Determine whether or not there exists a sequence of integers $a_1,a_2,\dots, a_{19}, a_{20}$ such that, the sum of all the terms is negative, and the sum of any three consecutive terms is positive.

Circles $\omega_1$ and $\omega_2$ are externally tangent to each other at $P$. A random line $\ell$ cuts $\omega_1$ at $A$ and $C$ and $\omega_2$ at $B$ and $D$ (points $A,C,B,D$ are in this order on $\ell$). Line $AP$ meets $\omega_2$ again at $E$ and line $BP$ meets $\omega_1$ again at $F$. Prove that the radical axis of circles $(PCD)$ and $(PEF)$ is parallel to $\ell$. \\ \\ [i](Vlad Robu)[/i]

Bob has invented the Very Normal Coin (VNC). When the VNC is flipped, it shows heads $\textstyle\frac{1}{2}$ of the time and tails $\textstyle\frac{1}{2}$ of the time - unless it has yielded the same result five times in a row, in which case it is guaranteed to yield the opposite result. For example, if Bob flips five heads in a row, then the next flip is guaranteed to be tails. Bob flips the VNC an infinite number of times. On the $n$th flip, Bob bets $2^{-n}$ dollars that the VNC will show heads (so if the second flip shows heads, Bob wins $\$0.25$, and if the third flip shows tails, Bob loses $\$0.125$). Assume that dollars are infinitely divisible. Given that the first flip is heads, the expected number of dollars Bob is expected to win can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a, b$. Compute $100a + b$. [i]Proposed by Lewis Chen[/i]

Four sequences of real numbers $(a_n), (b_n), (c_n), (d_n)$ satisfy for all $n$, $$a_{n+1} = a_n +b_n, b_{n+1} = b_n +c_n,$$ $$c_{n+1} = c_n +d_n, d_{n+1} = d_n +a_n.$$ Prove that if $a_{k+m} = a_m, b_{k+m} = b_m, c_{k+m} = c_m, d_{k+m} = d_m$ for some $k\ge 1,n \ge 1$, then $a_2 = b_2 = c_2 = d_2 = 0$.

a) Let $MNP$ be a triangle such that $\angle MNP> 60^o$. Show that the side $MP$ cannot be the smallest side of the triangle $MNP$. b) In a plane the equilateral triangle $ABC$ is considered. The point $V$ that does not belong to the plane $(ABC)$ is chosen so that $\angle VAB = \angle VBC = \angle VCA$. Show that if $VA = AB$, the tetrahedron $VABC$ is regular. Valentin Vornicu

Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent: [b]i)[/b] For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k \equal{} 1} a_k \cdot x_k \leq \sum^{n}_{k \equal{} 1} b_k \cdot x_k,$ [b]ii)[/b] We have $ \sum^{s}_{k \equal{} 1} a_k \leq \sum^{s}_{k \equal{} 1} b_k$ for every $ s\in\left\{1,2,...,n\minus{}1\right\}$ and $ \sum^{n}_{k \equal{} 1} a_k \equal{} \sum^{n}_{k \equal{} 1} b_k$.

The arithmetic function $\nu$ is defined by $$\nu (n) = \begin{cases}0, \,\,\,\,\, n=1 \\ k, \,\,\,\,\, n=p_1^{a_1} p_2^{a_2} ... p_k^{a_k}\end{cases}$$, where $n=p_1^{a_1} p_2^{a_2} ... p_k^{a_k}$ represents the prime factorization of the number. Prove that for any naturals $m,n$, $$\tau (n^m) = \sum_{d | n} m^{\nu (d)}.$$ [i](PP-nine)[/i]

Let $n\geqslant 2$ be a positive integer and $a_1,a_2, \ldots ,a_n$ be real numbers such that \[a_1+a_2+\dots+a_n=0.\] Define the set $A$ by \[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\] Prove that, if $A$ is not empty, then \[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]

The unfolding of the lateral surface of a cone is a sector of angle $120^o$. The angles at the base of a pyramid constitute an arithmetic progression with a difference of $15^o$. The pyramid is inscribed in the cone. Consider a lateral face of the pyramid with the smallest area. Find the angle $\alpha$ between the plane of this face and the base.

A projectile in flight is observed simultaneously from four radio stations which are situated at the corners of a square of side $b$. The distances of the projectile from the four stations, taken in order around the square, are found to be $R_1 , R_2 , R_3 $ and $R_4$. Show that $$R_{1}^{2}+ R_{3}^{2}= R_{2}^{2}+ R_{4}^{2}.$$ Show also that the height $h$ of the projectile above the ground is given by $$h^{2}=- \frac{1}{2} b^2 +\frac{1}{4}(R_{1}^{2}+R_{2}^{2}+R_{3}^{2}+R_{4}^{2}) -\frac{1}{8 b^{2}}(R_{1}^{4}+R_{2}^{4}+R_{3}^{4}+R_{4}^{4}- 2 R_{1}^{2}R_{3}^{2} -2 R_{2}^{2} R_{4}^{2}).$$

Find all continuous functions $f: R \to R$ such that for all reals $x$ and $y$, $f(x+f(y)) = y+f(x+1)$.

Determine all pairs of polynomials $f$ and $g$ with real coefficients such that \[ x^2 \cdot g(x) = f(g(x)). \]

In the coordinate plane, a point $A$ is chosen on the line $y =\frac32 x$ in the first quadrant. Two perpendicular lines $\ell_1$ and $\ell_2$ intersect at A where $\ell_1$ has slope $m > 1$. Let $\ell_1$ intersect the $ x$-axis at $B$, and $\ell_2$ intersects the $ x$ and $y$ axes at $C$ and $D$, respectively. Suppose that line $BD$ has slope $-m$ and $BD = 2$. Compute the length of $CD$.

There are 2023 natural written in a row. The first number is 12, and each number starting from the third is equal to the product of the previous two numbers, or to the previous number increased by 4. What is the largest number of perfect squares that can be among the 2023 numbers? [i]Based on the British Mathematical Olympiad[/i]

Any five points are taken inside or on a square with side length $1$. Let $a$ be the [i]smallest[/i] possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than $a$. Then $a$ is: ${{ \textbf{(A)}\ \sqrt{3}/3 \qquad\textbf{(B)}\ \sqrt{2}/2 \qquad\textbf{(C)}\ 2\sqrt{2}/3 \qquad\textbf{(D)}\ 1 }\qquad\textbf{(E)}\ \sqrt{2} } $

$\vartriangle ABC$ is a scalene triangle with circumcircle $\Omega$. For a arbitrary $X$ in the plane, define $D_x,E_x, F_x$ to be the intersection of tangent line of $X$ (with respect to $BXC$) and $BC,CA,AB$, respectively. Let the intersection of $AX$ with $\Omega$ be $S_x$ and $T_x = D_xS_x \cap \Omega$. Show that $\Omega$ and circumcircle of $\vartriangle T_xE_xF_x$ are tangent to each other. [img]https://2.bp.blogspot.com/-rTMODHbs5Ac/XnYNQYjYzBI/AAAAAAAALeg/576nGDQ6NDA0-W5XqiNczNtI07cEZxPeQCK4BGAYYCw/s1600/imoc2019g4.png[/img]