf(x,y,z) = zf(z,y,x)$,
(b) $f(x, ky, k^2z) = kf(x,y,z)$,
(c) $f(1, k, k+1) = k+1$.
([i]United Kingdom[/i])
2006 Lithuania National Olympiad, 1
Solve the system of equations:
$\left\{ \begin{aligned} x^4+y^2-xy^3-\frac{9}{8}x = 0 \\ y^4+x^2-yx^3-\frac{9}{8}y=0 \end{aligned} \right.$
2025 Turkey EGMO TST, 4
Find all positive integers $n$ such that the number
\[
\frac{3 + \sqrt{4n + 9}}{2}
\]
is the sixth smallest positive divisor of $n$.
2018 Brazil Team Selection Test, 1
Let $n \ge 1$ be an integer. For each subset $S \subset \{1, 2, \ldots , 3n\}$, let $f(S)$ be the sum of the elements of $S$, with $f(\emptyset) = 0$. Determine, as a function of $n$, the sum $$\sum_{\mathclap{\substack{S \subset \{1,2,\ldots,3n\}\\
3 \mid f(S)}}} f(S)$$
where $S$ runs through all subsets of $\{1, 2,\ldots, 3n\}$ such that $f(S)$ is a multiple of $3$.
1999 Estonia National Olympiad, 2
Find the value of the integral $\int_{-1}^{1} ln \left(x +\sqrt{1 + x^2}\right) dx$.
2019 AMC 10, 3
In a high school with $500$ students, $40\%$ of the seniors play a musical instrument, while $30\%$ of the non-seniors do not play a musical instrument. In all, $46.8\%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument?
$\textbf{(A) } 66 \qquad\textbf{(B) } 154 \qquad\textbf{(C) } 186 \qquad\textbf{(D) } 220 \qquad\textbf{(E) } 266$
2010 Gheorghe Vranceanu, 1
Let be a number $ x $ and three positive numbers $ a,b,c $ such that $ a^x+b^x=c^x. $
Prove that $ a^y,b^y,c^y $ are the lenghts of the sides of an obtuse triangle if and only if $ y<x<2y. $
2005 Bosnia and Herzegovina Team Selection Test, 4
On the line which contains diameter $PQ$ of circle $k(S,r)$, point $A$ is chosen outside the circle such that tangent $t$ from point $A$ touches the circle in point $T$. Tangents on circle $k$ in points $P$ and $Q$ are $p$ and $q$, respectively. If $PT \cap q={N}$ and $QT \cap p={M}$, prove that points $A$, $M$ and $N$ are collinear.
2023 Purple Comet Problems, 16
A sequence of $28$ letters consists of $14$ of each of the letters $A$ and $B$ arranged in random order. The expected number of times that $ABBA$ appears as four consecutive letters in that sequence is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2014 ELMO Shortlist, 6
Let $ABCD$ be a cyclic quadrilateral with center $O$.
Suppose the circumcircles of triangles $AOB$ and $COD$ meet again at $G$, while the circumcircles of triangles $AOD$ and $BOC$ meet again at $H$.
Let $\omega_1$ denote the circle passing through $G$ as well as the feet of the perpendiculars from $G$ to $AB$ and $CD$.
Define $\omega_2$ analogously as the circle passing through $H$ and the feet of the perpendiculars from $H$ to $BC$ and $DA$.
Show that the midpoint of $GH$ lies on the radical axis of $\omega_1$ and $\omega_2$.
[i]Proposed by Yang Liu[/i]
2000 AMC 10, 19
Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the trangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is $m$ times the area of the square. The ratio of the area of the other small right triangle to the area of the square is
$\text{(A)}\ \frac1{2m+1}\qquad\text{(B)}\ m \qquad\text{(C)}\ 1-m\qquad\text{(D)}\ \frac1{4m} \qquad\text{(E)}\ \frac1{8m^2}$
1977 AMC 12/AHSME, 8
For every triple $(a,b,c)$ of non-zero real numbers, form the number \[ \frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}. \] The set of all numbers formed is
$\textbf{(A)}\ {0} \qquad
\textbf{(B)}\ \{-4,0,4\} \qquad
\textbf{(C)}\ \{-4,-2,0,2,4\} \qquad
\textbf{(D)}\ \{-4,-2,2,4\} \qquad
\textbf{(E)}\ \text{none of these}$
2016 India Regional Mathematical Olympiad, 3
Two circles $C_1$ and $C_2$ intersect each other at points $A$ and $B$. Their external common tangent (closer to $B$) touches $C_1$ at $P$ and $C_2$ at $Q$. Let $C$ be the reflection of $B$ in line $PQ$. Prove that $\angle CAP=\angle BAQ$.
2020 Online Math Open Problems, 12
At a party, there are $100$ cats. Each pair of cats flips a coin, and they shake paws if and only if the coin comes up heads. It is known that exactly $4900$ pairs of cats shook paws. After the party, each cat is independently assigned a ``happiness index" uniformly at random in the interval $[0,1]$. We say a cat is [i]practical[/i] if it has a happiness index that is strictly greater than the index of every cat with which it shook paws. The expected value of the number of practical cats is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m + n$.
[i]Proposed by Brandon Wang[/i]
2022 CMIMC, 2.8 1.4
Let $z$ be a complex number that satisfies the equation \[\frac{z-4}{z^2-5z+1} + \frac{2z-4}{2z^2-5z+1} + \frac{z-2}{z^2-3z+1} = \frac{3}{z}.\] Over all possible values of $z$, find the sum of the values of \[\left| \frac{1}{z^2-5z+1} + \frac{1}{2z^2-5z+1} + \frac{1}{z^2-3z+1} \right|.\]
[i]Proposed by Justin Hsieh[/i]
2008 Danube Mathematical Competition, 2
In a triangle $ABC$ let $A_1$ be the midpoint of side $BC$. Draw circles with centers $A, A1$ and radii $AA_1, BC$ respectively and let $A'A''$ be their common chord. Similarly denote the segments $B'B''$ and $C'C''$. Show that lines $A'A'', B'B'''$ and $C'C''$ are concurrent.
2013 AMC 10, 17
Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?
${ \textbf{(A)}\ 62 \qquad\textbf{(B)}\ 82 \qquad\textbf{(C)}\ 83\qquad\textbf{(D}}\ 102\qquad\textbf{(E)}\ 103 $
2003 All-Russian Olympiad Regional Round, 10.4
On the plane we mark $n$ ($n > 2$) straight lines passing through one point $O$ in such a way that for any two of them there is a marked straight line that bisects one of the pairs of vertical angles, formed by these straight lines. Prove that the drawn straight lines divide full angle into equal parts.
2012 IMC, 5
Let $a$ be a rational number and let $n$ be a positive integer. Prove that the polynomial $X^{2^n}(X+a)^{2^n}+1$ is irreducible in the ring $\mathbb{Q}[X]$ of polynomials with rational coefficients.
[i]Proposed by Vincent Jugé, École Polytechnique, Paris.[/i]
1994 AMC 12/AHSME, 9
If $\angle A$ is four times $\angle B$, and the complement of $\angle B$ is four times the complement of $\angle A$, then $\angle B=$
$ \textbf{(A)}\ 10^{\circ} \qquad\textbf{(B)}\ 12^{\circ} \qquad\textbf{(C)}\ 15^{\circ} \qquad\textbf{(D)}\ 18^{\circ} \qquad\textbf{(E)}\ 22.5^{\circ} $
2001 AMC 12/AHSME, 12
How many positive integers not exceeding 2001 are multiple of 3 or 4 but not 5?
$ \textbf{(A)} \ 768 \qquad \textbf{(B)} \ 801 \qquad \textbf{(C)} \ 934 \qquad \textbf{(D)} \ 1067 \qquad \textbf{(E)} \ 1167$
Russian TST 2018, P2
Inside the acute-angled triangle $ABC$, the points $P{}$ and $Q{}$ are chosen so that $\angle ACP = \angle BCQ$ and $\angle CBP =\angle ABQ$. The point $Z{}$ is the projection of $P{}$ onto the line $BC$. The point $Q'$ is symmetric to $Q{}$ with respect to $Z{}$. The points $K{}$ and $L{}$ are chosen on the rays $AB$ and $AC$ respectively, so that $Q'K \parallel QC$ and $Q'L \parallel QB$. Prove that $\angle KPL=\angle BPC$.
2001 Switzerland Team Selection Test, 1
The $2001 \times 2001$ trees in a park form a square grid. What is the largest number of trees that can be cut so that no tree stump can be seen from any other? (Each tree has zero width.)
1990 IMO Longlists, 39
Let $a, b, c$ be integers. Prove that there exist integers $p_1, q_1, r_1, p_2, q_2$ and $r_2$, satisfying $a = q_1r_2 - q_2r_1, b = r_1p_2 - r_2p_1$ and $c = p_1q_2 - p_2q_1.$
2024 Polish Junior MO Finals, 2
Determine the smallest integer $n \ge 1$ such that a $n \times n$ square can be cut into square pieces of size $1 \times 1$ and $2 \times 2$ with both types occuring the same number of times.