Let $n$ be a positive integer, and let $f_n(z) = n + (n-1)z + (n-2)z^2 + \dots + z^{n-1}$. Prove that $f_n$ has no roots in the closed unit disk $\{z \in \mathbb{C}: |z| \le 1\}$.

Let $n > 1$ be an integer. Find, with proof, all sequences $x_1 , x_2 , \ldots , x_{n-1}$ of positive integers with the following three properties: (a). $x_1 < x_2 < \cdots < x_{n-1}$ ; (b). $x_i + x_{n-i} = 2n$ for all $i = 1, 2, \ldots , n - 1$; (c). given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.

We inscribe a circle $\omega$ in equilateral triangle $ABC$ with radius $1$. What is the area of the region inside the triangle but outside the circle?

Consider the complex numbers $a_1, a_2,\cdots , a_n$, $n \geq 2$. Which have the following properties: [list] [*] $|a_i|=1$ $\forall$ $i=1,2,\cdots , n$ [*] $\sum \limits_{k=1}^n arg(a_k)\leq \pi$ [/list] Show that the inequality$$\left(n^2\cot \left(\frac{\pi}{2n}\right)\right)^{-1}\left |\sum \limits_{k=0}^n(-1)^k\left[3n^2-(8k+5)n+4k(k+1)\sigma_k\right]\right |\geq \sqrt{\left(1+\frac{1}{n}\right)^2\cot^2 \left(\frac{\pi}{2n}\right)}+16\left |\sum \limits_{k=0}^n(-1)^k\sigma_k\right |$$where $\sigma_0=1$, $\sigma_k=\sum \limits_{1\leq i_1\leq i_2\leq \cdots \leq i_k\leq n}a_{i_1}a_{i_2}\cdots a_{i_k}$, $\forall$ $k=1,2,\cdots , n$

Alf, the alien from the $1980$s TV show, has a big appetite for the mineral apatite. However, he’s currently on a diet, so for each integer $k \ge 1,$ he can eat exactly $k$ pieces of apatite on day $k.$ Additionally, if he eats apatite on day $k,$ he cannot eat on any of days $k + 1, k + 2, \ldots, 2k - 1.$ Compute the maximum total number of pieces of apatite Alf could eat over days $1,2, \ldots,99,100.$

What is the maximal number of solutions can the equation have $$\max \{a_1x+b_1, a_2x+b_2, \ldots, a_{10}x+b_{10}\}=0$$ where $a_1,b_1, a_2, b_2, \ldots , a_{10},b_{10}$ are real numbers, all $a_i$ not equal to $0$.

Consider a $3 \times 3$ grid of squares. A circle is inscribed in the lower left corner, the middle square of the top row, and the rightmost square of the middle row, and a circle $O$ with radius $r$ is drawn such that $O$ is externally tangent to each of the three inscribed circles. If the side length of each square is 1, compute $r$.

There are $2^n$ words of length $n$ over the alphabet $\{0, 1\}$. Prove that the following algorithm generates the sequence $w_0, w_1, \ldots, w_{2^n-1}$ of all these words such that any two consecutive words differ in exactly one digit. (1) $w_0 = 00 \ldots 0$ ($n$ zeros). (2) Suppose $w_{m-1} = a_1a_2 \ldots a_n,\quad a_i \in \{0, 1\}$. Let $e(m)$ be the exponent of $2$ in the representation of $n$ as a product of primes, and let $j = 1 + e(m)$. Replace the digit $a_j$ in the word $w_{m-1}$ by $1 - a_j$. The obtained word is $w_m$.

Let $ABC$ be a triangle such that $AB = 26, AC = 30,$ and $BC = 28$. Let $C'$ and $B'$ be the reflections of the circumcenter $O$ over $AB$ and $AC$, respectively. The length of the portion of line segment $B'C'$ inside triangle $ABC$ can be written as $\frac{p}{q}$, where $p,q$ are relatively prime positive integers. Compute $p+q$.

Euler is selling Mathematician cards to Gauss. Three Fermat cards plus $5$ Newton cards costs $95$ Euros, while $5$ Fermat cards plus $2$ Newton cards also costs $95$ Euros. How many Euroes does one Fermat card cost? $$ \mathrm a. ~ 10\qquad \mathrm b.~15\qquad \mathrm c. ~20 \qquad \mathrm d. ~30 \qquad \mathrm e. ~35 $$

Let $ n$ be a positive integer such that $ n\geq 3$. Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2n$ positive real numbers satisfying the equations \[ a_1 \plus{} a_2 \plus{} ... \plus{} a_n \equal{} 1, \quad \text{and} \quad b_1^2 \plus{} b_2^2 \plus{} ... \plus{} b_n^2 \equal{} 1.\] Prove the inequality \[a_1\left(b_1 \plus{} a_2\right) \plus{} a_2\left(b_2 \plus{} a_3\right) \plus{} ... \plus{} a_{n \minus{} 1}\left(b_{n \minus{} 1} \plus{} a_n\right) \plus{} a_n\left(b_n \plus{} a_1\right) < 1.\]

In the following, the point of intersection of two lines $ g$ and $ h$ will be abbreviated as $ g\cap h$. Suppose $ ABC$ is a triangle in which $ \angle A \equal{} 90^{\circ}$ and $ \angle B > \angle C$. Let $ O$ be the circumcircle of the triangle $ ABC$. Let $ l_{A}$ and $ l_{B}$ be the tangents to the circle $ O$ at $ A$ and $ B$, respectively. Let $ BC \cap l_{A} \equal{} S$ and $ AC \cap l_{B} \equal{} D$. Furthermore, let $ AB \cap DS \equal{} E$, and let $ CE \cap l_{A} \equal{} T$. Denote by $ P$ the foot of the perpendicular from $ E$ on $ l_{A}$. Denote by $ Q$ the point of intersection of the line $ CP$ with the circle $ O$ (different from $ C$). Denote by $ R$ be the point of intersection of the line $ QT$ with the circle $ O$ (different from $ Q$). Finally, define $ U \equal{} BR \cap l_{A}$. Prove that \[ \frac {SU \cdot SP}{TU \cdot TP} \equal{} \frac {SA^{2}}{TA^{2}}. \]

Prove that in a group of $50$ people there are always two who have an even number (possibly zero) of common acquaintances within the group. (SI Tokarev)

Find the number of subsets of $\{1,2,\ldots,7\}$ that do not contain two consecutive integers.

We string $2n$ white balls and $2n$ black balls, forming a continuous chain. Demonstrate that, in whatever order the balls are placed, it is always possible to cut a segment of the chain to contain exactly $n$ white balls and $n$ black balls.

An acute triangle $\triangle ABC$ has incenter $I$, and the incircle hits $BC, CA, AB$ at $D, E, F$. Lines $BI, CI, BC, DI$ hits $EF$ at $K, L, M, Q$ and the line connecting the midpoint of segment $CL$ and $M$ hits the line segment $CK$ at $P$. Prove that $$PQ=\frac{AB \cdot KQ}{BI}$$

For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.

The natural numbers are written in sequence, in increasing order, and by this we get an infinite sequence of digits. Find the least natural $k$, for which among the first $k$ digits of this sequence, any two nonzero digits have been written a different number of times. [i]Aleksandar Ivanov, Emil Kolev [/i]

Let $n$ be an integer greater than two, and let $A_1,A_2, \cdots , A_{2n}$ be pairwise distinct subsets of $\{1, 2, ,n\}$. Determine the maximum value of \[\sum_{i=1}^{2n} \dfrac{|A_i \cap A_{i+1}|}{|A_i| \cdot |A_{i+1}|}\] Where $A_{2n+1}=A_1$ and $|X|$ denote the number of elements in $X.$

If $a$, $b$ and $c$ are real numbers such that $ab = 31$, $ac = 13$, and $bc = 5$, compute the product of all possible values of $abc$.

Let $ a,b,c $ be positive real numbers such that $ a^4+b^4+c^4=3 $. Prove that \[ \frac{9}{a^2+b^4+c^6}+\frac{9}{a^4+b^6+c^2}+\frac{9}{a^6+b^2+c^4}\leq\ a^6+b^6+c^6+6 \]

Vanya has chosen two positive numbers, x and y. He wrote the numbers x+y, x-y, x/y, and xy, and has shown these numbers to Petya. However, he didn't say which of the numbers was obtained from which operation. Show that Petya can uniquely recover x and y.

Prove that if the function $f$ is defined on the set of positive real numbers, its values are real, and $f$ satisfies the equation $$f\left( \frac{x+y}{2}\right) + f\left(\frac{2xy}{x+y} \right) =f(x)+f(y)$$ for all positive $x,y$, then $$2f(\sqrt{xy})=f(x)+f(y)$$ for every pair $x,y$ of positive numbers.

Given a prime number $p$ such that $2p$ is equal to the sum of the squares of some four consecutive positive integers. Prove that $p-7$ is divisible by 36.

Let $p > 3$ be a given prime number. For a set $S \subseteq \mathbb{Z}$ and $a \in \mathbb{N}$ , define $S_a = \{ x \in \{ 0,1, 2,...,p-1 \}$ | $(\exists_s \in S) x \equiv_p a \cdot s \}$ . $(a)$ How many sets $S \subseteq \{ 1, 2,...,p-1 \} $ are there for which the sequence $S_1 , S_2 , ..., S_{p-1}$ contains exactly two distinct terms? $(b)$ Determine all numbers $k \in \mathbb{N}$ for which there is a set $ S \subseteq \{ 1, 2,...,p-1 \} $ such that the sequence $S_1 , S_2 , ..., S_{p-1} $ contains exactly $k$ distinct terms. [i]Proposed by Milan Basic and Milos Milosavljevic[/i]