For $n$ distinct positive integers all their $n(n-1)/2$ pairwise sums are considered. For each of these sums Ivan has written on the board the number of original integers which are less than that sum and divide it. What is the maximum possible sum of the numbers written by Ivan?

How many factors of $2020$ have more than $3$ factors? (As an example, $12$ has $6$ factors, namely $1$, $2$, $3$, $4$, $6$, and $12$.) $\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 10$

a) Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that\[f(x-f(y))=f(f(x))-f(y)-1\]holds for all $x,y\in\mathbb{Z}$. (It is [url=https://artofproblemsolving.com/community/c6h1268817p6621849]2015 IMO Shortlist A2 [/url]) b) The same question for if \[f(x-f(y))=f(f(x))-f(y)-2\] for all integers $x,y$

Among all ordered pairs of real numbers $(a, b)$ satisfying $a^4 + 2a^2b + 2ab + b^2 = 960$, find the smallest possible value for $a$.

Let $p$ and $k$ be positive integers such that $p$ is prime and $k>1$. Prove that there is at most one pair $(x,y)$ of positive integers such that $x^k+px=y^k$.

The figure below is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\sqrt{m}+\sqrt{n},$ where $m$ and $n$ are positive integers. What is $m+n?$ [asy] /* Made by samrocksnature */ pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-2,2); pair G=(-3,2); draw(A--B--C--D--E--A); draw(A--F--A--G); draw(B--F--C); draw(E--G--D); label("A",A,N); label("B",B,W); label("C",C,S); label("D",D,S); label("E",E,dir(0)); dot(A^^B^^C^^D^^E^^F^^G); [/asy] $\textbf{(A) }20 \qquad \textbf{(B) }21 \qquad \textbf{(C) }22\qquad \textbf{(D) }23 \qquad \textbf{(E) }24$ Proposed by [b]djmathman[/b]

$n$ is an odd positive integer and $x,y$ are two rational numbers satisfying $$x^n+2y=y^n+2x.$$Prove that $x=y$.

Let $n$ be a positive integer. Each number $1, 2, ..., 1000$ has been colored with one of $n$ colours. Each two numbers , such that one is a divisor of second of them, are colored with different colours. Determine minimal number $n$ for which it is possible.

Find all functions $ f\colon \mathbb{R} \to \mathbb{R} $ such that $ f(x^3+y^3+xy)=x^2f(x)+y^2f(y)+f(xy) $, for all $ x,y \in \mathbb{R} $.

For numbers $a,b \in \mathbb{R}$ we consider the sets: $$A=\{a^n | n \in \mathbb{N}\} , B=\{b^n | n \in \mathbb{N}\}$$ Find all $a,b > 1$ for which there exists two real , non-constant polynomials $P,Q$ with positive leading coefficients st for each $r \in \mathbb{R}$: $$ P(r) \in A \iff Q(r) \in B$$

Terri produces a sequence of positive integers by following three rules. She starts with a positive integer, then applies the appropriate rule to the result, and continues in this fashion. Rule 1: If the integer is less than 10, multiply it by 9. Rule 2: If the integer is even and greater than 9, divide it by 2. Rule 3: If the integer is odd and greater than 9, subtract 5 from it. Find the $98th$ term of the sequence that begins $ 98, 49,\ldots . $ $ \text{(A)}\ 6\qquad\text{(B)}\ 11\qquad\text{(C)}\ 22\qquad\text{(D)}\ 27\qquad\text{(E)}\ 54 $

Which of the following represents the result when the figure shown below is rotated clockwise $120^\circ$ about its center? [asy] unitsize(6); draw(circle((0,0),5)); draw((-1,2.5)--(1,2.5)--(0,2.5+sqrt(3))--cycle); draw(circle((-2.5,-1.5),1)); draw((1.5,-1)--(3,0)--(4,-1.5)--(2.5,-2.5)--cycle); [/asy] [asy] unitsize(6); for (int i = 0; i < 5; ++i) { draw(circle((12*i,0),5)); } draw((-1,2.5)--(1,2.5)--(0,2.5+sqrt(3))--cycle); draw(circle((-2.5,-1.5),1)); draw((1.5,-1)--(3,0)--(4,-1.5)--(2.5,-2.5)--cycle); draw((14,-2)--(16,-2)--(15,-2+sqrt(3))--cycle); draw(circle((12,3),1)); draw((10.5,-1)--(9,0)--(8,-1.5)--(9.5,-2.5)--cycle); draw((22,-2)--(20,-2)--(21,-2+sqrt(3))--cycle); draw(circle((27,-1),1)); draw((24,1.5)--(22.75,2.75)--(24,4)--(25.25,2.75)--cycle); draw((35,2.5)--(37,2.5)--(36,2.5+sqrt(3))--cycle); draw(circle((39,-1),1)); draw((34.5,-1)--(33,0)--(32,-1.5)--(33.5,-2.5)--cycle); draw((50,-2)--(52,-2)--(51,-2+sqrt(3))--cycle); draw(circle((45.5,-1.5),1)); draw((48,1.5)--(46.75,2.75)--(48,4)--(49.25,2.75)--cycle); label("(A)",(0,5),N); label("(B)",(12,5),N); label("(C)",(24,5),N); label("(D)",(36,5),N); label("(E)",(48,5),N); [/asy]

We write the decimal expressions of $\sqrt{2}$ and $\sqrt{3}$ as \[\sqrt{2}=1.a_1a_2a_3\dots\quad\quad\sqrt{3}=1.b_1b_2b_3\dots\] where each $a_i$ or $b_i$ is a digit between 0 and 9. Prove that there exist at least 1000 values of $i$ between $1$ and $10^{1000}$ such that $a_i\neq b_i$.

The cells of a $5\times5$ grid are each colored red, white, or blue. Sam starts at the bottom-left cell of the grid and walks to the top-right cell by taking steps one cell either up or to the right. Thus, he passes through $9$ cells on his path, including the start and end cells. Compute the number of colorings for which Sam is guaranteed to pass through a total of exactly $3$ red cells, exactly $3$ white cells, and exactly $3$ blue cells no matter which route he takes.

In many computer languages, the division operation ignores remainders. Let’s denote this operation by $//$, so for instance $13//3 = 4$. If, for some $b$, $a//b = c$, then we say that $c$ is a [i]near factor[/i] of $a$. Thus, the near factors of $13$ are $1$, $2$, $3$, $4$, and $6$. Let $a$ be a positive integer. Prove that every positive integer less than or equal to $\sqrt{a}$ is a near factor of $a$.

Let $M$ be a finite set of lattice points and $n$ be a positive integer. A $\textit{mine-avoiding path}$ is a path of lattice points with length $n$, beginning at $(0,0)$ and ending at a point on the line $x+y=n,$ that does not contain any point in $M$. Prove that if there exists a mine-avoiding path, then there exist at least $2^{n-|M|}$ mine-avoiding paths. [hide=*]A lattice point is a point $(x,y)$ where $x$ and $y$ are integers. A path of lattice points with length $n$ is a sequence of lattice points $P_0,P_1,\ldots, P_n$ in which any two adjacent points in the sequence have distance 1 from each other.[/hide] [i]Ankit Bisain and Holden Mui[/i]

Find all angles $0 < \theta < 90^\circ$ for which there exists an angle $0 < \beta < 90^\circ$ such that a right triangle with angles $90^\circ, \theta, 90^\circ - \theta$ can be tiled by a finite number of isosceles triangles with angles $\beta, \beta, 180^\circ - 2\beta$. (The isosceles triangles are not necessarily pairwise congruent, but they are pairwise similar.)

Find all natural numbers $n$ such that the number $n(n+1)(n+2)(n+3)$ has exactly three different prime divisors.

Find the largest constant $c>0$ such that for every positive integer $n\ge 2$, there always exist a positive divisor $d$ of $n$ such that $$d\le \sqrt{n}\hspace{0.5cm} \text{and} \hspace{0.5cm} \tau(d)\ge c\sqrt{\tau(n)}$$ where $\tau(n)$ is the number of divisors of $n$. [i]Proposed by Mohd. Suhaimi Ramly[/i]

For a natural number $n$, let $$C_n=\frac{1}{n+1}\binom{2n}{n}=\frac{(2n)!}{n!(n+1)!}$$ be the $n$-th Catalan number. Prove that for any natural number $m$, $$\sum_{i+j+k=m} C_{i+j}C_{j+k}C_{k+i}=\frac{3}{2m+3}C_{2m+1}.$$ [i]Proposed by Bin Wang[/i]

For how many integer $ n$, $ P \equal{} n^4 \plus{} 4n^3 \plus{} 3n^2 \minus{} 2n \plus{} 7$ is prime? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{Infinitely many}$

Given is a set of $2n$ cards numbered $1,2, \cdots, n$, each number appears twice. The cards are put on a table with the face down. A set of cards is called good if no card appears twice. Baron Munchausen claims that he can specify $80$ sets of $n$ cards, of which at least one is sure to be good. What is the maximal $n$ for which the Baron's words could be true?

Find the period of the repetend of the fraction $\frac{39}{1428}$ by using [i]binary[/i] numbers, i.e. its binary decimal representation. (Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finite steps, or it is of the form $0.b_1b_2\cdots b_sa_1a_2\cdots a_ka_1a_2\cdots a_ka_1a_2 \cdots a_k \cdots$. Here the repeated sequence $a_1a_2\cdots a_k$ is called the [i]repetend[/i] of the fraction, and the smallest length of the repetend, $k$, is called the [i]period[/i] of the decimal number.)

$(a)$ Let $a_1,a_2,a_3$ be three real numbers. Prove that $(a_1-a_2)(a_1-a_3)+(a_2-a_1)(a_2-a_3)+(a_3-a_1)(a_2-a_2)\geq 0$. $(b)$ Prove that the inequality above doesn't hold if we use four number instead of three.

As shown in the figure, there are two rectangles $ABCD$ and $PQRD$ with the same area, and with parallel corresponding edges. Let points $N,$ $M$ and $T$ be the midpoints of segments $QR,$ $PC$ and $AB$, respectively. Prove that points $N,M$ and $T$ lie on the same line. [img]http://s4.picofile.com/file/8372959484/E02.png[/img] [i]Proposed by Morteza Saghafian[/i]