The quadrilateral $ABCD$ is inscribed in the circle ω. The diagonals $AC$ and $BD$ intersect at the point $O$. On the segments $AO$ and $DO$, the points $E$ and $F$ are chosen, respectively. The straight line $EF$ intersects ω at the points $E_1$ and $F_1$. The circumscribed circles of the triangles $ADE$ and $BCF$ intersect the segment $EF$ at the points $E_2$ and $F_2$ respectively (assume that all the points $E, F, E_1, F_1, E_2$ and $F_2$ are different). Prove that $E_1E_2 = F_1F_2$. $(N. Sedrakyan)$

Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard: [list] [*]In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin. [*]In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are two (not necessarily distinct) numbers from the first line. [*]In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line. [/list] Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.

$n$ coins lies in the circle. If two neighbour coins lies both head up or both tail up, then we can flip both. How many variants of coins are available that can not be obtained from each other by applying such operations?

Call an arrangement of n not necessarily distinct nonnegative integers in a circle [i]wholesome[/i] when, for any subset of the integers such that no pair of them is adjacent in the circle, their average is an integer. Over all wholesome arrangements of $n$ integers where at least two of them are distinct, let $M(n)$ denote the smallest possible value for the maximum of the integers in the arrangement. What is the largest integer $n < 2023$ such that $M(n+1)$ is strictly greater than $M(n)$?

Find all pairs of primes $(p, q)$ such that $p^3 - q^5 = (p + q)^2$ .

Evaluate $$\sum_{i=0}^{\infty}\frac{7^i}{(7^i+1)(7^i+7)}$$ [i]Proposed by Connor Gordon[/i]

There is child camp with some rooms. Call room as $4-$room, if $4$ children live here. Not less then half of all rooms are $4-$rooms , other rooms are $3-$rooms. Not less than $2/3$ girls live in $3-$rooms. Prove that not less than $35\%$ of all children are boys.

Hari is obsessed with cubics. He comes up with a cubic with leading coefficient 1, rational coefficients and real roots $0 < a < b < c < 1$. He knows the following three facts: $P(0) = -\frac{1}{8}$, the roots form a geometric progression in the order $a,b,c$, and \[ \sum_{k=1}^{\infty} (a^k + b^k + c^k) = \dfrac{9}{2} \] The value $a + b + c$ can be expressed as $\frac{m}{n}$, where $m,n$ are relatively prime positive integers. Find $m + n$. [i]Proposed by Akshar Yeccherla (TopNotchMath)[/i]

We have a table in the form of a regular polygon with $1000$ sides, where each side has length $1$. At one of the vertices is a beetle (consider this vertex to be fixed). The $1000$ vertices must be numbered, in some order, using the numbers $1, 2,\ldots ,1000$ such that the beetle is at vertex $1$. The beetle can only move along the edge of the table and always moves clockwise. The beetle moves from vertex $1$ to vertex $2$ and stops there. then it moves from vertex $2$ to vertex $3$, and stops there. So on, until the beetle ends its journey at vertex $1000$. Find the number of ways the numbers can be assigned to the vertices so that the total length of the beetle's journey is $2017$.

Calculate $ \displaystyle \left|\frac {\int_0^{\frac {\pi}{2}} (x\cos x + 1)e^{\sin x}\ dx}{\int_0^{\frac {\pi}{2}} (x\sin x - 1)e^{\cos x}\ dx}\right|$.

Let $ABC$ be a triangle in a plane $\pi$. Find the set of all points $P$ (distinct from $A,B,C$ ) in the plane $\pi$ such that the circumcircles of triangles $ABP$, $BCP$, $CAP$ have the same radii.

Find all non empty subset $ S$ of $ \mathbb{N}: \equal{} \{0,1,2,\ldots\}$ such that $ 0 \in S$ and exist two function $ h(\cdot): S \times S \to S$ and $ k(\cdot): S \to S$ which respect the following rules: i) $ k(x) \equal{} h(0,x)$ for all $ x \in S$ ii) $ k(0) \equal{} 0$ iii) $ h(k(x_1),x_2) \equal{} x_1$ for all $ x_1,x_2 \in S$. [i](Pierfrancesco Carlucci)[/i]

For a function satisfying $ f'(x) > 0$ for $ a\leq x\leq b$, let $ F(x) \equal{} \int_a^b |f(t) \minus{} f(x)|\ dt$. For what value of $ x$ is $ F(x)$ is minimized?

Find all the pairs of integers $ (a,b)$ satisfying $ ab(a \minus{} b)\not \equal{} 0$ such that there exists a subset $ Z_{0}$ of set of integers $ Z,$ for any integer $ n$, exactly one among three integers $ n,n \plus{} a,n \plus{} b$ belongs to $ Z_{0}$.

suppose that $a=3^{100}$ and $b=5454$. how many $z$s in $[1,3^{99})$ exist such that for every $c$ that $gcd(c,3)=1$, two equations $x^z\equiv c$ and $x^b\equiv c$ (mod $a$) have the same number of answers?($\frac{100}{6}$ points)

A right prism \(ABCA_1B_1C_1\) is given. It is known that triangles \(A_1BC\), \(AB_1C\), \(ABC_1\), and \(ABC\) are all acute. Prove that the orthocenters of these triangles, together with the centroid of triangle \(ABC\), lie on the same sphere.

What is the value of \[(\log 5)^{3}+(\log 20)^{3}+(\log 8)(\log 0.25)\] where $\log$ denotes the base-ten logarithm? $\textbf{(A)}~\displaystyle\frac{3}{2}\qquad\textbf{(B)}~\displaystyle\frac{7}{4}\qquad\textbf{(C)}~2\qquad\textbf{(D)}~\displaystyle\frac{9}{4}\qquad\textbf{(E)}~3$

Alex has one-pound red bricks and two-pound blue bricks, and has 360 total pounds of brick. He observes that it is impossible to rearrange the bricks into piles that all weigh three pounds, but he can put them in piles that each weigh five pounds. Finally, when he tries to put them into piles that all have three bricks, he has one left over. If Alex has $r$ red bricks, find the number of values $r$ could take on.

Let be a finite group $ G $ having an endomorphism $ \eta $ that has exactly one fixed point. [b]a)[/b] Demonstrate that the function $ f:G\longrightarrow G $ defined as $ f(x)=x^{-1}\cdot\eta (x) $ is bijective. [b]b)[/b] Show that $ G $ is commutative if the composition of the function $ f $ from [b]a)[/b] with itself is the identity function.

In regular tetrahedron $ABCD$, $E,F,G$ are midpoints of $AB,BC,CD$. Dihedral angle $C-FG-E$ is equal to $\text{(A)}\arcsin\frac{\sqrt6}{3}\qquad\text{(B)}\frac{\pi}{2}+\arccos\frac{\sqrt3}{3}\qquad\text{(C)}\frac{\pi}{2}-\arctan{\sqrt2}\qquad\text{(D)}\pi-\text{arccot}\frac{\sqrt2}{2}$

Let $x,y,z$ be non-negative numbers such that $x+y+z \leq 3$. Prove that $$\frac{2}{1+x}+\frac{2}{1+y}+\frac{2}{1+z} \geq 3$$

A positive integer is [i]dapper[/i] if at least one of its multiples begins with $ 2008$. For example, $ 7$ is dapper because $ 200858$ is a multiple of $ 7$ and begins with $ 2008$. Observe that $ 200858 \equal{} 28694\times 7$. Prove that every positive integer is dapper.

How many integers $ x $ satisfy $ x^2 - 9x + 18 < 0 $?

Let $S$ be the set of positive integers $n$ such that \[3\cdot \varphi (n)=n,\] where $\varphi (n)$ is the number of positive integers $k\leq n$ such that $\gcd (k, n)=1$. Find \[\sum_{n\in S} \, \frac{1}{n}.\] [i]Proposed by Nathan Ramesh

Let $[ABC]$ be a triangle and $D$ a point between $A$ and $B$. If the triangles $[ABC], [ACD]$ and $[BCD]$ are all isosceles, what are the possible values of $\angle ABC$?