Find the sum of all positive integers $n$ such that $n+2$ divides the product $3(n+3)(n^2+9)$.

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

Let $a_1=2020$ and let $a_{n+1}=\sqrt{2020+a_n}$ for $n\ge 1$. How much is $\left\lfloor a_{2020}\right\rfloor$? Note: $\lfloor x\rfloor$ denotes the integer part of a number, that is that is, the immediate integer less than $x$. For example, $\lfloor 2.71\rfloor=2$ and $\lfloor \pi\rfloor=3$.

Sally randomly chooses three different numbers from the set $\left\{1,2,\ldots,14\right\}$. What is the probability that the sum of her smallest number and her biggest number is at least $15$?

Let Akbar and Birbal together have $n$ marbles, where $n > 0$. Akbar says to Birbal, “ If I give you some marbles then you will have twice as many marbles as I will have.” Birbal says to Akbar, “ If I give you some marbles then you will have thrice as many marbles as I will have.” What is the minimum possible value of $n$ for which the above statements are true?

Charles has two six-sided dice. One of the dice is fair, and the other die is biased so that it comes up six with probability $\tfrac23,$ and each of the other five sides has probability $\tfrac{1}{15}.$ Charles chooses one of the two dice at random and rolls it three times. Given that the first two rolls are both sixes, the probability that the third roll will also be a six is $\tfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Let $f(x)$ and $g(x)$ be given by $f(x) = \frac{1}{x} + \frac{1}{x-2} + \frac{1}{x-4} + \cdots + \frac{1}{x-2018}$ $g(x) = \frac{1}{x-1} + \frac{1}{x-3} + \frac{1}{x-5} + \cdots + \frac{1}{x-2017}$. Prove that $|f(x)-g(x)| >2$ for any non-integer real number $x$ satisfying $0 < x < 2018$.

A circle is divided into $432$ congruent arcs by $432$ points. The points are colored in four colors such that some $108$ points are colored Red, some $108$ points are colored Green, some $108$ points are colored Blue, and the remaining $108$ points are colored Yellow. Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.

Denote by $d(n)$ the number of distinct positive divisors of a positive integer $n$ (including $1$ and $n$). Let $a>1$ and $n>0$ be integers such that $a^n+1$ is a prime. Prove that $d(a^n-1)\ge n$.

$A$ and $B$ are on a circle of radius $20$ centered at $C$, and $\angle ACB = 60^\circ$. $D$ is chosen so that $D$ is also on the circle, $\angle ACD = 160^\circ$, and $\angle DCB = 100^\circ$. Let $E$ be the intersection of lines $AC$ and $BD$. What is $DE$?

The value of $ x$ at the intersection of $ y\equal{}\frac{8}{x^2\plus{}4}$ and $ x\plus{}y\equal{}2$ is: $ \textbf{(A)}\ \minus{}2\plus{}\sqrt{5} \qquad\textbf{(B)}\ \minus{}2\minus{}\sqrt{5} \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{none of these}$

Let $f=f_0+f_1z+f_2z^2+\ldots+f_{2n}z^{2n}$ and $f_k=f_{2n-k}$ for each $k$. Prove that $f(z)=z^ng(z+z^{-1})$, where $g$ is a polynomial of degree $n$.

Give a square of side $1$. Show that for each finite set of points of the sides of the square you can find a vertex of the square with the following property: the arithmetic mean of the squares of the distances from this vertex to the points of the set is greater than or equal to $3/4$.

Let $A_0,A_1, \ldots , A_n$ be points in a plane such that (i) $A_0A_1 \leq \frac{1}{ 2} A_1A_2 \leq \cdots \leq \frac{1}{2^{n-1} } A_{n-1}A_n$ and (ii) $0 < \measuredangle A_0A_1A_2 < \measuredangle A_1A_2A_3 < \cdots < \measuredangle A_{n-2}A_{n-1}A_n < 180^\circ,$ where all these angles have the same orientation. Prove that the segments $A_kA_{k+1},A_mA_{m+1}$ do not intersect for each $k$ and $n$ such that $0 \leq k \leq m - 2 < n- 2.$

[b]p1.[/b] Al usually arrives at the train station on the commuter train at $6:00$, where his wife Jane meets him and drives him home. Today Al caught the early train and arrived at $5:00$. Rather than waiting for Jane, he decided to jog along the route he knew Jane would take and hail her when he saw her. As a result, Al and Jane arrived home $12$ minutes earlier than usual. If Al was jogging at a constant speed of $5$ miles per hour, and Jane always drives at the constant speed that would put her at the station at $6:00$, what was her speed, in miles per hour? [b]p2.[/b] In the figure, points $M$ and $N$ are the respective midpoints of the sides $AB$ and $CD$ of quadrilateral $ABCD$. Diagonal $AC$ meets segment $MN$ at $P$, which is the midpoint of $MN$, and $AP$ is twice as long as $PC$. The area of triangle $ABC$ is $6$ square feet. (a) Find, with proof, the area of triangle $AMP$. (b) Find, with proof, the area of triangle $CNP$. (c) Find, with proof, the area of quadrilateral $ABCD$. [img]https://cdn.artofproblemsolving.com/attachments/a/c/4bdcd8390bae26bc90fc7eae398ace06900a67.png[/img] [b]p3.[/b] (a) Show that there is a triangle whose angles have measure $\tan^{-1}1$, $\tan^{-1}2$ and $\tan^{-1}3$. (b) Find all values of $k$ for which there is a triangle whose angles have measure $\tan^{-1}\left(\frac12 \right)$, $\tan^{-1}\left(\frac12 +k\right)$, and $\tan^{-1}\left(\frac12 +2k\right)$ [b]p4.[/b] (a) Find $19$ consecutive integers whose sum is as close to $1000$ as possible. (b) Find the longest possible sequence of consecutive odd integers whose sum is exactly $1000$, and prove that your sequence is the longest. [b]p5.[/b] Let $AB$ and $CD$ be chords of a circle which meet at a point $X$ inside the circle. (a) Suppose that $\frac{AX}{BX}=\frac{CX}{DX}$. Prove that $|AB|=|CD|$. (b) Suppose that $\frac{AX}{BX}>\frac{CX}{DX}>1$. Prove that $|AB|>|CD|$. ($|PQ|$ means the length of the segment $PQ$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $A, B, C, D$ be points that lie on the same circle . Let $F$ be such that the arc $AF$ is congruent with the arc $BF$. Let $P$ be the intersection point of the segments $DF$ and $AC$. Let $Q$ be intersection point of the $CF$ and $BD$ segments. Prove that $PQ \parallel AB$.

Forty-nine students solve a set of 3 problems. The score for each problem is a whole number of points from 0 to 7. Prove that there exist two students $ A$ and $ B$ such that, for each problem, $ A$ will score at least as many points as $ B.$

Given is a simple connected graph with $2n$ vertices. Prove that its vertices can be colored with two colors so that if there are $k$ edges connecting vertices with different colors and $m$ edges connecting vertices with the same color, then $k-m \geq n$.

Find the maximum value of $\frac{xyz}{(1 + 5x)(4x + 3y)(5y + 6z)(z + 18)}$ as $x, y$ and $z$ range over the set of all positive real numbers. Justify your answer.

The volume of a certain rectangular solid is $ 8 \text{ cm}^3$, its total surface area is $ 32 \text{ cm}^3$, and its three dimensions are in geometric progression. The sums of the lengths in cm of all the edges of this solid is $ \textbf{(A)}\ 28 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 40 \qquad \textbf{(E)}\ 44$

How maay zeros are there at the end of the number $$1000! = 1 \cdot 2 \cdot 3 \cdot ... \cdot 999 \cdot 1000?$$

A circle is drawn on the board, and $2n$ points are marked on it, dividing it into $2n$ equal arcs. Petya and Vasya are playing the following game. Petya chooses a positive integer $d \leqslant n$ and announces this number to Vasya. To win the game, Vasya needs to color all marked points using $n$ colors, such that each color is assigned to exactly two points, and for each pair of same-colored points, one of the arcs between them contains exactly $(d - 1)$ marked points. Find all $n$ for which Petya will be able to prevent Vasya from winning.

Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$. Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$. [i]Proposed by Carlos Yuzo Shine, Brazil[/i]

We color all points of a plane using $3$ colors. Prove that there are at least two points of the plane having same colours and with distance among them $1$.

Find all four-element sets of positive integers $\{w,x,y,z\}$ such that $w^x+w^y=w^z$.