Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The probability that both marbles are black is $27/50,$ and the probability that both marbles are white is $m/n,$ where $m$ and $n$ are relatively prime positive integers. What is $m+n?$

Let $a, b, c$ be positive reals such that $4(a+b+c)\geq\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$. Define \begin{align*} &A =\sqrt{\frac{3a}{a+2\sqrt{bc}}}+\sqrt{\frac{3b}{b+2\sqrt{ca}}}+\sqrt{\frac{3c}{c+2\sqrt{ab}}} \\ &B =\sqrt{a}+\sqrt{b}+\sqrt{c} \\ &C =\frac{a}{\sqrt{a+b}}+\frac{b}{\sqrt{b+c}}+\frac{c}{\sqrt{c+a}}. \end{align*} Prove that $$A\leq 2B\leq 4C.$$

Fourteen friends met at a party. One of them, Fredek, wanted to go to bed early. He said goodbye to 10 of his friends, forgot about the remaining 3, and went to bed. After a while he returned to the party, said goodbye to 10 of his friends (not necessarily the same as before), and went to bed. Later Fredek came back a number of times, each time saying goodbye to exactly 10 of his friends, and then went back to bed. As soon as he had said goodbye to each of his friends at least once, he did not come back again. In the morning Fredek realized that he had said goodbye a different number of times to each of his thirteen friends! What is the smallest possible number of times that Fredek returned to the party?

There are $n$ line segments on the plane, no three intersecting at a point, and each pair intersecting once in their respective interiors. Tony and his $2n - 1$ friends each stand at a distinct endpoint of a line segment. Tony wishes to send Christmas presents to each of his friends as follows: First, he chooses an endpoint of each segment as a “sink”. Then he places the present at the endpoint of the segment he is at. The present moves as follows : $\bullet$ If it is on a line segment, it moves towards the sink. $\bullet$ When it reaches an intersection of two segments, it changes the line segment it travels on and starts moving towards the new sink. If the present reaches an endpoint, the friend on that endpoint can receive their present. Prove that Tony can send presents to exactly $n$ of his $2n - 1$ friends.

a) Does there exist a quadrilateral with (both of) the following properties: three of its edges are of the same length, but the fourth one is different, and three of its angles are equal, but the fourth one is different? b) Does there exist a pentagon with (both of) the following properties: four of its edges are of the same length, but the fifth one is different, and four of its angles are equal, but the fifth one is different?

The integers $n>1$ is given . The positive integer $a_1,a_2,\cdots,a_n$ satisfing condition : (1) $a_1<a_2<\cdots<a_n$; (2) $\frac{a^2_1+a^2_2}{2},\frac{a^2_2+a^2_3}{2},\cdots,\frac{a^2_{n-1}+a^2_n}{2}$ are all perfect squares . Prove that :$a_n\ge 2n^2-1.$

Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$

Three circles $\mathcal C_1(O_1)$, $\mathcal C_2(O_2)$ and $\mathcal C_3(O_3)$ share a common point and meet again pairwise at the points $A$, $B$ and $C$. Show that if the points $A$, $B$, $C$ are collinear then the points $Q$, $O_1$, $O_2$ and $O_3$ lie on the same circle.

A point $P$ lies in the interior of the triangle $ABC$. The lines $AP, BP$, and $CP$ intersect $BC, CA$, and $AB$ at points $D, E$, and $F$, respectively. Prove that if two of the quadrilaterals $ABDE, BCEF, CAFD, AEPF, BFPD$, and $CDPE$ are concyclic, then all six are concyclic.

Determine for which natural numbers $n$ it is possible to completely cover a board of $ n \times n$, divided into $1 \times 1$ squares, with pieces like the one in the figure, without gaps or overlays and without leaving the board. Each of the pieces covers exactly six boxes. Note: Parts can be rotated. [img]https://cdn.artofproblemsolving.com/attachments/c/2/d87d234b7f9799da873bebec845c721e4567f9.png[/img]

Find the number of pairs $(n,C)$ of positive integers such that $C\leq 100$ and $n^2+n+C$ is a perfect square.

Given a circle with diameter $AB$ and a point $X$ on the circle different from $A$ and $B$, let $t_{a}$, $t_{b}$ and $t_{x}$ be the tangents to the circle at $A$, $B$ and $X$ respectively. Let $Z$ be the point where line $AX$ meets $t_{b}$ and $Y$ the point where line $BX$ meets $t_{a}$. Show that the three lines $YZ$, $t_{x}$ and $AB$ are either concurrent (i.e., all pass through the same point) or parallel. [img]6762[/img]

Let $n,a,b,c$ be natural numbers. Every point on the coordinate plane with integer coordinates is colored in one of $n$ colors. Prove there exists $c$ triangles whose vertices are colored in the same color, which are pairwise congruent, and which have a side whose lenght is divisible by $a$ and a side whose lenght is divisible by $b$.

Two circles $\omega_1$ and $\omega_2$ with equal radius intersect at two points $E$ and $X$. Arbitrary points $C, D$ lie on $\omega_1, \omega_2$. Parallel lines to $XC, XD$ from $E$ intersect $\omega_2, \omega_1$ at $A, B$, respectively. Suppose that $CD$ intersect $\omega_1, \omega_2$ again at $P, Q$, respectively. Prove that $ABPQ$ is cyclic. [i]Proposed by Ali Zamani[/i]

A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon? $\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qquad\textbf{(D)}~\frac{\sqrt{5}+1}{2}\qquad\textbf{(E)}~\frac{2+\sqrt{5}}{3}$

Let $x_1$, $x_2$, …, $x_{10}$ be 10 numbers. Suppose that $x_i + 2 x_{i + 1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512 x_{10}$?

Find all functions $f : R \to R$ such that, for all real numbers $x, y,$ $$(x - y)(f(x) - f(y)) = f(x - f(y))f(f(x) - y).$$

It turned out for some triangle with sides $a, b$ and $c$, that a circle of radius $r = \frac{a+b+c}{2}$ touches side $c$ and extensions of sides $a$ and $b$. Prove that a circle of radius $ \frac{a+c-b}{2}$ is tangent to $a$ and the extensions of $b$ and $c$.

4. There are 5 tables in a classroom. Each table has 4 chairs with a child sitting on it. All the children get up and randomly sit in a seat. Two people that sat at the same table before are not allowed to sit at the same table again. Assuming tables and chairs are distinguishable, if the number of different classroom arrangements can be written as $2^a3^b5^c$, what is $a+b+c$? [i]Proposed by Tragic[/i]

Let be two $ 2\times 2 $ real matrices $A,B$ having the property that all their natural powers are not real multiples of the identity. Prove that if some natural power of $ A $ is equal to some natural power of $ B, $ then, $ A,B $ commute. Is the converse statement true? [i]Cosmin Nitu[/i]

Graph $G_1$ of a quadratic trinomial $y = px^2 + qx + r$ with real coefficients intersects the graph $G_2$ of a quadratic trinomial $y = x^2$ in points $A$, $B$. The intersection of tangents to $G_2$ in points $A$, $B$ is point $C$. If $C \in G_1$, find all possible values of $p$.

Find all real polynomials $ f$ with $ x,y \in \mathbb{R}$ such that \[ 2 y f(x \plus{} y) \plus{} (x \minus{} y)(f(x) \plus{} f(y)) \geq 0. \]

Let $f:\mathbb{Z}^2\rightarrow\mathbb{Z}$ be a function satisfying \[f(x+1,y)+f(x,y+1)+1=f(x,y)+f(x+1,y+1)\] for all integers $x$ and $y$. Can it happen that $|f(x,y)|\leq 2024$ for all $x,y\in\mathbb{Z}$?

Prove the following inequality. \[\int_{-1}^1 \frac{e^x+e^{-x}}{e^{e^{e^x}}}dx<e-\frac{1}{e}\] Own

For all non-zero numbers $x$ and $y$ such that $x = 1/y$, \[\left(x-\frac{1}{x}\right)\left(y+\frac{1}{y}\right)\] equals $\textbf{(A) }2x^2\qquad\textbf{(B) }2y^2\qquad\textbf{(C) }x^2+y^2\qquad\textbf{(D) }x^2-y^2\qquad \textbf{(E) }y^2-x^2$