How many primes $p$ are there such that $5p(2^{p+1}-1)$ is a perfect square? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None of the preceding} $

A line with negative slope passing through the point $(18,8)$ intersects the $x$ and $y$ axes at $(a,0)$ and $(0,b)$, respectively. What is the smallest possible value of $a+b$?

Given is a tree $G$ with $2023$ vertices. The longest path in the graph has length $2n$. A vertex is called good if it has degree at most $6$. Find the smallest possible value of $n$ if there doesn't exist a vertex having $6$ good neighbors.

Let $C_1$ and $C_2$ be circles of radius $1$ that are in the same plane and tangent to each other. How many circles of radius $3$ are in this plane and tangent to both $C_1$ and $C_2$? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8 $

Recall that a palindrome is a word which is the same when we read it forward or backward. (a) We have an infinite number of cards with words $\{abc, bca, cab\}$. A word is made from them in the following way. The initial word is an arbitrary card. At each step we obtain a new word either gluing a card (from the right or from the left) to the existing word or making a cut between any two of its letters and gluing a card between both parts. Is it possible to obtain a palindrome this way? [i](4 points)[/i] (b) We have an infinite number of red cards with words $\{abc, bca, cab\}$ and of blue cards with words $\{cba, acb, bac\}$. A palindrome was formed from them in the same way as in part (a). Is it necessarily true that the number of red and blue cards used was equal? [i](6 points)[/i] [i]Alexandr Gribalko, Ivan Mitrofanov [/i]

Quadrilateral $ABCD$ is circumscribed around a circle. Diagonals $AC,BD$ are not perpendicular to each other. The angle bisectors of angles between these diagonals, intersect the segments $AB,BC,CD$ and $DA$ at points $K,L,M$ and $N$. Given that $KLMN$ is cyclic, prove that so is $ABCD$. Proposed by Nikolai Beluhov (Bulgaria)

The perimeter of one square is $3$ times the perimeter of another square. The area of the larger square is how many times the area of the smaller square? $\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 9$

Let $\ell$ be a line in the plane. Two circles with respective radii $2$ and $4$ are tangent to $\ell$ on the same side so that their points of tangency are distance $9$ apart. The two common internal tangents to both circles are drawn. What is the area of the triangle formed by the line $\ell$ and the two internal tangents? $\text{(A) }\frac{25}{3}\qquad\text{(B) }\frac{26}{3}\qquad\text{(C) }9\qquad\text{(D) }\frac{28}{3}\qquad\text{(E) }\frac{29}{3}$

Let $ABC$ be a triangle with $AB < AC$ and circumcenter $O$. The angle bisector of $\angle BAC$ meets the side $BC$ at $D$. The line through $D$ perpendicular to $BC$ meets the segment $AO$ at $X$. Furthermore, let $Y$ be the midpoint of segment $AD$. Prove that points $B, C, X, Y$ are concyclic.

Let $ABC$ be a given acute triangle, in which $BC$ is the longest side. Let $H$ be the orthocenter of the triangle, and let $D$ and $E$ be the feet of the altitudes from $B$ and $C$, respectively. Let $F$ and $G$ be the midpoints of sides $AB$ and $AC$, respectively. $X$ is the point of intersection of lines $DF$ and $EG$. Let $O_1$ and $O_2$ be the circumcenters of triangles $EFX$ and $DGX$, respectively. Finally, $M$ is the midpoint of line segment $O_1O_2$. Prove that points $X, H$ and $M$ are collinear.

Let a real number $\lambda > 1$ be given and a sequence $(n_k)$ of positive integers such that $\frac{n_{k+1}}{n_k}> \lambda$ for $k = 1, 2,\ldots$ Prove that there exists a positive integer $c$ such that no positive integer $n$ can be represented in more than $c$ ways in the form $n = n_k + n_j$ or $n = n_r - n_s$.

Let $f : R \to R$ and $g : R \to R$ be strictly increasing linear functions such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that $f(x) - g(x)$ is an integer for any $x \in R$.

Two circles $ \Omega_{1}$ and $ \Omega_{2}$ are externally tangent to each other at a point $ I$, and both of these circles are tangent to a third circle $ \Omega$ which encloses the two circles $ \Omega_{1}$ and $ \Omega_{2}$. The common tangent to the two circles $ \Omega_{1}$ and $ \Omega_{2}$ at the point $ I$ meets the circle $ \Omega$ at a point $ A$. One common tangent to the circles $ \Omega_{1}$ and $ \Omega_{2}$ which doesn't pass through $ I$ meets the circle $ \Omega$ at the points $ B$ and $ C$ such that the points $ A$ and $ I$ lie on the same side of the line $ BC$. Prove that the point $ I$ is the incenter of triangle $ ABC$. [i]Alternative formulation.[/i] Two circles touch externally at a point $ I$. The two circles lie inside a large circle and both touch it. The chord $ BC$ of the large circle touches both smaller circles (not at $ I$). The common tangent to the two smaller circles at the point $ I$ meets the large circle at a point $ A$, where the points $ A$ and $ I$ are on the same side of the chord $ BC$. Show that the point $ I$ is the incenter of triangle $ ABC$.

The three roots of the cubic $ 30 x^3 \minus{} 50x^2 \plus{} 22x \minus{} 1$ are distinct real numbers between $ 0$ and $ 1$. For every nonnegative integer $ n$, let $ s_n$ be the sum of the $ n$th powers of these three roots. What is the value of the infinite series \[ s_0 \plus{} s_1 \plus{} s_2 \plus{} s_3 \plus{} \dots \, ?\]

Given two acute-angled triangles $ABC$ and $A'B'C'$ with the points $O$ and $O'$ inside. Three pairs of the perpendiculars are drawn: $[OA_1]$ to the side $[BC]$, $[O'A'_1]$ to the side $[B'C']$, $[OB_1]$ to the side $[AC]$, $[O'B'_1]$ to the side $[A'C']$, $[OC_1] $ to the side $[AB]$, $[O'C'_1]$ to the side $[A'B']$; It is known that $$[OA_1] \parallel [O'A'], [OB_1] \parallel [O'B'], [OC_1] \parallel [O'C'] $$ and $$|OA_1|\cdot|O'A'| = |OB_1|\cdot |O'B'| = |OC_1|\cdot |O'C'|$$ Prove that $$[O'A'_1] \parallel [OA], [O'B'_1] \parallel[OB], [O'C'_1] \parallel[OC]$$ and $$|O'A'_1|\cdot|OA| = |O'B'_1|\cdot|OB| = |O'C'_1|\cdot|OC|$$

Find all pairs of positive integers $ m$ and $ n$ that satisfy (both) following conditions: (i) $ m^{2}\minus{}n$ divides $ m\plus{}n^{2}$ (ii) $ n^{2}\minus{}m$ divides $ n\plus{}m^{2}$

Adrian has $2n$ cards numbered from $ 1$ to $2n$. He gets rid of $n$ cards that are consecutively numbered. The sum of the numbers of the remaining papers is $1615$. Find all the possible values of $n$.

Determine the largest $k\ge0$ such that the inequality \[\left(\sum_{j=1}^n x_j\right)^2\left(\sum_{j=1}^n x_jx_{j+1}\right)\ge k\sum_{j=1}^n x_j^2x_{j+1}^2\] holds for every $n\ge2$ and any $n$-tuple $x_1,\ldots,x_n$ of non-negative numbers (given that $x_{n+1}=x_1$)

Emily draws six dots on a piece of paper such that no three lie on a straight line, then draws a line segment connecting each pair of dots. She then colors five of these segments red. Her coloring is said to be $\emph{red-triangle-free}$ if for every set of three points from her six drawn points there exists an uncolored segment connecting two of the three points. In how many ways can Emily color her drawing such that it is red-triangle-free?

Suppose we have $10$ balls and $10$ colors. For each ball, we (independently) color it one of the $10$ colors, then group the balls together by color at the end. If $S$ is the expected value of the square of the number of distinct colors used on the balls, find the sum of the digits of $S$ written as a decimal. [i]Proposed by Michael Kural[/i]

Let $ABC$ be a triangle with $AB = 9$, $BC = 10$, and $CA = 17$. Let $B'$ be the reflection of the point $B$ over the line $CA$. Let $G$ be the centroid of triangle $ABC$, and let $G'$ be the centroid of triangle $AB'C$. Determine the length of segment $GG'$.

A convex hexagon $ABCDEF$ is given where $ \measuredangle FAB + \measuredangle BCD + \measuredangle DEF = 360^{\circ}$ and $ \measuredangle AEB = \measuredangle ADB$. Suppose the lines $AB$ and $DE$ are not parallel. Prove that the circumcenters of the triangles $ \triangle AFE, \triangle BCD$ and the intersection of the lines $AB$ and $DE$ are collinear.

Square trinomials $P(x) = x^2 + ax + b$ and $Q(x) = x^2 + cx + d$ are such that the equation $P(Q(x)) = Q(P(x))$ has no real roots. Prove that $b \ne d$.

Let $P$ be a variable point on arc $BC$ of the circumcircle of triangle $ABC$ not containing $A$. Let $I_1$ and $I_2$ be the incenters of the triangles $PAB$ and $PAC$, respectively. Prove that: [b](a)[/b] The circumcircle of $?PI_1I_2$ passes through a fixed point. [b](b)[/b] The circle with diameter $I_1I_2$ passes through a fixed point. [b](c)[/b] The midpoint of $I_1I_2$ lies on a fixed circle.

Find all the pairs of positive integers $(x,y)$ such that $x\leq y$ and \[\frac{(x+y)(xy-1)}{xy+1}=p,\]where $p$ is a prime number.