Describe all the ways to color each natural number as one of three colors so that the following condition is satisfied: if the numbers $a$, $b$ and $c$ (not necessarily different) satisfy the condition $2000(a + b) = c$, then they either all the same color or three different colors

Find the number of integers $n$ with $1\le n\le 100$ for which $n-\phi(n)$ is prime. Here $\phi(n)$ denotes the number of positive integers less than $n$ which are relatively prime to $n$. [i]Proposed by Mehtaab Sawhney[/i]

Positive integers $a,b,c$ satisfying the equation $$a^3+4b+c = abc,$$ where $a \geq c$ and the number $p = a^2+2a+2$ is a prime. Prove that $p$ divides $a+2b+2$.

Let $f(x)$ be an odd function on $\mathbb{R}$, such that $f(x)=x^2$ when $x\ge 0$. Knowing that for all $x\in [a,a+2]$, the inequality $f(x+a)\ge 2f(x)$ holds, find the range of real number $a$.

Let $A,B,C$ be three points on a circle $\gamma$, and let $L$ denote the midpoint of segment $BC$. The perpendicular bisector of $BC$ intersects the circle $\gamma$ at two points $M$ and $N$, such that $A$ and $M$ are on different sides of line $BC$. Let $S$ denote the point where the segments $BC$ and $AM$ intersect. Line $NS$ intersects the circumcircle of $\triangle ALM$ at two points $D$ and $E$, with $D$ lying in the interior of the circle $\gamma$. (a) Prove that $M$ is the circumcentre of $\triangle BCD$. (b) Prove that the circumcircles of $\triangle BCD$ and $\triangle ADN$ are tangent at the point $D$.

In a right angled-triangle $ABC$, $\angle{ACB} = 90^o$. Its incircle $O$ meets $BC$, $AC$, $AB$ at $D$,$E$,$F$ respectively. $AD$ cuts $O$ at $P$. If $\angle{BPC} = 90^o$, prove $AE + AP = PD$.

Pairwise distinct prime numbers $p, q, r$ satisfy the equality $$rp^3 + p^2 + p = 2rq^2 +q^2 + q.$$ Determine all possible values of the product $pqr$.

A regular hexagon with side length $ 1 $ is given. There are $ m $ points in its interior such that no $ 3 $ are collinear. The hexagon is divided into triangles (triangulated), such that every point of the $ m $ given and every vertex of the hexagon is a vertex of such a triangle. The triangles don't have common interior points. Prove that there exists a triangle with area not greater than $ \frac{3 \sqrt{3}}{4(m+2)}$.

Let $L$ be the number formed by $2022$ digits equal to $1$, that is, $L=1111\dots 111$. Compute the sum of the digits of the number $9L^2+2L$.

How many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different. [asy] fill((0,2)--(1,2)--(1,3)--(0,3)--cycle,gray); fill((1,2)--(2,2)--(2,3)--(1,3)--cycle,gray); draw((0,0)--(3,0)--(3,3)--(0,3)--cycle,linewidth(1)); draw((2,0)--(2,3),linewidth(1)); draw((0,1)--(3,1),linewidth(1)); draw((1,0)--(1,3),linewidth(1)); draw((0,2)--(3,2),linewidth(1)); fill((6,0)--(8,0)--(8,1)--(6,1)--cycle,gray); draw((6,0)--(9,0)--(9,3)--(6,3)--cycle,linewidth(1)); draw((8,0)--(8,3),linewidth(1)); draw((6,1)--(9,1),linewidth(1)); draw((7,0)--(7,3),linewidth(1)); draw((6,2)--(9,2),linewidth(1)); fill((14,1)--(15,1)--(15,3)--(14,3)--cycle,gray); draw((12,0)--(15,0)--(15,3)--(12,3)--cycle,linewidth(1)); draw((14,0)--(14,3),linewidth(1)); draw((12,1)--(15,1),linewidth(1)); draw((13,0)--(13,3),linewidth(1)); draw((12,2)--(15,2),linewidth(1)); fill((18,1)--(19,1)--(19,3)--(18,3)--cycle,gray); draw((18,0)--(21,0)--(21,3)--(18,3)--cycle,linewidth(1)); draw((20,0)--(20,3),linewidth(1)); draw((18,1)--(21,1),linewidth(1)); draw((19,0)--(19,3),linewidth(1)); draw((18,2)--(21,2),linewidth(1));[/asy] $ \text{(A)}\ 3\qquad\text{(B)}\ 6\qquad\text{(C)}\ 8\qquad\text{(D)}\ 12\qquad\text{(E)}\ 18 $

Let $A$ and $B$ be fixed points in the Euclidean plane with $AB = 6$. Let $R$ be the region of points in the plane such that, for each $P \in R$, there exists a point $C$ such that $AC = 3$ and $P$ does not lie outside $\vartriangle ABC$. Compute the greatest integer less than or equal to the area of $R$.

Answer the following questions: (1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$ (2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$ 35 points

Assume that the set of all positive integers is decomposed into $r$ disjoint subsets $A_{1}, A_{2}, \cdots, A_{r}$ $A_{1} \cup A_{2} \cup \cdots \cup A_{r}= \mathbb{N}$. Prove that one of them, say $A_{i}$, has the following property: There exist a positive integer $m$ such that for any $k$ one can find numbers $a_{1}, \cdots, a_{k}$ in $A_{i}$ with $0 < a_{j+1}-a_{j} \le m \; (1\le j \le k-1)$.

A prime is called [i]perfect[/i] if there is a permutation $a_1, a_2, \cdots, a_{\frac{p-1}{2}}, b_1, b_2, \cdots, b_{\frac{p-1}{2}}$ of $1, 2, \cdots, p-1$ satisfies $$b_i \equiv a_i + \frac{1}{a_i} \pmod p$$ for all $1 \le i \le \frac{p-1}{2}$. Show that there are infinitely many primes that are not perfect. [i]Proposed By - CSJL[/i]

Determine all prime numbers $ p $ and natural numbers $ x, y $ for which $ p^x-y^3 = 1 $.

What is the value of $ { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=odd} $ $ - { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=even} $

Let $p(x)$ be a polynomial with integer coefficients such that $p(0)=p(1)=1$. We define the sequence $a_0, a_1, a_2, \ldots, a_n, \ldots$ that starts with an arbitrary nonzero integer $a_0$ and satisfies $a_{n+1}=p(a_n)$ for all $n \in \mathbb N\cup \{0\}$. Prove that $\gcd(a_i,a_j)=1$ for all $i,j \in \mathbb N \cup \{0\}$.

How many permutations of the string $0123456$ are there such that no contiguous substrings of lengths $1<\ell<7$ have a sum of digits divisible by $7$? [i]Proposed by Srinivasan Sathiamurthy[/i]

Find all positive integers $n$ such that there exist non-constant polynomials with integer coefficients $f_1(x),...,f_n(x)$ (not necessarily distinct) and $g(x)$ such that $$1 + \prod_{k=1}^{n}\left(f^2_k(x)-1\right)=(x^2+2013)^2g^2(x)$$

In how many ways can we enter numbers from the set $\{1,2,3,4\}$ into a $4\times 4$ array so that all of the following conditions hold? (a) Each row contains all four numbers. (b) Each column contains all four numbers. (c) Each "quadrant" contains all four numbers. (The quadrants are the four corner $2\times 2$ squares.)

If $x,y$ are real numbers with $x+y\geqslant 0$, determine the minimum value of the expression \[K=x^5+y^5-x^4y-xy^4+x^2+4x+7\] For which values of $x,y$ does $K$ take its minimum value?

Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.

For all positive integers $ x$, let \[ f(x) \equal{} \begin{cases}1 & \text{if }x \equal{} 1 \\ \frac x{10} & \text{if }x\text{ is divisible by 10} \\ x \plus{} 1 & \text{otherwise}\end{cases}\]and define a sequence as follows: $ x_1 \equal{} x$ and $ x_{n \plus{} 1} \equal{} f(x_n)$ for all positive integers $ n$. Let $ d(x)$ be the smallest $ n$ such that $ x_n \equal{} 1$. (For example, $ d(100) \equal{} 3$ and $ d(87) \equal{} 7$.) Let $ m$ be the number of positive integers $ x$ such that $ d(x) \equal{} 20$. Find the sum of the distinct prime factors of $ m$.

Prove that the locus of the point of intersection of three mutually perpendicular planes tangent to the surface $$ax^2 + by^2 +cz^2 =1\;\;\; (\text{where}\;\;abc \ne 0)$$ is the sphere $$x^2 +y^2 +z^2 =\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$$

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.