A twin prime pair is a pair of primes $(p,q)$ such that $q = p + 2$. The Twin Prime Conjecture states that there are infinitely many twin prime pairs. What is the arithmetic mean of the two primes in the smallest twin prime pair? (1 is not a prime.) $\textbf{(A) }4$

Function $f : \mathbb{R^+} \rightarrow \mathbb{R^+}$ satisfies the following condition. (Condition) For each positive real number $x$, there exists a positive real number $y$ such that $(x + f(y))(y + f(x)) \leq 4$, and the number of $y$ is finite. Prove $f(x) > f(y)$ for any positive real numbers $x < y$. ($\mathbb{R^+}$ is a set for all positive real numbers.)

Points $A$ and $B$ are fixed on a circle $c_1$. Circle $c_2$, whose centre lies on $c_1$, touches line $AB$ at $B$. Another line through $A$ intersects $c_2$ at points $D$ and $E$, where $D$ lies between $A$ and $E$. Line $BD$ intersects $c_1$ again at $F$. Prove that line $EB$ is tangent to $c_1$ if and only if $D$ is the midpoint of the segment $BF$.

For what positive integers $n$ are there positive rational, but not integer, numbers $a$ and $b$ such that both numbers $a + b$ and $a^n + b^n$ are integers?

Denote by $l(n)$ the largest prime divisor of $n$. Let $a_{n+1} = a_n + l(a_n)$ be a recursively defined sequence of integers with $a_1 = 2$. Determine all natural numbers $m$ such that there exists some $i \in \mathbb{N}$ with $a_i = m^2$. [i]Proposed by Nikola Velov, North Macedonia[/i]

If $ x$, $ y$, and $ z$ are positive numbers satisfying \[x \plus{} 1/y \equal{} 4,\quad y \plus{} 1/z \equal{} 1,\quad\text{and}\quad z \plus{} 1/x \equal{} 7/3,\] then $ xyz \equal{}$ $ \textbf{(A)}\ 2/3 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 4/3 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 7/3$

Let $n$ be a fixed positive integer. There are $n \geq 1$ lamps in a row, some of them are on and some are off. In a single move, we choose a positive integer $i$ ($1 \leq i \leq n$) and switch the state of the first $i$ lamps from the left. Determine the smallest number $k$ with the property that we can make all of the lamps be switched on using at most $k$ moves, no matter what the initial configuration was. [i]Proposed by Viktor Simjanoski and Nikola Velov[/i]

Calculate the product $\tfrac13 \times \tfrac24 \times \tfrac35 \times \cdots \times \tfrac{18}{20} \times \tfrac{19}{21}$. $\textbf{(A) }\dfrac{1}{210}\qquad\textbf{(B) }\dfrac{1}{190}\qquad\textbf{(C) }\dfrac{1}{21}\qquad\textbf{(D) }\dfrac{1}{20}\qquad\textbf{(E) }\dfrac{1}{10}$

$ABCD$ is a rectangle. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ respectively so that $KL$ is parallel to $MN$, and $KM$ is perpendicular to $LN$. Show that the intersection of $KM$ and $LN$ lies on $BD$.

For every positive integer $x$, let $k(x)$ denote the number of composite numbers that do not exceed $x$. Find all positive integers $n$ for which $(k (n))! $ lcm $(1, 2,..., n)> (n - 1) !$ .

A line $d_i~(i=1,2,3)$ intersects two opposite sides of a square $ABCD$ at points $M_i$ and $N_i$. Prove that if $M_1N_1=M_2N_2=M_3N_3$, then two of the lines $d_i$ are either parallel or perpendicular.

Within an equilateral triangle $ABC$, take any point $P$. Let $L, M, N$ be the projections of $P$ on sides $AB, BC, CA$ respectively. Prove that $\frac{AP}{NL}=\frac{BP}{LM}=\frac{CP}{MN}$.

(a) Prove that there is a unique function $f : Q \to Q$ satisfying: (i) $f(q)= 1 + f\left(\frac{q}{1-2q}\right)$ for $0<q< \frac12$ (ii) $f(q)= 1 + f(q-1)$ for $1<q\le 2$ (iii) $f(q)f\left(\frac{1}{q}\right)=1$ for all $q\in Q^+$ (b) For this function $f$ , find all $r\in Q^+$ such that $f(r) = r$

Let $I $be the incenter of triangle $ABC$and $J$ the excenter of the side $BC$: Let $M$ be the midpoint of $CB$ and $N$ the midpoint of arc $BAC$ of circle $(ABC)$. If $T$ is the symmetric of the point $N$ by the point $A$, prove that the quadrilateral $JMIT$ is cyclic

Let $g:\mathbb{N}\to \mathbb{N}$ be a bijective function and suppose that $f:\mathbb{N}\to \mathbb{N}$ is a function such that: [list] [*] For all naturals $x$, $$\underbrace{f(\cdots (f}_{x^{2023}\;f\text{'s}}(x)))=x. $$ [*] For all naturals $x,y$ such that $x|y$, we have $f(x)|g(y)$. [/list] Prove that $f(x)=x$. [i]Proposed by Pulkit Sinha[/i]

Rhombus $ABCD$ has side length $2$ and $\angle B = 120 ^\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$? $ \textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad \textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad \textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad \textbf{(D)}\ 1+\frac{\sqrt{3}}{3} \qquad \textbf{(E)}\ 2 $

Find all positive integers $n$ and $k$ such that $(n+1)^n=2n^k+3n+1$.

Which of the following statements is (are) equivalent to the statement "If the pink elephant on planet alpha has purple eyes, then the wild pig on planet beta does not have a long nose"? $\textbf{I. }$ "If the wild pig on planet beta has a long nose, then the pink elephant on planet alpha has purple eyes." $\textbf{II. }$ "If the pink elephant on planet alpha does not have purple eyes, then the wild pig on planet beta does not have a long nose. $\textbf{III. }$ "If the wild pig on planet beta has a long nose, then the pink elephant on planet alpha does not have purple eyes." $ \textbf{IV. }$ "The pink elephant on planet alpha does not have purple eyes, or the wild pig on planet beta does not have a long nose." $\textbf{(A) }\textbf{I. }\text{and }\textbf{II. }\text{only}\qquad\textbf{(B) }\textbf{III. }\text{and }\textbf{IV. }\text{only}\qquad\textbf{(C) }\textbf{II. }\text{and }\textbf{IV. }\text{only}\qquad\textbf{(D) }\textbf{II. }\text{and }\textbf{III. }\text{only}\qquad \textbf{(E) }\text{and }\textbf{III. }\text{only}$

Does there exist a function $f : \mathbb{R} \rightarrow \mathbb{R}$, which satisfies both conditions : [b]a)[/b] $f( x + y + z) \leq 3(xy + yz + zx)$ for all real numbers $x , y , z$ and [b]b)[/b] there exist function $g$ and natural number $n$, such that $g(g(x)) = x ^ {2n + 1}$ and $f(g(x)) = (g(x)) ^2$ for every real number $x$ ?

Given sets $A = \{1, 4, 5, 6, 7, 9, 11, 16, 17\}$, $B = \{2, 3, 8, 10, 12, 13, 14, 15, 18\}$, if a positive integer leaves a remainder (the smallest non-negative remainder) that belongs to $A$ when divided by 19, then that positive integer is called an $\alpha$ number. If a positive integer leaves a remainder that belongs to $B$ when divided by 19, then that positive integer is called a $\beta$ number. (1) For what positive integer $n$, among all its positive divisors, are the numbers of $\alpha$ divisors and $\beta$ divisors equal? (2) For which positive integers $k$, are the numbers of $\alpha$ divisors less than the numbers of $\beta$ divisors? For which positive integers $l$, are the numbers of $\alpha$ divisors greater than the numbers of $\beta$ divisors?

In the plane, there are $n$ points ($n\ge 4$) where no 3 of them are collinear. Let $A(n)$ be the number of parallelograms whose vertices are those points with area $1$. Prove the following inequality: $A(n)\leq \frac{n^2-3n}{4}$ for all $n\ge 4$

It is given a triangle $ABC$. Let $R$ be the radius of the circumcircle of the triangle and $O_1,O_2,O_3$ be the centers of excircles of the triangle $ABC$ and $q$ is the perimeter of the triangle $O_1O_2O_3$. Prove that $q\le6R\sqrt3$. When does equality hold?

Given a cube with edge $ AB = a $ cm. Point $ M $ of segment $ AB $ is distant from the diagonal of the cube, which is oblique to $ AB $, by $ k $ cm. Find the distance of point $ M $ from the midpoint $ S $ of segment $ AB $.

Let $ a\geq 2$ be a real number; with the roots $ x_{1}$ and $ x_{2}$ of the equation $ x^2\minus{}ax\plus{}1\equal{}0$ we build the sequence with $ S_{n}\equal{}x_{1}^n \plus{} x_{2}^n$. [b]a)[/b]Prove that the sequence $ \frac{S_{n}}{S_{n\plus{}1}}$, where $ n$ takes value from $ 1$ up to infinity, is strictly non increasing. [b]b)[/b]Find all value of $ a$ for the which this inequality hold for all natural values of $ n$ $ \frac{S_{1}}{S_{2}}\plus{}\cdots \plus{}\frac{S_{n}}{S_{n\plus{}1}}>n\minus{}1$

A wall contains three switches $A,B,C$, each of which powers a light when flipped on. Every $20$ seconds, switch $A$ is turned on and then immediately turned off again. The same occurs for switch $B$ every $21$ seconds and switch $C$ every $22$ seconds. At time $t = 0$, all three switches are simultaneously on. Let $t = T > 0$ be the earliest time that all three switches are once again simultaneously on. Compute the number of times $t > 0$ before $T$ when at least two switches are simultaneously on.