Ten vertices of a regular $20$-gon $A_1A_2....A_{20}$ are painted black and the other ten vertices are painted blue. Consider the set consisting of diagonal $A_1A_4$ and all other diagonals of the same length. 1. Prove that in this set, the number of diagonals with two black endpoints is equal to the number of diagonals with two blue endpoints. 2. Find all possible numbers of the diagonals with two black endpoints.

Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{(n/2)}$ when $n$ is even and by $t_n=\frac{1}{t_{(n-1)}}$ when $n$ is odd. Given that $t_n=\frac{19}{87}$, the sum of the digits of $n$ is $ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 17 \qquad\textbf{(C)}\ 19 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 23$

Let $x$ be a solution to the equation $\lfloor x \lfloor x + 2\rfloor + 2\rfloor = 10$. Compute the smallest $C$ such that for any solution $x$, $x < C$. Here, $\lfloor m \rfloor$ is defined as the greatest integer less than or equal to $m$. For example, $\lfloor 3\rfloor = 3$ and $\lfloor -4.25\rfloor = -5$.

Let $P$ be a point inside the acute angle $\angle BAC$. Suppose that $\angle ABP = \angle ACP = 90^{\circ}$. The points $D$ and $E$ are on the segments $BA$ and $CA$, respectively, such that $BD = BP$ and $CP = CE$. The points $F$ and $G$ are on the segments $AC$ and $AB$, respectively, such that $DF$ is perpendicular to $AB$ and $EG$ is perpendicular to $AC$. Show that $PF = PG$.

Let $A$ be a real $n\times n$ matrix such that $A^3=0$ a) prove that there is unique real $n\times n$ matrix $X$ that satisfied the equation $X+AX+XA^2=A$ b) Express $X$ in terms of $A$

What is the smallest positive multiple of $225$ that can be written using digits $0$ and $1$ only?

Let the major axis of an ellipse be $AB$, let $O$ be its center, and let $F$ be one of its foci. $P$ is a point on the ellipse, and $CD$ a chord through $O$, such that $CD$ is parallel to the tangent of the ellipse at $P$. $PF$ and $CD$ intersect at $Q$. Compare the lengths of $PQ$ and $OA$.

Calculate the sum: $1+ 2q + 3q^2 +...+nq^{n-1}$

Let $d_n$ denote the number of derangements of the integers $1, 2, \ldots n$ so that no integer $i$ is in the $i^{th}$ position. It is possible to write a recurrence relation $d_{n}=f(n)d_{n-1}+g(n)d_{n-2}$; what is $f(n)+g(n)$?

How many real numbers $a \in (1,9)$ such that the corresponding number $a- \frac1a$ is an integer? (A): $0$, (B): $1$, (C): $8$, (D): $9$, (E) None of the above.

Six straight lines are drawn in a plane with no two parallel and no three concurrent. The number of regions into which they divide the plane is: $\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 20\qquad \textbf{(C)}\ 22 \qquad \textbf{(D)}\ 24 \qquad \textbf{(E)}\ 26$

Let $a, b>1$ be reals such that $a+\frac{1}{a^2} \geq 5b-\frac{3}{b^2}$. Show that $a>5b-\frac{4}{b^2}$.

On the $BE$ side of a regular $ABE$ triangle, a $BCDE$ rhombus is built outside it. The segments $AC$ and $BD$ intersect at point $F$. Prove that $AF <BD$.

[u]Round 5[/u] [b]p13.[/b] You have a $26 \times 26$ grid of squares. Color each randomly with red, yellow, or blue. What is the expected number (to the nearest integer) of $2 \times 2$ squares that are entirely red? [b]p14.[/b] Four snakes are boarding a plane with four seats. Each snake has been assigned to a different seat. The first snake sits in the wrong seat. Any subsequent snake will sit in their assigned seat if vacant, if not, they will choose a random seat that is available. What is the expected number of snakes who sit in their correct seats? [b]p15.[/b] Let $n \ge 1$ be an integer and $a > 0$ a real number. In terms of n, find the number of solutions $(x_1, ..., x_n)$ of the equation $\sum^n_{i=1}(x^2_i + (a - x_i)^2) = na^2$ such that $x_i$ belongs to the interval $[0, a]$ , for $i = 1, 2, . . . , n$. [u]Round 6 [/u] [b]p16.[/b] All roots of $$\prod^{25}_{n=1} \prod^{2n}_{k=0}(-1)^k \cdot x^k = 0$$ are written in the form $r(\cos \phi + i\sin \phi)$ for $i^2 = -1$, $r > 0$, and $0 \le \phi < 2\pi$. What is the smallest positive value of $\phi$ in radians? [b]p17.[/b] Find the sum of the distinct real roots of the equation $$\sqrt[3]{x^2 - 2x + 1} + \sqrt[3]{x^2 - x - 6} = \sqrt[3]{2x^2 - 3x - 5}.$$ [b]p18.[/b] If $a$ and $b$ satisfy the property that $a2^n + b$ is a square for all positive integers $n$, find all possible value(s) of $a$. [u]Round 7 [/u] [b]p19.[/b] Compute $(1 - \cot 19^o)(1 - \cot 26^o)$. [b]p20.[/b] Consider triangle $ABC$ with $AB = 3$, $BC = 5$, and $\angle ABC = 120^o$. Let point $E$ be any point inside $ABC$. The minimum of the sum of the squares of the distances from $E$ to the three sides of $ABC$ can be written in the form $a/b$ , where a and b are natural numbers such that the greatest common divisor of $a$ and $b$ is $1$. Find $a + b$. [b]p21.[/b] Let $m \ne 1$ be a square-free number (an integer – possibly negative – such that no square divides $m$). We denote $Q(\sqrt{m})$ to be the set of all $a + b\sqrt{m}$ where $a$ and $b$ are rational numbers. Now for a fixed $m$, let $S$ be the set of all numbers $x$ in $Q(\sqrt{m})$ such that x is a solution to a polynomial of the form: $x^n + a_1x^{n-1} + .... + a_n = 0$, where $a_0$, $...$, $a_n$ are integers. For many integers m, $S = Z[\frac{m}] = \{a + b\sqrt{m}\}$ where $a$ and $b$ are integers. Give a classification of the integers for which this is not true. (Hint: It is true for $ m = -1$ and $2$.) PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782002p24434611]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A circle intersects square $ABCD$ at points $A, E$, and $F$, where $E$ lies on $AB$ and $F$ lies on $AD$, such that $AE + AF = 2(BE + DF)$. If the square and the circle each have area $ 1$, determine the area of the union of the circle and square.

The number $2017$ is prime. Let $S=\sum_{k=0}^{62}\binom{2014}{k}$. What is the remainder when $S$ is divided by $2017$? $\textbf{(A) }32\qquad \textbf{(B) }684\qquad \textbf{(C) }1024\qquad \textbf{(D) }1576\qquad \textbf{(E) }2016\qquad$

Find all positive integers $ k$ for which the following statement is true: If $ F(x)$ is a polynomial with integer coefficients satisfying the condition $ 0 \leq F(c) \leq k$ for each $ c\in \{0,1,\ldots,k \plus{} 1\}$, then $ F(0) \equal{} F(1) \equal{} \ldots \equal{} F(k \plus{} 1)$.

A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might begin with the terms $ 247,$ $ 275,$ and $ 756$ and end with the term $ 824.$ Let $ \mathcal{S}$ be the sum of all the terms in the sequence. What is the largest prime factor that always divides $ \mathcal{S}?$ $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 37 \qquad \textbf{(E)}\ 43$

Let $A_1, B_1$ and $C_1$ be points on sides $BC$, $CA$ and $AB$ of an acute triangle $ABC$ respectively, such that $AA_1$, $BB_1$ and $CC_1$ are the internal angle bisectors of triangle $ABC$. Let $I$ be the incentre of triangle $ABC$, and $H$ be the orthocentre of triangle $A_1B_1C_1$. Show that $$AH + BH + CH \geq AI + BI + CI.$$

Determine the greatest possible value of the constant $c$ that satisfies the following condition: for any convex heptagon the sum of the lengthes of all it’s diagonals is greater than $cP$, where $P$ is the perimeter of the heptagon. (I. Zhuk)

From the two cities of Dobruisk and Bodruisk, the distance between which is $40$ km, two cyclists Dobi and Bodi simultaneously rode towards each other. Dobie was traveling at $23$ km/h and Bodie was traveling at $17$ km/h. Before departure, a fly landed on Dobie’s nose, which, at the moment of his departure from the city, flew towards Bodruisk at a speed of $40$ km/h. The fly met Bodie, immediately turned back and flew towards Dobruisk at a speed of $30$ km/h. (The fact is that the wind was blowing from Dobruisk to Bodruisk.) Having met Doby, the fly turned back again, etc. Determine the total path that the fly flew until the moment the cyclists met. (The speed of the fly in each direction did not change.)

Inside a given convex quadrilateral, find a point such that the segments connecting this point with the midpoints of the quadrilateral's sides divide the quadrilateral into four parts with equal areas.

Let $C$ denote the family of continuous functions on the real axis. Let $T$ be a mapping of $C$ into $C$ which has the following properties: 1. $T$ is linear, i.e. $T(c_1\psi _1+c_2\psi _2)= c_1T\psi _1+c_2T\psi _2$ for $c_1$ and $c_2$ real and $\psi_1$ and $\psi_2$ in $C$. 2. $T$ is local, i.e. if $\psi_1 \equiv \psi_2$ in some interval $I$ then also $T\psi_1 \equiv T\psi_2$ holds in $I$. Show that $T$ must necessarily be of the form $T\psi(x)=f(x)\psi(x)$ where $f(x)$ is a suitable continuous function.

Let $a$ and $b$ be relatively prime positive integers, $b$ even. For each positive integer $q$, let $p=p(q)$ be chosen so that $$ \left| \frac{p}{q} - \frac{a}{b} \right|$$ is a minimum. Prove that $$ \lim_{n \to \infty} \sum_{q=1 }^{n} \frac{ q\left| \frac{p}{q} - \frac{a}{b} \right|}{n} = \frac{1}{4}.$$

Let $ABC$ be an acute triangle with $AB<AC$. Point $M{}$ from the side $(BC)$ is the foot of the bisector from the vertex $A{}$. The perpendicular bisector of the segment $[AM]$ intersects the side $(AC)$ in $E{}$, the side $(AB)$ in $D$ and the line $(BC)$ in $F{}$. Prove that $\frac{DB}{CE}=\frac{FB}{FC}=\left(\frac{AB}{AC}\right)^2$.