If $a$ and $b$ are randomly selected real numbers between $0$ and $1$, find the probability that the nearest integer to $\frac{a-b}{a+b}$ is odd.

Gandalf (the wizard) and Bilbo (the assistant) are presenting a magic trick to Nitzan (the audience). While Gandalf leaves the room, Nitzan chooses a number $1\leq x\leq 2^{2022}$ and shows it to Bilbo. Now bilbo writes on the board a long row of $N$ digits, each of which is $0$ or $1$. After this Nitzan can, if he wishes, switch the order of two consecutive digits in the row, but only once. Then Gandalf returns to the room, looks at the row, and guesses the number $x$. Can Bilbo and Gandalf come up with a strategy that allows Gandalf to guess $x$ correctly no matter how Nitzan acts, if [b]a)[/b] $N=2500$? [b]b)[/b] $N=2030$? [b]c)[/b] $N=2040$?

Given a table in a form of the regular $1000$-gon with sidelength $1$. A Beetle initially is in one of its vertices. All $1000$ vertices are numbered in some order by numbers $1$, $2$, $\ldots$, $1000$ so that initially the Beetle is in the vertex $1$. The Beetle can move only along the edges of $1000$-gon and only clockwise. He starts to move from vertex $1$ and he is moving without stops until he reaches vertex $2$ where he has a stop. Then he continues his journey clockwise from vertex $2$ until he reaches the vertex $3$ where he has a stop, and so on. The Beetle finishes his journey at vertex $1000$. Find the number of ways to enumerate all vertices so that the total length of the Beetle's journey is equal to $2017$.

$(FRA 4)$ A right-angled triangle $OAB$ has its right angle at the point $B.$ An arbitrary circle with center on the line $OB$ is tangent to the line $OA.$ Let $AT$ be the tangent to the circle different from $OA$ ($T$ is the point of tangency). Prove that the median from $B$ of the triangle $OAB$ intersects $AT$ at a point $M$ such that $MB = MT.$

For the reaction: $2X + 3Y \rightarrow 3Z$, the combination of $2.00$ moles of $X$ with $2.00$ moles of $Y$ produces $1.75 $ moles of $Z$. What is the percent yield of this reaction? $\textbf{(A)}\hspace{.05in}43.8\%\qquad\textbf{(B)}\hspace{.05in}58.3\%\qquad\textbf{(C)}\hspace{.05in}66.7\%\qquad\textbf{(D)}\hspace{.05in}87.5\%\qquad $

Find the maximum number of points on a sphere of radius $1$ in $\mathbb{R}^n$ such that the distance between any two of these points is strictly greater than $\sqrt{2}$.

Mary chose an even $4$-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,...,\tfrac{n}{2},n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$? $\textbf{(A) } 324 \qquad \textbf{(B) } 330 \qquad \textbf{(C) } 340 \qquad \textbf{(D) } 361 \qquad \textbf{(E) } 646$

Find all surjective functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(n) \ge n+(-1)^{n}.\]

Find all functions $f:[0,1] \to \mathbb{R}$ such that the inequality \[(x-y)^2\leq|f(x) -f(y)|\leq|x-y|\] is satisfied for all $x,y\in [0,1]$

For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.

For square $ABCD$, $M$ is a midpoint of segment $CD$ and $E$ is a point on $AD$ satisfying $\angle BEM = \angle MED$. $P$ is an intersection of $AM$, $BE$. Find the value of $\frac{PE}{BP}$

For each integer $n>1$, let $p(n)$ denote the largest prime factor of $n$. Determine all triples $(x, y, z)$ of distinct positive integers satisfying [list] [*] $x, y, z$ are in arithmetic progression, [*] $p(xyz) \le 3$. [/list]

The natural numbers from $1$ to $50$ are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime?

Prove that a convex polygon can be cut by disjoint diagonals into acute triangles in at least one way.

Let $ n\ge 2 $ be a positive integer and $ S= \{1,2,\cdots ,n\} $. Let two functions $ f:S \rightarrow \{1,-1\} $ and $ g:S \rightarrow S $ satisfy: i) $ f(x)f(y)=f(x+y) , \forall x,y \in S $ \\ ii) $ f(g(x))=f(x) , \forall x \in S $\\ iii) $f(x+n)=f(x) ,\forall x \in S$\\ iv) $ g $ is bijective.\\ Find the number of pair of such functions $ (f,g)$ for every $n$.

In an acute-angled triangle $ABC$, $M$ and $N$ are points on the sides $AC$ and $BC$ respectively, and $K$ the midpoint of $MN$. The circumcircles of triangles $ACN$ and $BCM$ meet again at a point $D$. Prove that the line $CD$ contains the circumcenter $O$ of $\triangle ABC$ if and only if $K$ is on the perpendicular bisector of $AB.$

Let $F$ be the collection of subsets of a set with $n$ elements such that no element of $F$ is a subset of another of its elements. Prove that $$|F|\le\binom n{\lfloor n/2\rfloor}.$$

Given quadrilateral $ABCD$ with $AB=BC=CD$. Let $AC\cap BD=O$, $X,Y$ are symmetry points of $O$ respect to midpoints of $BC$, $AD$, and $Z$ is intersection point of lines, which perpendicular bisects of $AC$, $BD$. Prove that $X,Y,Z$ are collinear.

On each day of their tour of the West Indies, Sourav and Srinath have either an apple or an orange for breakfast. Sourav has oranges for the first $m$ days, apples for the next $m$ days, followed by oranges for the next $m$ days, and so on. Srinath has oranges for the first $n$ days, apples for the next $n$ days, followed by oranges for the next $n$ days, and so on. If $\gcd(m,n)=1$, and if the tour lasted for $mn$ days, on how many days did they eat the same kind of fruit?

For each $ x$ in $ [0,1]$, define \[ f(x)=\begin{cases}2x, &\text { if } 0 \leq x \leq \frac {1}{2}; \\ 2 - 2x, &\text { if } \frac {1}{2} < x \leq 1. \end{cases} \]Let $ f^{[2]}(x) = f(f(x))$, and $ f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $ n \geq 2$. For how many values of $ x$ in $ [0,1]$ is $ f^{[2005]}(x) = \frac {1}{2}$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2005 \qquad \textbf{(C)}\ 4010 \qquad \textbf{(D)}\ 2005^2 \qquad \textbf{(E)}\ 2^{2005}$

Into how many regions do $n$ great circles, no three of which meet at a point, divide a sphere?

Show that in a tournament of $799$ teams (every team plays with every other team for a win or loss), there exist $14$ teams such that the first seven teams have each defeated the remaining teams.

We call a monic polynomial $P(x) \in \mathbb{Z}[x]$ [i]square-free mod n[/i] if there [u]dose not[/u] exist polynomials $Q(x),R(x) \in \mathbb{Z}[x]$ with $Q$ being non-constant and $P(x) \equiv Q(x)^2 R(x) \mod n$. Given a prime $p$ and integer $m \geq 2$. Find the number of monic [i]square-free mod p[/i] $P(x)$ with degree $m$ and coeeficients in $\{0,1,2,3,...,p-1\}$. [i]Proposed by Masud Shafaie[/i]

In acute-angled triangle $ABC$, $BH$ is the altitude of the vertex $B$. The points $D$ and $E$ are midpoints of $AB$ and $AC$ respectively. Suppose that $F$ be the reflection of $H$ with respect to $ED$. Prove that the line $BF$ passes through circumcenter of $ABC$. by Davood Vakili

Complex numbers $|z_1|=2,|z_2|=3$, and the intersection angle between the vectors corresponding to $z_1,z_2$ is $60^{\circ}$, then $\frac{|z_1+z_2|}{|z_1-z_2|}=$________.