Is it is possible to choose 24 distinct points in the space such that no three of them lie on the same line and choose 2019 distinct planes in a way that each plane passes through at least 3 of the chosen points and each triple belongs to one of the chosen planes?

Find the lowest possible values from the function \[ f(x) \equal{} x^{2008} \minus{} 2x^{2007} \plus{} 3x^{2006} \minus{} 4x^{2005} \plus{} 5x^{2004} \minus{} \cdots \minus{} 2006x^3 \plus{} 2007x^2 \minus{} 2008x \plus{} 2009\] for any real numbers $ x$.

If $ xy \equal{} 6$ and $ x^2 y \plus{} xy^2 \plus{} x \plus{} y \equal{} 63,$ find $ x^2\plus{}y^2$.

A class contains $80$ boys and $80$ girls. On each weekday (Monday to Friday) of the week before final exams, the teacher has $16$ books for the students to borrow, where a book can only be borrowed for one day at a time, and each student can only borrow once during the entire week. Show that there are two days and two books such that one of the following two statements is true: (i) Both books were not borrowed on both days (ii) Both books were borrowed on both days, and the four students who borrowed the books on these days are either all boys or all girls.

Andy wishes to open an electronic lock with a keypad containing all digits from $0$ to $9$. He knows that the password registered in the system is $2469$. Unfortunately, he is also aware that exactly two different buttons (but he does not know which ones) $\underline{a}$ and $\underline{b}$ on the keypad are broken $-$ when $\underline{a}$ is pressed the digit $b$ is registered in the system, and when $\underline{b}$ is pressed the digit $a$ is registered in the system. Find the least number of attempts Andy needs to surely be able to open the lock. [i]Proposed by Andrew Wen[/i]

Three speed skaters have a friendly "race" on a skating oval. They all start from the same point and skate in the same direction, but with different speeds that they maintain throughout the race. The slowest skater does $1$ lap per minute, the fastest one does $3.14$ laps per minute, and the middle one does $L$ laps a minute for some $1 < L < 3.14$. The race ends at the moment when all three skaters again come together to the same point on the oval (which may differ from the starting point.) Determine the number of different choices for $L$ such that exactly $117$ passings occur before the end of the race. Note: A passing is defined as when one skater passes another one. The beginning and the end of the race when all three skaters are together are not counted as passings.

Let $\mathcal A = A_0A_1A_2A_3 \cdots A_{2013}A_{2014}$ be a [i]regular 2014-simplex[/i], meaning the $2015$ vertices of $\mathcal A$ lie in $2014$-dimensional Euclidean space and there exists a constant $c > 0$ such that $A_iA_j = c$ for any $0 \le i < j \le 2014$. Let $O = (0,0,0,\dots,0)$, $A_0 = (1,0,0,\dots,0)$, and suppose $A_iO$ has length $1$ for $i=0,1,\dots,2014$. Set $P=(20,14,20,14,\dots,20,14)$. Find the remainder when \[PA_0^2 + PA_1^2 + \dots + PA_{2014}^2 \] is divided by $10^6$. [i]Proposed by Robin Park[/i]

When $4 \cos \theta - 3 \sin \theta = \tfrac{13}{3},$ it follows that $7 \cos 2\theta - 24 \sin 2\theta = \tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Let $N>= 2$ be an integer. Show that $4n(N-n)+1$ is never a perfect square for each natural number $n$ less than $N$ if and only if $N^2+1$ is prime.

$ F(x)$ is polynomial with real coefficients. $ F(x) \equal{} x^{4}\plus{}a_{1}x^{3}\plus{}a_{2}x^{2}\plus{}a_{1}x^{1}\plus{}a_{0}$. $ M$ is local maximum and $ m$ is minimum. Prove that $ \frac{3}{10}(\frac{a_{1}^{2}}{4}\minus{}\frac{2a_{2}}{3^{2}})^{2}< M\minus{}m < 3(\frac{a_{1}^{2}}{4}\minus{}\frac{2a_{2}}{3^{2}})^{2}$

Let $BM$ be the median of the triangle $ABC$ with $AB > BC$. The point $P$ is chosen so that $AB\parallel PC$ and $PM \perp BM$. Prove that $\angle ABM = \angle MBP$. [i]Proposed by Mykhailo Shandenko[/i]

Given two sequences of positive numbers $\{a_k\}$ and $\{b_k\} \ (k \in \mathbb N)$ such that: [b](i)[/b] $a_k < b_k,$ [b](ii) [/b] $\cos a_kx + \cos b_kx \geq -\frac 1k $ for all $k \in \mathbb N$ and $x \in \mathbb R,$ prove the existence of $\lim_{k \to \infty} \frac{a_k}{b_k}$ and find this limit.

In country there are two capitals ($A$ and $B$) and finite number of towns. Some towns (or town with one of capital) connected with roads (one-way). (between every two towns or capital and town there are arbitrary number of roads) such that exist at least one way from $A$ to $B$. Given, that any two ways from $A$ to $B$ have at least one common road. Prove, that exist one road, such that all ways from $A$ to $B$ pass through this road.

[b][i](a) [/i][/b] Show that the set $\mathbb N$ of all positive integers can be partitioned into three disjoint subsets $A, B$, and $C$ satisfying the following conditions: \[A^2 = A, B^2 = C, C^2 = B,\] \[AB = B, AC = C, BC = A,\] where $HK$ stands for $\{hk | h \in H, k \in K\}$ for any two subsets $H, K$ of $\mathbb N$, and $H^2$ denotes $HH.$ [b][i](b)[/i][/b] Show that for every such partition of $\mathbb N$, $\min\{n \in N | n \in A \text{ and } n + 1 \in A\}$ is less than or equal to $77.$

Let $M$ be a set containing positive integers with the following three properties: (1) $2018 \in M$. (2) If $m \in M$, then all positive divisors of m are also elements of $M$. (3) For all elements $k, m \in M$ with $1 < k < m$, the number $km + 1$ is also an element of $M$. Prove that $M = Z_{\ge 1}$. [i](Proposed by Walther Janous)[/i]

[b]p1.[/b] The total cost of $1$ football, $3$ tennis balls and $7$ golf balls is $\$14$ , while that of $1$ football, $4$ tennis balls and $10$ golf balls is $\$17$.If one has $\$20$ to spend, is this sufficient to buy a) $3$ footballs and $2$ tennis balls? b) $2$ footballs and $3$ tennis balls? [b]p2.[/b] Let $\overline{AB}$ and $\overline{CD}$ be two chords in a circle intersecting at a point $P$ (inside the circle). a) Prove that $AP \cdot PB = CP\cdot PD$. b) If $\overline{AB}$ is perpendicular to $\overline{CD}$ and the length of $\overline{AP}$ is $2$, the length of $\overline{PB}$ is $6$, and the length of $\overline{PD}$ is $3$, find the radius of the circle. [b]p3.[/b] A polynomial $P(x)$ of degree greater than one has the remainder $2$ when divided by $x-2$ and the remainder $3$ when divided by $x-3$. Find the remainder when $P(x)$ is divided by $x^2-5x+6$. [b]p4.[/b] Let $x_1= 2$ and $x_{n+1}=x_n+ (3n+2)$ for all $n$ greater than or equal to one. a) Find a formula expressing $x_n$ as a function of$ n$. b) Prove your result. [b]p5.[/b] The point $M$ is the midpoint of side $\overline{BC}$ of a triangle $ABC$. a) Prove that $AM \le \frac12 AB + \frac12 AC$. b) A fly takes off from a certain point and flies a total distance of $4$ meters, returning to the starting point. Explain why the fly never gets outside of some sphere with a radius of one meter. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Three schools each have $n$ students. Each student knows a total of $n + 1$ students at the other two schools. Show that there must be three students, one from each school, who know each other.

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that: $\bullet$ $f(x)<2$ for all $x\in (0,1)$; $\bullet$ for all real numbers $x,y$ we have: $$max\{f(x+y),f(x-y)\}=f(x)+f(y)$$ Proposed by Navid Safaei

Show that for any real numbers $x, y > 1$, we have $$\frac{x^2}{y - 1}+ \frac{y^2}{x - 1} \ge 8$$

Given that $a,b,c$ are integers with $abc = 60$, and that complex number $\omega \neq 1$ satisfies $\omega^3=1$, find the minimum possible value of $|a + b\omega + c\omega^2|$.  

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

p1. Known points $A (-1.2)$, $B (0,2)$, $C (3,0)$, and $D (3, -1)$ as seen in the following picture. Determine the measure of the angle $AOD$ . [img]https://cdn.artofproblemsolving.com/attachments/f/2/ca857aaf54c803db34d8d52505ef9a80e7130f.png[/img] p2. Determine all prime numbers $p> 2$ until $p$ divides $71^2 - 37^2 - 51$. p3. A ball if dropped perpendicular to the ground from a height then it will bounce back perpendicular along the high third again, down back upright and bouncing back a third of its height, and next. If the distance traveled by the ball when it touches the ground the fourth time is equal to $106$ meters. From what height is the ball was dropped? p4. The beam $ABCD.EFGH$ is obtained by pasting two unit cubes $ABCD.PQRS$ and $PQRS.EFGH$. The point K is the midpoint of the edge $AB$, while the point $L$ is the midpoint of the edge $SH$. What is the length of the line segment $KL$? p5. How many integer numbers are no greater than $2004$, with remainder $1$ when divided by $2$, with remainder $2$ when divided by $3$, with remainder $3$ when divided by $4$, and with remainder $4$ when divided by $5$?

The inscribed circle of the triangle $ABC$ touches its sides $AB, BC, CA$, at points $K,N, M$ respectively. It is known that $\angle ANM = \angle CKM$. Prove that the triangle $ABC$ is isosceles. (Vyacheslav Yasinsky)

$$F(x)=x^{6}+15x^{5}+85x^{4}+225x^{3}+274x^{2}+120x+1$$

Find all natural numbers $a>1$, with the property that every prime divisor of $a^6-1$ divides also at least one of the numbers $a^3-1$, $a^2-1$. [i]K. Dochev[/i]