Let $\gamma_1$ and $\gamma_2$ be two circles tangent at point $T$, and let $\ell_1$ and $\ell_2$ be two lines through $T$. The lines $\ell_1$ and $\ell_2$ meet again $\gamma_1$ at points $A$ and $B$, respectively, and $\gamma_2$ at points $A_1$ and $B_1$, respectively. Let further $X$ be a point in the complement of $\gamma_1 \cup \gamma_2 \cup \ell_1 \cup \ell_2$. The circles $ATX$ and $BTX$ meet again $\gamma_2$ at points $A_2$ and $B_2$, respectively. Prove that the lines $TX, A_1B_2$ and $A_2B_1$ are concurrent.
Let $VABCD$ be a regular pyramid, having the square $ABCD$ as basis. Suppose that on the line $AC$ lies a point $M$ such that $VM = MB$ and $(VMB) \perp (VAB)$. Prove that $4AM = 3AC$.
Let $ABC$ be a triangle such that $AB \neq AC$. The internal bisector lines of the angles $ABC$ and $ACB$ meet the opposite sides of the triangle at points $B_0$ and $C_0$, respectively, and the circumcircle $ABC$ at points $B_1$ and $C_1$, respectively. Further, let $I$ be the incenter of the triangle $ABC$. Prove that the lines $B_0C_0$ and $B_1C_1$ meet at some point lying on the parallel through $I$ to the line $BC$.
Radu Gologan
Let $a$ and $n$ be two positive integer numbers such that the (positive) prime factors of $a$ be all greater than $n$.
Prove that $n!$ divides $(a-1)(a^2-1)\cdots(a^{n-1}-1)$.
Let $a \lt c \lt b$ be real numbers and let $f : [a, b] \to \mathbb{R}$ be a function continuous at $c$. Show that if $f$ is the derivative of a function $F_a$ on $[a, c)$, and also the derivative of a function $F_b$ on $(c, b]$, then $f$ is the derivative of some function $F$ on the entire interval $[a, b]$.
Let $ABC$ be a scalene triangle. The tangents to the nine-point circle at the foot of the perpendicular dropped from $A$ on the line $BC$ and at the midpoint of the side $BC$ meet at the point $A'$; the points $B'$ and $C'$ are defined similarly. Prove that the lines $AA'$, $BB'$ and $CC'$ are concurrent.
Let $f : [0, 1] \to \mathbb{R}$ be an integrable function. If $f(1) = 0$ and $f$ is differentiable in $1$, prove that
$$
\lim_{n \to \infty} n^2 \int_0^1 x^n f(x) dx = -f'(1).
$$
Let $n$ be an integer, $n \ge 2$. For each $k = 1, 2, \dots, n$, let $a_k$ be the number of multiples of $k$ in the set $\{1, 2, \dots, n\}$, and let $x_k = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{a_k}$. Show that $\frac{x_1 + x_2 + \dots + x_n}{n} \le \frac{1}{1^2} + \frac{1}{2^2} + \dots + \frac{1}{n^2}$.
Let $G$ be a finite group of order $n$. Define the set
$$
H = \{x : x \in G \text{ and } x^2 = e\},
$$
where $e$ is the neutral element of $G$. Let $p = |H|$ be the cardinal of $H$. Prove that
a) $|H \cap xH| \geq 2p - n$, for any $x \in G$, where $xH = \{xh : h \in H\}$.
b) If $p \gt \frac{3n}{4}$, then $G$ is commutative.
c) If $\frac{n}{2} \lt p \leq \frac{3n}{4}$, then $G$ is non-commutative.
Let $ABC$ be a triangle and let $D, E, F$ be the midpoints of the sides $BC, CA, AB$. Prove that $\angle DAC = \angle ABE$ if and only if $\angle AFC = \angle BDA$.
Let $n$ be a positive integer and let $x_1, x_2, \dots, x_n$ be positive real numbers such that $x_1x_2 \cdots x_n = 1$. Prove that
$$
\sum_{i=1}^{n} x_{i}^{n}(1 + x_{i}) \geq \frac{n}{2^{n-1}} \prod_{i=1}^{n}(1 + x_{i}).
$$
Let $n$ be an integer number greater than or equal to $2$, and let $K$ be a closed convex set of area greater than or equal to $n$, contained in the open square $(0, n) \times (0, n)$. Prove that $K$ contains some point of the integral lattice $\mathbb{Z} \times \mathbb{Z}$.
Consider the cube $ABCDA'B'C'D'$. The angle bisectors of the angles $\angle A'C'A$ and $\angle A'AC'$ meet $AA'$ and $A'C'$ at points $P$ and $Q$, respectively. The point $M$ is the foot of the perpendicular from $A'$ onto $C'P$ while $N$ is the foot of the perpendicular from $A'$ onto $AS$. The point $O$ is the center of the face $ABB'A'$.
a) Prove that the planes $(MNO)$ and $(AC'B)$ are parallel.
b) Given that $AB = 1$, find the distance between the planes $(MNO)$ and $(AC'B)$.
Find all continuous functions $f : \mathbb{R} \to \mathbb{R}$ with the property that for every $a \in \mathbb{R}$ there exists $b \in \mathbb{R}$ such that, for every $x \in \mathbb{R}$,
$$
f(f(x) + a) = f(x) + b.
$$
Let $f : [0, 1] \to \mathbb{R}$ be a differentiable function with $f(0) = f(1)$, $\int_{0}^{1} f(x) dx = 0$ and $f'(x) \neq 1$, for every $x \in [0, 1]$.
a) Prove that the function $g : [0, 1] \to \mathbb{R}$ given by $g(x) = f(x) - x$ is decreasing.
b) Prove that for any integer $n \ge 1$ the following inequality holds
$$
\left| \sum_{k=0}^{n-1} f \left( \frac{k}{n} \right) \right| \lt \frac{1}{2}.
$$
Given a positive integer number $n$, determine the minimum of
$$
\max \left\{ \frac{x_1}{1+x_1}, \frac{x_2}{1+x_1+x_2}, \dots, \frac{x_n}{1+x_1+x_2+\dots+x_n} \right\},
$$
as $x_1, x_2, \dots, x_n$ run through all non-negative real numbers which add up to 1.