Let $a_1,a_2,...,a_9 \ge - 1$ and $a^3_1+a^3_2+...+a^3_9= 0$. Determine the maximal value of $M = a_1 + a_2 + ... + a_9$.

Prove that there exists an absolute constant $c$ such that any set $H$ of $n$ points of the plane in general position can be coloured with $c\log n$ colours in such a way that any disk of the plane containing at least one point of $H$ intersects some colour class of $H$ in exactly one point.

Let $n$ and $m$ be positive integers in the range $[1, 10^{10}]$. Let $R$ be the rectangle with corners at $(0, 0), (n, 0), (n, m), (0, m)$ in the coordinate plane. A simple non-self-intersecting quadrilateral with vertices at integer coordinates is called [i]far-reaching[/i] if each of its vertices lie on or inside $R$, but each side of $R$ contains at least one vertex of the quadrilateral. Show that there is a far-reaching quadrilateral with area at most $10^6$.

In the triangle $ABC$ on the sides $AB$ and $AC$ outward constructed equilateral triangles $ABD$ and $ACE$. The segments $CD$ and $BE$ intersect at point $F$. It turns out that point $A$ is the center of the circle inscribed in triangle $ DEF$. Find the angle $BAC$. (Rozhkova Maria)

Let $P_1P_2\dotsb P_{100}$ be a cyclic $100$-gon and let $P_i = P_{i+100}$ for all $i$. Define $Q_i$ as the intersection of diagonals $\overline{P_{i-2}P_{i+1}}$ and $\overline{P_{i-1}P_{i+2}}$ for all integers $i$. Suppose there exists a point $P$ satisfying $\overline{PP_i}\perp\overline{P_{i-1}P_{i+1}}$ for all integers $i$. Prove that the points $Q_1,Q_2,\dots, Q_{100}$ are concyclic. [i]Michael Ren[/i]

Given a prime number $p$ and let $\overline{v_1},\overline{v_2},\dotsc ,\overline{v_n}$ be $n$ distinct vectors of length $p$ with integer coordinates in an $\mathbb{R}^3$ Cartesian coordinate system. Suppose that for any $1\leqslant j<k\leqslant n$, there exists an integer $0<\ell <p$ such that all three coordinates of $\overline{v_j} -\ell \cdot \overline{v_k} $ is divisible by $p$. Prove that $n\leqslant 6$.

Let the incircle of an acute triangle $\triangle ABC$ touches $BC,CA$, and $AB$ at points $D,E$, and $F$, respectively. Place point $K$ on the side $AB$ so that $DF$ bisects $\angle ADK$, and place point $L$ on the side $AB$ so that $EF$ bisects $\angle BEL$. [list=a] [*]Prove that $\triangle ALE\sim\triangle AEB$. [*]Prove that $FK=FL$. [/list]

Give the answer about the following propositions $(p),\ (q)$ whether they are true or not. If the answer is true, then give the proof and if the answer is false, then give the proof by giving the counter example. $(p)$ If we can form a triangle such that one of inner angles of the triangle is $60^\circ$ by choosing 3 points from the vertices of a regular $n$-polygon, then $n$ is a multiple of 3. $(q)$ In $\triangle{ABC},\ \triangle{ABD}$, if $AC<AD$ and $BC<BD$, then $\angle{C}>\angle{D}$. 35 points

Find all polynomials $f, g, h$ with real coefficients, such that $f(x)^2+(x+1)g(x)^2=(x^3+x)h(x)^2$

Let $(a_1, a_2,\ldots , a_n)$ be a permutation of $1, 2, \ldots , n,$ where $n  \geq 2.$ For each $k = 1, \ldots , n$, we know that $a_k$ apples are placed at the point $k$ on the real axis. Children named $A,B,C$ are assigned respective points $x_A, x_B, x_C \in \{1, \ldots , n\}.$ For each $k,$ the children whose points are closest to $ k$ divide $a_k$ apples equally among themselves. We call $(x_A, x_B, x_C)$ a [i]stable configuration[/i] if no child’s total share can be increased by assigning a new point to this child and not changing the points of the other two. Determine the values of $n$ for which a stable configuration exists for some distribution $(a_1, \ldots, a_n)$ of the apples.

A computer science teacher has asked his students to write a program that, given a list of $n$ numbers $a_1, a_2, ..., a_n$, calculates the list $b_1, b_2, ..., b_n$ where $b_k$ is the number of times the number $a_k$ appears in the list. So, for example, for the list $1,2,3,1$, the program returns the list $2,1,1,2$. Next, the teacher asked Alexandre to run the program for a list of $2025$ numbers. Then he asked him to apply the program to the resulting list, and so on, until a number greater than or equal to $k$ appears in the list. Find the largest value of $k$ for which, whatever the initial list of $2025$ positive integers $a_1, a_2, ..., a_{2025}$, it is possible for Alexander to do what the teacher asked him to do.

Prove that for $\forall$ $z\in \mathbb{C}$ the following inequality is true: $|z|^2+2|z-1|\geq 1$, where $"="$ is reached when $z=1$.

Let $n$ be positive integer. Define a sequence $\{a_k\}$ by \[a_1=\frac{1}{n(n+1)},\ a_{k+1}=-\frac{1}{k+n+1}+\frac{n}{k}\sum_{i=1}^k a_i\ \ (k=1,\ 2,\ 3,\ \cdots).\] (1) Find $a_2$ and $a_3$. (2) Find the general term $a_k$. (3) Let $b_n=\sum_{k=1}^n \sqrt{a_k}$. Prove that $\lim_{n\to\infty} b_n=\ln 2$. 50 points

Consider the polynomial $p(x)=a_nx^n+\ldots+a_1x+a_0$ with real coefficients such that $0\le a_i\le a_0$ for each $i=1,2,\ldots,n$. If $a$ is the coefficient of $x^{n+1}$ in the polynomial $q(x)=p(x)^2$, prove that $2a\le p(1)^2$.

Find the smallest positive integer $n$ that satisfies the following condition: For every finite set of points on the plane, if for any $n$ points from this set there exist two lines containing all the $n$ points, then there exist two lines containing all points from the set.

Let $(a_n)_{n \ge 1}$ be a sequence of real numbers satisfying the properties: [list=1] [*] the sequence $x_n=\sum\limits_{k=1}^n a_k^2$ is convergent; [*] the sequence $y_n=\sum\limits_{k=1}^n a_k$ is unbounded. [/list] Prove that the sequence $(b_n)_{n \ge 1}$ given by $b_n=\{y_n\}$ is divergent. Note: $\{ x \}$ denotes the fractional part of $x.$

In a right $\Delta ABC$ ($\angle C = 90^{\circ} $), $CD$ is the height. Let $r_1$ and $r_2$ be the radii of inscribed circles of $\Delta ACD$ and $\Delta DCB$. Find the radius of inscribed circle of $\Delta ABC$

In a triangle $ABC$, the angle bisector at $A$ intersects $BC$ at $D$. The tangents at $D$ to the circumcircles of the triangles $ABD$ and $ACD$ meet $AC$ and $AB$ at $N$ and $M$, respectively. Prove that the quadrilateral $AMDN$ is inscribed in a circle tangent to $BC$.

$ 15 $ minors of order $ 3 $ of a $ 4\times 4 $ real matrix whose determinant is a nonzero rational number, are rational. Prove that this matrix is rational.

Find all $m,n\in\mathbb N$ and primes $p\geq 5$ satisfying \[m(4m^2+m+12)=3(p^n-1).\]

Find all prime numbers $p$ and $q$ such that $\frac{(7^{p}-2^{p})(7^{q}-2^{q})}{pq}$ is an integer.

[u]Set 6[/u] [b]p16.[/b] Let $n! = n \times (n - 1) \times ... \times 2 \times 1$. Find the maximum positive integer value of $x$ such that the quotient $\frac{160!}{160^x}$ is an integer. [b]p17.[/b] Let $\vartriangle OAB$ be a triangle with $\angle OAB = 90^o$ . Draw points $C, D, E, F, G$ in its plane so that $$\vartriangle OAB \sim \vartriangle OBC \sim \vartriangle OCD \sim \vartriangle ODE \sim \vartriangle OEF \sim \vartriangle OFG,$$ and none of these triangles overlap. If points $O, A, G$ lie on the same line, then let $x$ be the sum of all possible values of $\frac{OG}{OA }$. Then, $x$ can be expressed in the form $m/n$ for relatively prime positive integers $m, n$. Compute $m + n$. [b]p18.[/b] Let $f(x)$ denote the least integer greater than or equal to $x^{\sqrt{x}}$. Compute $f(1)+f(2)+f(3)+f(4)$. [u]Set 7[/u] The Fibonacci sequence $\{F_n\}$ is defined as $F_0 = 0$, $F_1 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all integers $n \ge 0$. [b]p19.[/b] Find the least odd prime factor of $(F_3)^{20} + (F_4)^{20} + (F_5)^{20}$. [b]p20.[/b] Let $$S = \frac{1}{F_3F_5}+\frac{1}{F_4F_6}+\frac{1}{F_5F_7}+\frac{1}{F_6F_8}+...$$ Compute $420S$. [b]p21.[/b] Consider the number $$Q = 0.000101020305080130210340550890144... ,$$ the decimal created by concatenating every Fibonacci number and placing a 0 right after the decimal point and between each Fibonacci number. Find the greatest integer less than or equal to $\frac{1}{Q}$. [u]Set 8[/u] [b]p22.[/b] In five dimensional hyperspace, consider a hypercube $C_0$ of side length $2$. Around it, circumscribe a hypersphere $S_0$, so all $32$ vertices of $C_0$ are on the surface of $S_0$. Around $S_0$, circumscribe a hypercube $C_1$, so that $S_0$ is tangent to all hyperfaces of $C_1$. Continue in this same fashion for $S_1$, $C_2$, $S_2$, and so on. Find the side length of $C_4$. [b]p23.[/b] Suppose $\vartriangle ABC$ satisfies $AC = 10\sqrt2$, $BC = 15$, $\angle C = 45^o$. Let $D, E, F$ be the feet of the altitudes in $\vartriangle ABC$, and let $U, V , W$ be the points where the incircle of $\vartriangle DEF$ is tangent to the sides of $\vartriangle DEF$. Find the area of $\vartriangle UVW$. [b]p24.[/b] A polynomial $P(x)$ is called spicy if all of its coefficients are nonnegative integers less than $9$. How many spicy polynomials satisfy $P(3) = 2019$? [i]The next set will consist of three estimation problems.[/i] [u]Set 9[/u] Points will be awarded based on the formulae below. Answers are nonnegative integers that may exceed $1,000,000$. [b]p25.[/b] Suppose a circle of radius $20192019$ has area $A$. Let s be the side length of a square with area $A$. Compute the greatest integer less than or equal to $s$. If $n$ is the correct answer, an estimate of $e$ gives $\max \{ 0, \left\lfloor 1030 ( min \{ \frac{n}{e},\frac{e}{n}\}^{18}\right\rfloor -1000 \}$ points. [b]p26.[/b] Given a $50 \times 50$ grid of squares, initially all white, define an operation as picking a square and coloring it and the four squares horizontally or vertically adjacent to it blue, if they exist. If a square is already colored blue, it will remain blue if colored again. What is the minimum number of operations necessary to color the entire grid blue? If $n$ is the correct answer, an estimate of $e$ gives $\left\lfloor \frac{180}{5|n-e|+6}\right\rfloor$ points. [b]p27.[/b] The sphere packing problem asks what percent of space can be filled with equally sized spheres without overlap. In three dimensions, the answer is $\frac{\pi}{3\sqrt2} \approx 74.05\%$ of space (confirmed as recently as $2017!$), so we say that the packing density of spheres in three dimensions is about $0.74$. In fact, mathematicians have found optimal packing densities for certain other dimensions as well, one being eight-dimensional space. Let d be the packing density of eight-dimensional hyperspheres in eightdimensional hyperspace. Compute the greatest integer less than $10^8 \times d$. If $n$ is the correct answer, an estimate of e gives $\max \left\{ \lfloor 30-10^{-5}|n - e|\rfloor, 0 \right\}$ points. PS. You had better use hide for answers. First sets have be posted [url=https://artofproblemsolving.com/community/c4h2777330p24370124]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x$ and $y$ be integers satisfying both $x^2 - 16x + 3y = 20$ and $y^2 + 4y - x = -12$. Find $x + y$.

Let $x,y,z$ be real numbers. Find the minimum value of the sum \begin{align*}|\cos(x)|+|\cos(y)|+|\cos(z)|+|\cos(x-y)|+|\cos(y-z)|+|\cos(z-x)|.\end{align*}

Let $C$ be a cube with edges of length 2. Construct a solid with fourteen faces by cutting off all eight corners at $C,$ keeping the new faces perpendicular to the diagonals of the cube, and keeping the newly formed faces indentical. If at the conclusion of this process the fourteen faces so have the same area, find the area of each of face of the new solid.