Given a circle $\Gamma$ with center $I$, and a triangle $\triangle ABC$ with all its sides tangent to $\Gamma$. A line $\ell$ is drawn such that it bisects both the area and the perimeter of $\triangle ABC$. Prove that line $\ell$ passes through $I$.

The triangle $ABC$ has $\angle ACB = 30^o$, $BC = 4$ cm and $AB = 3$ cm . Compute the altitudes of the triangle.

a) Let $ax^3 + bx^2 + cx + d$ be divisible by $5$ for given positive integers $a, b, c, d$ and any integer $x$. Prove that $a, b, c$ and $d$ are all divisible by $5$. b) Let $ax^4 + bx^3 + cx^2 + dx + e$ be divisible by $7$ for given positive integers $a, b, c, d, e$ and all integers $x$. Prove that $a, b, c, d$ and $e$ are all divisible by $7$.

$9$ horizontal and $9$ vertical lines are drawn through a square. Prove that it is possible to select $20$ rectangles so that the sides of each rectangle is a segment of one of the given lines (including the sides of the square), and for any two of the $20$ rectangles, it is possible to cover one of them with the other (rotations are allowed).

Find all $n \in \mathbb{N}$ such that $ 2^{n-1}$ divides $n!$.

For some integers $a$ and $b$ the function $f(x)=ax+b$ has the properties that $f(f(0))=0$ and $f(f(f(4)))=9$. Find $f(f(f(f(10))))$.

We denote by $S(k)$ the sum of digits of a positive integer number $k$. We say that the positive integer $a$ is $n$-good, if there is a sequence of positive integers $a_0$, $a_1, \dots , a_n$, so that $a_n = a$ and $a_{i + 1} = a_i -S (a_i)$ for all $i = 0, 1,. . . , n-1$. Is it true that for any positive integer $n$ there exists a positive integer $b$, which is $n$-good, but not $(n + 1)$-good? A. Antropov

Compute the sum of all positive integers $x$ such that $(x-17)\sqrt{x-1}+(x-1)\sqrt{x+15}$ is an integer.

There are $12$ acrobats who are assigned a distinct number ($1, 2, \cdots , 12$) respectively. Half of them stand around forming a circle (called circle A); the rest form another circle (called circle B) by standing on the shoulders of every two adjacent acrobats in circle A respectively. Then circle A and circle B make up a formation. We call a formation a “[i]tower[/i]” if the number of any acrobat in circle B is equal to the sum of the numbers of the two acrobats whom he stands on. How many heterogeneous [i]towers[/i] are there? (Note: two [i]towers[/i] are homogeneous if either they are symmetrical or one may become the other one by rotation. We present an example of $8$ acrobats (see attachment). Numbers inside the circle represent the circle A; numbers outside the circle represent the circle B. All these three formations are “[i]towers[/i]”, however they are homogeneous [i]towers[/i].)

Find the smallest real number $p$ such that the inequality $\sqrt{1^2+1}+\sqrt{2^2+1}+...+\sqrt{n^2+1} \le \frac{1}{2}n(n+p)$ holds for all natural numbers $n$.

Integers $a, b, c$ satisfy $a+b-c=1$ and $a^2+b^2-c^2=-1$. What is the sum of all possible values of $a^2+b^2+c^2$ ?

Which of the following sets of whole numbers has the largest average? \[ \textbf{(A)}\ \text{multiples of 2 between 1 and 101} \qquad \textbf{(B)}\ \text{multiples of 3 between 1 and 101} \\ \textbf{(C)}\ \text{multiples of 4 between 1 and 101} \qquad \textbf{(D)}\ \text{multiples of 5 between 1 and 101} \\ \textbf{(E)}\ \text{multiples of 6 between 1 and 101} \]

There is a complex number $K$ such that the quadratic polynomial $7x^2 +Kx + 12 - 5i$ has exactly one root, where $i =\sqrt{-1}$. Find $|K|^2$.

Let $B$ be a set of more than $\tfrac{2^{n+1}}{n}$ distinct points with coordinates of the form $(\pm 1, \pm 1, \cdots, \pm 1)$ in $n$-dimensional space with $n \ge 3$. Show that there are three distinct points in $B$ which are the vertices of an equilateral triangle.

Let $n\geq 4$ be an integer, and let $a_1,a_2,\ldots,a_n$ be positive real numbers such that \[ a_1^2+a_2^2+\cdots +a_n^2=1 . \] Prove that the following inequality takes place \[ \frac{a_1}{a_2^2+1}+\cdots +\frac{a_n}{a_1^2+1} \geq \frac{4}{5}\left( a_1 \sqrt{a_1}+\cdots +a_n \sqrt{a_n} \right)^2 . \] [i]Bogdan Enescu, Mircea Becheanu[/i]

On $\vartriangle ABC$ let $D$ be a point on side $\overline{AB}$, $F$ be a point on side $\overline{AC}$, and $E$ be a point inside the triangle so that $\overline{DE}\parallel \overline{AC}$ and $\overline{EF} \parallel \overline{AB}$. Given that $AF = 6, AC = 33, AD = 7, AB = 26$, and the area of quadrilateral $ADEF$ is $14$, nd the area of $\vartriangle ABC$.

How many sequences of real numbers $a_1,a_2,\ldots a_9$ satisfy \[|a_1-1|=|a_2-a_1|=\cdots=|a_9-a_8|=|1-a_9|=1?\] [i]Proposed by Evan Chang [/i]

Prove that for any real numbers $a,b,c$ satisfying $abc = 1$ the following inequality holds \[\dfrac{1} {2a^2+b^2+3}+\dfrac {1} {2b^2+c^2+3}+\dfrac{1} {2c^2+a^2+3}\leq \dfrac {1} {2}.\] [i]Proposed by V. Aksenov[/i]

A sit of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one? $ \text{(A)}\ 10 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 19 \qquad \text{(E)}\ 20$

Let $p$ be an odd prime number. For every integer $a,$ define the number $S_a = \sum^{p-1}_{j=1} \frac{a^j}{j}.$ Let $m,n \in \mathbb{Z},$ such that $S_3 + S_4 - 3S_2 = \frac{m}{n}.$ Prove that $p$ divides $m.$ [i]Proposed by Romeo Meštrović, Montenegro[/i]

Writing down the first $4$ rows of the Pascal triangle in the usual way and then adding up the numbers in vertical columns, we obtain $7$ numbers as shown above. If we repeat this procedure with the first $1024$ rows of the Pascal triangle, how many of the $2047$ numbers thus obtained will be odd? [img]https://cdn.artofproblemsolving.com/attachments/8/a/4dc4a815d8b002c9f36a6da7ad6e1c11a848e9.png[/img]

Given a positive integer $n > 1$. Find the smallest integer of the form $\frac{n^a-n^b}{n^c-n^d}$ for all positive integers $a,b,c,d$.

Prove that the volume $V$ and the lateral area $S$ of a right circular cone satisfy the inequality \[\left( \frac{6V}{\pi}\right)^2 \leq \left( \frac{2S}{\pi \sqrt 3}\right)^3\] When does equality occur?

Three cevians divided the triangle into six triangles, the areas of which are marked in the figure. 1) Prove that $S_1 \cdot S_2 \cdot S_3 =Q_1 \cdot Q_2 \cdot Q_3$. 2) Determine whether it is true that if $S_1 = S_2 = S_3$, then $Q_1 = Q_2 = Q_3$. [img]https://cdn.artofproblemsolving.com/attachments/c/d/3e847223b24f783551373e612283e10e477e62.png[/img]

The integer 287 is exactly divisible by $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 6$