Consider $A, B$ and $C$ three collinear points of the plane such that $B$ is between $A$ and $C$. Let $S$ be the circle of diameter $AB$ and $L$ a line that passes through $C$, which does not intersect $S$ and is not perpendicular to line $AC$. The points $M$ and $N$ are, respectively, the feet of the altitudes drawn from $A$ and $B$ on the line $L$. From $C$ draw the two tangent lines to $S$, where $P$ is the closest tangency point to $L$. Prove that the quadrilateral $MPBC$ is cyclic if and only if the lines $MB$ and $AN$ are perpendicular.

Determine all functions $ f$ defined in the set of rational numbers and taking their values in the same set such that the equation $ f(x + y) + f(x - y) = 2f(x) + 2f(y)$ holds for all rational numbers $x$ and $y$.

Let $a$ and $ b$ be integers and $n$ a positive integer. Prove that \[\frac{b^{n-1}a(a + b)(a + 2b) \cdots (a + (n - 1)b)}{n!}\] is an integer.

The positive numbers $a$, $b$, and $c$ satisfy $abc = 1$. Show that: $$ \frac{1}{a^2+a}+\frac{1}{b^2+b}+\frac{1}{c^2+c} \ge \frac{3}{2}. $$

A point $P$ is chosen in the interior of $\triangle ABC$ so that when lines are drawn through $P$ parallel to the sides of $\triangle ABC$, the resulting smaller triangles, $t_1$, $t_2$, and $t_3$ in the figure, have areas 4, 9, and 49, respectively. Find the area of $\triangle ABC$. [asy] size(200); pathpen=black+linewidth(0.65);pointpen=black; pair A=(0,0),B=(12,0),C=(4,5); D(A--B--C--cycle); D(A+(B-A)*3/4--A+(C-A)*3/4); D(B+(C-B)*5/6--B+(A-B)*5/6);D(C+(B-C)*5/12--C+(A-C)*5/12); MP("A",C,N);MP("B",A,SW);MP("C",B,SE); /* sorry mixed up points according to resources diagram. */ MP("t_3",(A+B+(B-A)*3/4+(A-B)*5/6)/2+(-1,0.8),N); MP("t_2",(B+C+(B-C)*5/12+(C-B)*5/6)/2+(-0.3,0.1),WSW); MP("t_1",(A+C+(C-A)*3/4+(A-C)*5/12)/2+(0,0.15),ESE);[/asy]

Let $ABCDEF$ be a convex hexagon. In the hexagon there is a point $K$, such that $ABCK,DEFK$ are both parallelograms. Prove that the three lines connecting $A,B,C$ to the midpoints of segments $CE,DF,EA$ meet at one point.

Let $f(x)$ be a non-constant polynomial with integer coefficients such that $f(1) \neq 1$. For a positive integer $n$, define $\text{divs}(n)$ to be the set of positive divisors of $n$. A positive integer $m$ is $f$-cool if there exists a positive integer $n$ for which $$f[\text{divs}(m)]=\text{divs}(n).$$ Prove that for any such $f$, there are finitely many $f$-cool integers. (The notation $f[S]$ for some set $S$ denotes the set $\{f(s):s \in S\}$.)

Let $m,n,k$ be positive integers, $m> n> k$. An $1 \times m$ strip of paper is divided into the $1 \times 1$ cells. A teacher asks Bill and Pit to place numbers $0$ and $1$ in the cells of the strip so that the sum of the numbers in any $n$ consecutive cells is equal to $k$. After the task was performed it turned out that the sum $S(B)$ of all numbers on the strip of Bill was different from the sum $S(P)$ of Pit. Find the largest possible value of $|S(B) - S(P) |$. (I. Voronovich)

Find the pairs of real numbers $(a,b)$ such that the biggest of the numbers $x=b^2-\frac{a-1}{2}$ and $y=a^2+\frac{b+1}{2}$ is less than or equal to $\frac{7}{16}$

Let $h_a$, $h_b$ and $h_c$ be the heights and $r$ the inradius of a triangle. Prove that the triangle is equilateral if and only if $h_a + h_b + h_c = 9r$.

Let $x, y$ and $z$ be rational numbers satisfying $$x^3 + 3y^3 + 9z^3 - 9xyz = 0.$$ Prove that $x = y = z = 0$.

If $a_i >0$ ($i=1, 2, \cdots , n$) and $\sum \limits_{i=1}^n a_i^k=1$, where $1\leq k\leq n+1$, then $$\sum \limits_{i=1}^n a_i + \frac{1}{\prod \limits_{i=1}^n a_i} \geq n^{1-\frac{1}{k}}+n^{\frac{n}{k}}$$

What is the perimeter of the right triangle whose exradius of the hypotenuse is $ 30$ ? $\textbf{(A)}\ 40 \qquad\textbf{(B)}\ 45 \qquad\textbf{(C)}\ 50 \qquad\textbf{(D)}\ 60 \qquad\textbf{(E)}\ 75$

Find out if there is a convex pentagon $A_1A_2A_3A_4A_5$ such that, for each $i = 1, \dots , 5$, the lines $A_iA_{i+3}$ and $A_{i+1}A_{i+2}$ intersect at a point $B_i$ and the points $B_1,B_2,B_3,B_4,B_5$ are collinear. (Here $A_{i+5} = A_i$.)

Triangle $ABC$ is inscribed in a circle $\omega$. Let the bisector of angle $A$ meet $\omega$ at $D$ and $BC$ at $E$. Let the reflections of $A$ across $D$ and $C$ be $D'$ and $C'$, respectively. Suppose that $\angle A = 60^o$, $AB = 3$, and $AE = 4$. If the tangent to $\omega$ at $A$ meets line $BC$ at $P$, and the circumcircle of APD' meets line $BC$ at $F$ (other than $P$), compute $FC'$.

Prove that if $n$ is a natural number such that $1 + 2^n + 4^n$ is prime then $n = 3^k$ for some $k \in N_0$.

Suppose $n$ lines in plane are such that no two are parallel and no three are concurrent. For each two lines their angle is a real number in $[0,\frac{\pi}2]$. Find the largest value of the sum of the $\binom n2$ angles between line. [i]By Aliakbar Daemi[/i]

Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[ xf(y) + f(xf(y)) - xf(f(y)) - f(xy) = 2x + f(y) - f(x+y)\] holds for all $x,y \in \mathbb{R}$.

Determine the smallest positive integer $m$ with the property that $m^3-3m^2+2m$ is divisible by both $79$ and $83$.

Find all non-negative integers $a, b, c, d$ such that $7^a = 4^b + 5^c + 6^d$

Prove that $n^4 + 4^{n}$ is composite for all values of $n$ greater than $1$.

Let $ ABC$ be a triangle. Points $ D$ and $ E$ are on sides $ AB$ and $ AC,$ respectively, and point $ F$ is on line segment $ DE.$ Let $ \frac {AD}{AB} \equal{} x,$ $ \frac {AE}{AC} \equal{} y,$ $ \frac {DF}{DE} \equal{} z.$ Prove that (1) $ S_{\triangle BDF} \equal{} (1 \minus{} x)y S_{\triangle ABC}$ and $ S_{\triangle CEF} \equal{} x(1 \minus{} y) (1 \minus{} z)S_{\triangle ABC};$ (2) $ \sqrt [3]{S_{\triangle BDF}} \plus{} \sqrt [3]{S_{\triangle CEF}} \leq \sqrt [3]{S_{\triangle ABC}}.$

Let $ ABC$ be acute triangle. The circle with diameter $ AB$ intersects $ CA,\, CB$ at $ M,\, N,$ respectively. Draw $ CT\perp AB$ and intersects above circle at $ T$, where $ C$ and $ T$ lie on the same side of $ AB$. $ S$ is a point on $ AN$ such that $ BT = BS$. Prove that $ BS\perp SC$.

Consider the triangles $[ABC]$ and $[EDC]$, right at $A$ and $D$, respectively. Show that, if $E$ is the midpoint of $[AC]$, then $AB <BD$. [img]https://cdn.artofproblemsolving.com/attachments/c/3/75bc1bda1a22bcf00d4fe7680c80a81a9aaa4c.png[/img]

Let $G$ be the subgroup of $GL_{2}(\mathbb{R})$ generated by $A$ and $B$, where $$A=\begin{pmatrix} 2 &0\\ 0&1 \end{pmatrix},\; B=\begin{pmatrix} 1 &1\\ 0&1 \end{pmatrix}$$. Let $H$ consist of the matrices $\begin{pmatrix} a_{11} &a_{12}\\ a_{21}& a_{22} \end{pmatrix}$ in $G$ for which $a_{11}=a_{22}=1$. a) Show that $H$ is an abelian subgroup of $G$. b) Show that $H$ is not finitely generated.