Let $C_1$ be an arbitrary point on the side $AB$ of triangle $ABC$. Points $A_1$ and $B_1$ on the rays $BC$ and $AC$ are such that $\angle AC_1B_1 = \angle BC_1A_1 = \angle ACB$. The lines $AA_1$ and $BB_1$ meet in point $C_2$. Prove that all the lines $C_1C_2$ have a common point.

In a circle $S$, a chord $AB$ is drawn and let $M$ be the midpoint of the arc $AB$. Let $P$ be a point in segment $AB$ other than its midpoint. The extension of the segment $MP$ cuts $S$ in $Q$. Let $S_1$ be the circle that is tangent to the AP segments and $MP$, and also is tangent to $S$, and let $S_2$ be the circle that is tangent to the segments $BP$ and $MP$, and also tangent to $S$. The common outer tangent lines to the circles $S_1$ and $S_2$ are cut at $C$. Prove that $\angle MQC = 90^o$.

What is the sum of all primes $p$ such that $7^p - 6^p + 2$ is divisible by 43?

Find all positive integers $m$ and $n$ such that $n^m - m$ divides $m^2 + 2m$.

The roots of the equation \[ x^3-3ax^2+bx+18c=0 \] form a non-constant arithmetic progression and the roots of the equation \[ x^3+bx^2+x-c^3=0 \] form a non-constant geometric progression. Given that $a,b,c$ are real numbers, find all positive integral values $a$ and $b$.

Let $ABC$ be a triangle with $AB = 5$, $BC = 6$, and $AC = 7$. Let its orthocenter be $H$ and the feet of the altitudes from $A, B, C$ to the opposite sides be $D, E, F$ respectively. Let the line $DF$ intersect the circumcircle of $AHF$ again at $X$. Find the length of $EX$.

Let $ABCD$ be a circumscribed quadrilateral. Its incircle $\omega$ touches the sides $BC$ and $DA$ at points $E$ and $F$ respectively. It is known that lines $AB,FE$ and $CD$ concur. The circumcircles of triangles $AED$ and $BFC$ meet $\omega$ for the second time at points $E_1$ and $F_1$. Prove that $EF$ is parallel to $E_1 F_1$.

Alice picks a random three-digit number, from $100$ to $999,$ inclusive. The probability that her first digit is larger than the sum of her other two digits can be expressed as a common fraction $\tfrac{a}{b}.$ Find $a+b.$

$A, B, C, D$ are points along a circle, in that order. $AC$ intersects $BD$ at $X$. If $BC=6$, $BX=4$, $XD=5$, and $AC=11$, find $AB$

Let $a$ be a positive integer such that the number $a^n$ has an odd number of digits in the decimal representation for all $n > 0$. Prove that the number $a$ is an even power of $10$.

[list=1] [*] If $a$, $b$, $c$, $d > 0$, show inequality:$$\left(\tan^{-1}\left(\frac{ad-bc}{ac+bd}\right)\right)^2\geq 2\left(1-\frac{ac+bd}{\sqrt{\left(a^2+b^2\right)\left(c^2+d^2\right)}}\right)$$ [*] Calculate $$\lim \limits_{n \to \infty}n^{\alpha}\left(n- \sum \limits_{k=1}^n\frac{n^+k^2-k}{\sqrt{\left(n^2+k^2\right)\left(n^2+(k-1)^2\right)}}\right)$$where $\alpha \in \mathbb{R}$ [/list]

The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and $\frac{3}{4}$ of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class? $\textbf{(A)}\ 12\qquad \textbf{(B)}\ 17\qquad \textbf{(C)}\ 24\qquad \textbf{(D)}\ 27\qquad \textbf{(E)}\ 36$

In triangle $ABC$, $X$ and $Y$ are the tangency points of incircle (with center $I$) with sides $AB$ and $AC$ respectively. A tangent line to the circumcircle of triangle $ABC$ (with center $O$) at point $A$, intersects the extension of $BC$ at $D$. If $D,X$ and $Y$ are collinear then prove that $D,I$ and $O$ are also collinear. [i]proposed by Amirhossein Zabeti[/i]

Find the set of solutions for $x$ in the inequality $\dfrac{x+1}{x+2}>\dfrac{3x+4}{2x+9}$ when $x\neq 2$, $x\neq -\dfrac{9}{2}$.

Let $p$ be a prime number greater than $2$. Patricia wants $7$ not-necessarily different numbers from $\{1, 2, . . . , p\}$ to the black dots in the figure below, on such a way that the product of three numbers on a line or circle always has the same remainder when divided by $p$. [img]https://cdn.artofproblemsolving.com/attachments/3/1/ef0d63b8ff5341ffc340de0cc75b24c7229e23.png[/img] (a) Suppose Patricia uses the number $p$ at least once. How many times does she have the number $p$ then a minimum sum needed? (b) Suppose Patricia does not use the number $p$. In how many ways can she assign numbers? (Two ways are different if there is at least one black one dot different numbers are assigned. The figure is not rotated or mirrored.)

Positive numbers $a,\ b,\ c$ satisfy $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3$. Prove the inequality \[\dfrac{1}{\sqrt{a^3+1}}+\dfrac{1}{\sqrt{b^3+1}}+\dfrac{1}{\sqrt{c^3+1}}\le \dfrac{3}{\sqrt{2}}. \] [i](N. Alexandrov)[/i]

A number of flowers are distributed between $n$ persons so that the first of them, Andreas, gets one flower, the other gets two flowers, the third gets three flowers, etc., to $n$-th person who gets $n$ flowers. Andreas then walks around shaking hands with each other of the others, in any order. In order to do so, he receives a flower from everyone which he hangs on to and which has more flowers than himself at the moment they shake hands. Which is the smallest number of flowers Andreas can have after shaking hands with everyone?

In an inscribed quadrilateral $ABCD$ we have $BC = CD$. Prove that the area of the quadrilateral is equal to $\frac{(AC)^2 \sin A}{2}$ (D. Fomin, Leningrad)

A hemisphere is inscribed in a cone so that its base lies on the base of the cone. The ratio of the area of the entire surface of the cone to the area of the hemisphere (without the base) is $\frac{18}5$. Compute the angle at the vertex of the cone.

There are 20 students in a high school class, and each student has exactly three close friends in the class. Five of the students have bought tickets to an upcoming concert. If any student sees that at least two of their close friends have bought tickets, then they will buy a ticket too. Is it possible that the entire class buys tickets to the concert? (Assume that friendship is mutual; if student $A$ is close friends with student $B$, then $B$ is close friends with $A$.)

[b]a)[/b] Let $ h:\mathbb{R}\longrightarrow\mathbb{R} $ he a function that admits a primitive $ H $ such that the function $ h/H $ is constant. Prove that there is a real number $ \gamma $ such that $ h(x)=\gamma\cdot\exp \left( x\cdot\frac{h}{H} (x) \right) , $ for any real number $ x. $ [b]b)[/b] Find the functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ that admit the primitives $ F,G, $ respectively, that satisfy $ f=\frac{G+g}{2},g=\frac{F+f}{2} $ and $ f(0)=g(0)=0. $

Let $ S$ be a finite family of squares on a plane such that every point on that plane is contained in at most $ k$ squares in $ S$. Prove that $ P$ can be divided into $ 4(k\minus{}1)\plus{}1$ sub-family such that in each sub-family, each pair of squares are disjoint.

Find all functions $f : R \to R$ satisfying the equality $f (2^x + 2y) =2^y f ( f (x)) f (y) $for every $x, y \in R$.

Let $n \ge 2$ be a positive integer. Prove that the following assertions are equivalent: a) for all integer $x$ coprime with n the congruence $x^6 \equiv 1$ (mod $n$) hold, b) $n$ divides $504$.

Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that $$(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),$$ for all positive real numbers $x, y, z$. [i]Fajar Yuliawan, Indonesia[/i]