For how many pairs of primes $(p, q)$, is $p^2 + 2pq^2 + 1$ also a prime?

Points $B$,$D$ , and $J$ are midpoints of the sides of right triangle $ACG$ . Points $K$, $E$, $I$ are midpoints of the sides of triangle , etc. If the dividing and shading process is done 100 times (the first three are shown) and $ AC=CG=6 $, then the total area of the shaded triangles is nearest [asy] draw((0,0)--(6,0)--(6,6)--cycle); draw((3,0)--(3,3)--(6,3)); draw((4.5,3)--(4.5,4.5)--(6,4.5)); draw((5.25,4.5)--(5.25,5.25)--(6,5.25)); fill((3,0)--(6,0)--(6,3)--cycle,black); fill((4.5,3)--(6,3)--(6,4.5)--cycle,black); fill((5.25,4.5)--(6,4.5)--(6,5.25)--cycle,black); label("$A$",(0,0),SW); label("$B$",(3,0),S); label("$C$",(6,0),SE); label("$D$",(6,3),E); label("$E$",(6,4.5),E); label("$F$",(6,5.25),E); label("$G$",(6,6),NE); label("$H$",(5.25,5.25),NW); label("$I$",(4.5,4.5),NW); label("$J$",(3,3),NW); label("$K$",(4.5,3),S); label("$L$",(5.25,4.5),S);[/asy] $ \text{(A)}\ 6\qquad\text{(B)}\ 7\qquad\text{(C)}\ 8\qquad\text{(D)}\ 9\qquad\text{(E)}\ 10 $

In a group of $2n$ students, each student has exactly $3$ friends within the group. The friendships are mutual and for each two students $A$ and $B$ which are not friends, there is a sequence $C_1, C_2, ..., C_r$ of students such that $A$ is a friend of $C_1$, $C_1$ is a friend of $C_2$, et cetera, and $C_r$ is a friend of $B$. Every student was asked to assess each of his three friendships with: "acquaintance", "friend" and "BFF". It turned out that each student either gave the same assessment to all of his friends or gave every assessment exactly once. We say that a pair of students is in conflict if they gave each other different assessments. Let $D$ be the set of all possible values of the total number of conflicts. Prove that $|D| \geq 3n$ with equality if and only if the group can be partitioned into two subsets such that each student is separated from all of his friends.

Since counting the numbers from 1 to 100 wasn't enough to stymie Gauss, his teacher devised another clever problem that he was sure would stump Gauss. Defining $\zeta_{15} = e^{2\pi i/15}$ where $i = \sqrt{-1}$, the teacher wrote the 15 complex numbers $\zeta_{15}^k$ for integer $0 \le k < 15$ on the board. Then, he told Gauss: On every turn, erase two random numbers $a, b$, chosen uniformly randomly, from the board and then write the term $2ab - a - b + 1$ on the board instead. Repeat this until you have one number left. What is the expected value of the last number remaining on the board?

There are $200$ numbers on a blackboard: $ 1! , 2! , 3! , 4! , ... ... , 199! , 200!$. Julia erases one of the numbers. When Julia multiplies the remaining $199$ numbers, the product is a perfect square. Which number was erased?

Let $ H(n)$ be the number of simply connected subsets with $ n$ hexagons in an infinite hexagonal network. Also let $ P(n)$ be the number of paths starting from a fixed vertex (that do not connect itself) with lentgh $ n$ in this hexagonal network. a) Prove that the limits \[ \alpha: \equal{}\lim_{n\rightarrow\infty}H(n)^{\frac1n}, \beta: \equal{}\lim_{n\rightarrow\infty}P(n)^{\frac1n}\]exist. b) Prove the following inequalities: $ \sqrt2\leq\beta\leq2$ $ \alpha\leq 12.5$ $ \alpha\geq3.5$ $ \alpha\leq\beta^4$

Find all monic polynomials $f$ with integer coefficients satisfying the following condition: there exists a positive integer $N$ such that $p$ divides $2(f(p)!)+1$ for every prime $p>N$ for which $f(p)$ is a positive integer. [i]Note: A monic polynomial has a leading coefficient equal to 1.[/i] [i](Greece - Panagiotis Lolas and Silouanos Brazitikos)[/i]

Find all 6-digit numbers $a_1a_2a_3a_4a_5a_6$ formed by using the digits $1,2,3,4,5,6$ once each such that the number $a_1a_2a_2\ldots a_k$ is divisible by $k$ for $1 \leq k \leq 6$.

$h_a$, $h_b$ and $h_c$ are altitudes, $t_a$, $t_b$ and $t_c$ are medians of acute triangle, $r$ radius of incircle, and $R$ radius of circumcircle of acute triangle $ABC$. Prove that $$\frac{t_a}{h_a}+\frac{t_b}{h_b}+\frac{t_c}{h_c} \leq 1+ \frac{R}{r}$$

Determine all real numbers $x$ such that $$\frac{2002\lfloor x\rfloor}{\lfloor-x\rfloor+x}>\frac{\lfloor2x\rfloor}{x-\lfloor1+x\rfloor}.$$

Let $ABC$ be an isosceles right triangle at $A$ with $AB=AC$. Let $M$ and $N$ be on side $BC$, with $M$ between $B$ and $N,$ such that $$BM^2+ NC^2= MN^2.$$ Determine the measure of the angle $\angle MAN.$

In order to draw a graph of $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$, a table of values was constructed. These values of the function for a set of equally spaced increasing values of $ x$ were $ 3844$, $ 3969$, $ 4096$, $ 4227$, $ 4356$, $ 4489$, $ 4624$, and $ 4761$. The one which is incorrect is: $ \textbf{(A)}\ 4096 \qquad\textbf{(B)}\ 4356 \qquad\textbf{(C)}\ 4489 \qquad\textbf{(D)}\ 4761 \qquad\textbf{(E)}\ \text{none of these}$

Let $ABC$ be an isosceles triangle ($AB=AC$) with incenter $I$. Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$. $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$, respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$. Prove that $AD$, $MN$ and $BC$ are concurrent. [i]Proposed by Alireza Dadgarnia[/i]

Find all possible integer values of the sum: $$\frac{a}{b}+ \frac{2023 \times b}{4 \times a},$$ where $a$ and $b$ are positive integers with no prime factors in common.

Let $ABC$ be a triangle with $AB = AC$ and $\angle BAC = 30^{\circ}$, Let $A'$ be the reflection of $A$ in the line $BC$; $B'$ be the reflection of $B$ in the line $CA$; $C'$ be the reflection of $C$ in line $AB$, Show that $A'B'C'$ is an equilateral triangle.

Let $a,b,c$ positive real numbers and $a+b+c = 1$. Prove that \[a^2 + b^2 + c^2 + \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} \ge 2(ab + bc + ac)\]

Positive integers $a$, $b$ and $c$ are positive integers with greatest common divisor equal to $1$ (i.e. they have no common divisors greater than $1$), and $$\frac{ab}{a-b}=c$$ Prove that $a -b$ is a perfect square. (SL Berlov)

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

Determine if there is a non-natural natural number $n$ with the property that $\sqrt{n + 1} + \sqrt{n - 1}$ is rational.

(a) Let $f: \mathbb{R}^{n\times n}\rightarrow\mathbb{R}$ be a linear mapping. Prove that $\exists ! C\in\mathbb{R}^{n\times n}$ such that $f(A)=Tr(AC), \forall A \in \mathbb{R}^{n\times n}$. (b) Suppose in addtion that $\forall A,B \in \mathbb{R}^{n\times n}: f(AB)=f(BA)$. Prove that $\exists \lambda \in \mathbb{R}: f(A)=\lambda Tr(A)$

In a bag of marbles, $\frac{3}{5}$ of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red? $ \textbf{(A)}\ \dfrac{2}{5} \qquad\textbf{(B)}\ \dfrac{3}{7} \qquad\textbf{(C)}\ \dfrac{4}{7} \qquad\textbf{(D)}\ \dfrac{3}{5} \qquad\textbf{(E)}\ \dfrac{4}{5} $

A football league has $n$ teams. Each team plays one game with every other team. Each win is awarded $2$ points, each tie $1$ point, and each loss $0$ points. After the league is over, the following statement is true: for every subset $S$ of teams in the league, there is a team (which may or may not be in $S$) such that the total points the team obtained by playing all the teams in $S$ is odd. Prove that $n$ is even.

Let $ABCD$ be a square pyramid of height $\frac{1}{2}$ with square base $ABCD$ of side length $AB=12$ (so $E$ is the vertex of the pyramid, and the foot of the altitude from $E$ to $ABCD$ is the center of square $ABCD$). The faces $ADE$ and $CDE$ meet at an acute angle of measure $\alpha$ (so that $0^{\circ}<\alpha<90^{\circ}$). Find $\tan \alpha$.

Let $ABC$ be an acute triangle with $AB<AC < BC$, inscirbed in circle $\Gamma_1$, with center $O$. Circle $\Gamma_2$, with center point $A$ and radius $AC$ intersects $BC$ at point $D$ and the circle $\Gamma_1$ at point $E$. Line $AD$ intersects circle $\Gamma_1$ at point $F$. The circumscribed circle $\Gamma_3$ of triangle $DEF$, intersects $BC$ at point $G$. Prove that: a) Point $B$ is the center of circle $\Gamma_3$ b) Circumscribed circle of triangle $CEG$ is tangent to $AC$.

A square is inscribed in an ellipse such that two sides of the square respectively pass through the two foci of the ellipse. The square has a side length of $4$. The square of the length of the minor axis of the ellipse can be written in the form $a + b\sqrt{c}$ where $a, b$, and $c$ are integers, and $c$ is not divisible by the square of any prime. Find the sum $a + b + c$.