In the complex plane consider the unit circle with the origin as its center. Furthermore, consider inscribed regular 17-gon with one of its vertices being $1+0i.$ How many of its vertices lie in the (open) unit disc centered in $\sqrt{3/2}(1+i)$?

Let $x$ and $y$ be integers and let $p$ be a prime number. Suppose that there exist realatively prime positive integers $m$ and $n$ such that $$x^m \equiv y^n \pmod p$$ Prove that there exists an unique integer $z$ modulo $p$ such that $$x \equiv z^n \pmod p \quad \text{and} \quad y \equiv z^m \pmod p$$

For integers $n>1$, define $f(n)$ to be the sum of all postive divisors of $n$ that are less than $n$. Prove that for any positive integer $k$, there exists a positive integer $n>1$ such that $n<f(n)<f^2(n)<\cdots<f^k(n)$, where $f^i(n)=f(f^{i-1}(n))$ for $i>1$ and $f^1(n)=f(n)$.

Let $A,B,C\in \mathcal{M}_n(\mathbb{R})$ such that $ABC=O_n$ and $\text{rank}\ B=1$. Prove that $AB=O_n$ or $BC=O_n$.

Prove the following equalities of sets: \[ \text{i)} \{x\in \mathbb{R}\ |\ \log_2 \lfloor x \rfloor \equal{} \lfloor \log_2 x\rfloor \} \equal{} \bigcup_{m\in \mathbb{N}} \left[2^m,2^m \plus{} 1\right)\] \[ \text{ii)} \{x\in \mathbb{R}\ |\ 2^{\lfloor x\rfloor} \equal{} \left\lfloor 2^x\right\rfloor \} \equal{} \bigcup_{m\in \mathbb{N}} \left[m, \log_2 (2^m \plus{} 1) \right)\]

We consider triangle $ABC$ with $\angle BAC = 90^{\circ}$ and $\angle ABC = 60^{\circ}.$ Let $ D \in (AC) , E \in (AB),$ such that $CD = 2 \cdot DA$ and $DE $ is bisector of $\angle ADB.$ Denote by $M$ the intersection of $CE$ and $BD$, and by $P$ the intersection of $DE$ and $AM$. a) Show that $AM \perp BD$. b) Show that $3 \cdot PB = 2 \cdot CM$.

Show that \[A_n=\prod_{j=0}^{n-1}\cfrac{(3j+1)!}{(n+j)!}\] is an integer, for any positive integer \(n\).

Any of $6$ gossips has her own news. From time to time one of them makes a telephone call to some other gossip and they discuss fill the news they know. What the minimum number of the calls is necessary so as (for) all of them to know all the news?

Let $ABC$ be a triangle. Suppose that $X,Y$ are points in the plane such that $BX,CY$ are tangent to the circumcircle of $ABC$, $AB=BX,AC=CY$ and $X,Y,A$ are in the same side of $BC$. If $I$ be the incenter of $ABC$ prove that $\angle BAC+\angle XIY=180$.

Decide if there exists any number of $10$ digits such that rearranging $10,000$ times its digits results in $10,000$ different numbers that are multiples of $7$.

In quadrilateral $ABCD$, $m\angle B=m\angle C=120^\circ$, $AB=3$, $BC=4$, and $CD=5$. Find the area of $ABCD$. $\textbf{(A) }15\qquad\textbf{(B) }9\sqrt3\qquad\textbf{(C) }\dfrac{45\sqrt3}4\qquad\textbf{(D) }\dfrac{47\sqrt3}4\qquad\textbf{(E) }15\sqrt3$

If $ x \plus{} y \equal{} 1$, then the largest value of $ xy$ is: $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 0.5\qquad \textbf{(C)}\ \text{an irrational number about }{0.4}\qquad \textbf{(D)}\ 0.25\qquad \textbf{(E)}\ 0$

Given $a, b,c \ge 0, a + b + c = 1$, prove that $(a^2 + b^2 + c^2)^2 + 6abc \ge ab + bc + ac$ (I. Voronovich)

Ralph is standing along a road which heads straight east. If you go nine miles east, make a left turn, and travel seven miles north, you will find Pamela with her mountain bike. At exactly the same time that Ralph begins running eastward along the road at 6 miles per hour, Pamela begins biking in a straight line at 10 miles per hour. Pamela’s direction is chosen so that she will reach a point on the road where Ralph is running at exactly the same time Ralph reaches that same point. Let $M$ and $N$ be relatively prime positive integers such that $\frac{M}{N}$ is the number of hours that it takes Pamela and Ralph to meet. Find $M+N$.

Let $\alpha_{1}, \alpha_{2}, \cdots , \alpha_{n}$ be numbers in the interval $[0, 2\pi]$ such that the number $\displaystyle\sum_{i=1}^n (1 + \cos \alpha_{i})$ is an odd integer. Prove that \[\displaystyle\sum_{i=1}^n \sin \alpha_i \ge 1\]

An inscribed into a circle quadraliteral $ABCD$ is given. Points $M$ and $N$ lie on sides $AB$ and $CD$ such that $AK:KB=DM:MC$ and points $L$ and $N$ lie on sides $BC$ and $DA$ such that $BL:LC=AN:ND$. The circumcircle of the triangle $CML$ intersects diagonal $AC$ for the second time in point $P$. The circumcircle of triangle $DNM$ intersects diagonal $BD$ for the second time in point $Q$. Circumcircles of triangles $AKN$ and $BLK$ intersect for the second time in point $R$. Prove that the circumcircle of $PQR$ passes through the intersection of $AC$ and $BD$

Let $\Gamma_1$ and $\Gamma_2$ be concentric circles with radii $1$ and $2$, respectively. Four points are chosen on the circumference of $\Gamma_2$ independently and uniformly at random, and are then connected to form a convex quadrilateral. What is the probability that the perimeter of this quadrilateral intersects $\Gamma_1$? [i]Proposed by Yuan Yao.[/i]

The circles $C_1$ and $C_2$ with radii $r_1$ and $r_2$ touch the line $p$ at the point $P$. $C_1$ lies entirely within $C_2$. Line $q \perp p$ intersects $p$ at $S$ and touches $C_1$ at $R$. $q$ intersects $C_2$ at $M$ and $N$, where $N$ is between $R$ and $S$. a) Prove that line $PR$ bisects angle $\angle MPN$. b) Calculate the ratio $r_1 : r_2$ if line $PN$ bisects angle $\angle RPS$.

Show that for any $n$ we can find a set $X$ of $n$ distinct integers greater than 1, such that the average of the elements of any subset of $X$ is a square, cube or higher power.

Let be two natural numbers $ m,n\ge 2, $ two increasing finite sequences of real numbers $ \left( a_i \right)_{1\le i\le n} ,\left( b_j \right)_{1\le j\le m} , $ and the set $$ \left\{ a_i+b_j| 1\le i\le n,1\le j\le m \right\} . $$ Show that the set above has $ n+m-1 $ elements if and only if the two sequences above are arithmetic progressions and these have the same ratio.

Let $n$ and $k$ be positive integers with $k<n$. Let $P(x)$ be a polynomial of degree $n$ with real coefficients, nonzero constant term, and no repeated roots. Suppose that for any real numbers $a_0,\,a_1,\,\ldots,\,a_k$ such that the polynomial $a_kx^k+\cdots+a_1x+a_0$ divides $P(x)$, the product $a_0a_1\cdots a_k$ is zero. Prove that $P(x)$ has a nonreal root.

Let $n$ be a positive integer. There are $2018n+1$ cities in the Kingdom of Sellke Arabia. King Mark wants to build two-way roads that connect certain pairs of cities such that for each city $C$ and integer $1\le i\le 2018,$ there are exactly $n$ cities that are a distance $i$ away from $C.$ (The [i]distance[/i] between two cities is the least number of roads on any path between the two cities.) For which $n$ is it possible for Mark to achieve this? [i]Proposed by Michael Ren[/i]

Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a [i]maximal grid-aligned configuration[/i] on the board if $C$ consists of $(n^2-1)/2$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find all possible values of $k(C)$ as a function of $n$. [i]Proposed by Holden Mui[/i]

Each student at a school divided 18 subjects into six disjoint triples. Could it happen that every triple of subjects is among the triples of exactly one student?

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.