a) Prove that there exists a sequence of digits $\{c_n\}_{n\geq 1}$ such that or each $n\geq 1$ no matter how we interlace $k_n$ digits, $1\leq k_n\leq 9$, between $c_n$ and $c_{n+1}$, the infinite sequence thus obtained does not represent the fractional part of a rational number. b) Prove that for $1\leq k_n\leq 10$ there is no such sequence $\{c_n\}_{n\geq 1}$. [i]Dan Schwartz[/i]

$P_1,P_2,\ldots,P_{100}$ are $100$ points on the plane, no three of them are collinear. For each three points, call their triangle [b]clockwise[/b] if the increasing order of them is in clockwise order. Can the number of [b]clockwise[/b] triangles be exactly $2017$? [i]Proposed by Morteza Saghafian[/i]

If $ x \equal{} \sqrt [3]{11 \plus{} \sqrt {337}} \plus{} \sqrt [3]{11 \minus{} \sqrt {337}}$, then $ x^3 \plus{} 18x$ = ? $\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 22 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 10$

We say that a positive integer $k$ is [i]fair [/i] if the number of $2021$-digit palindromes that are a multiple of $k$ is the same as the number of $2022$-digit palindromes that are a multiple of $k$. Does the set $M = \{1, 2,..,35\}$ contain more numbers that are fair or those that are not fair? (A palindrome is an integer that reads the same forward and backward.) [i](David Hruska)[/i]

Each term in a sequence $1,0,1,0,1,0...$starting with the seventh is the sum of the last 6 terms mod 10 .Prove that the sequence $...,0,1,0,1,0,1...$ never occurs

Let $ABC$ be an isosceles triangle with $AB=AC$. Let $D$ be a point on the segment $BC$ such that $BD=2DC$. Let $P$ be a point on the segment $AD$ such that $\angle BAC=\angle BPD$. Prove that $\angle BAC=2\angle DPC$.

In a convex pentagon $ ABCDE$ in which $ BC\equal{}DE$ following equalities hold: \[ \angle ABE \equal{}\angle CAB \equal{}\angle AED\minus{}90^{\circ},\qquad \angle ACB\equal{}\angle ADE\] Show that $ BCDE$ is a parallelogram.

$a, b, c$ are three distinct integers and $n$ is a positive integer. Show that $$\frac{a^n}{(a - b)(a - c)}+\frac{ b^n}{(b - a)(b - c)} +\frac{ c^n}{(c - a)(c - b)}$$ is an integer.

Find all pairs of positive integers \( (a, b) \) such that \( f(x) = x \) is the only function \( f : \mathbb{R} \to \mathbb{R} \) that satisfies \[ f^a(x)f^b(y) + f^b(x)f^a(y) = 2xy \quad \text{for all } x, y \in \mathbb{R}. \] Here, \( f^n(x) \) represents the function obtained by applying \( f \) \( n \) times to \( x \). That is, \( f^1(x) = f(x) \) and \( f^{n+1}(x) = f(f^n(x))\) for all \(n \geq 1\).

Each cell of an $100\times 100$ board is divided into two triangles by drawing some diagonal. What is the smallest number of colors in which it is always possible to paint these triangles so that any two triangles having a common side or vertex have different colors?

The volleyball championship with $16$ teams was held in one round (each team played with each exactly one times, there are no draws in volleyball). It turned out that some two teams won the same number of matches. Prove there are the three teams that beat each other in a round robin (i.e. A beat B, B beat C, and C beat A).

Let ${\triangle ABC}$ be a right triangle with $\angle A = 90^\circ$ and $BC = 38.$ There exist points $K$ and $L$ inside the triangle such \[AK = AL = BK = CL = KL = 14.\] The area of the quadrilateral $BKLC$ can be expressed as $n\sqrt3$ for some positive integer $n.$ Find $n.$

Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area? $ \textbf{(A)} \ \frac{553}{715} \qquad \textbf{(B)} \ \frac{443}{572} \qquad \textbf{(C)} \ \frac{111}{143} \qquad \textbf{(D)} \ \frac{81}{104} \qquad \textbf{(E)} \ \frac{223}{286}$

For a non-constant function $f:\mathbb{R}\to \mathbb{R}$ prove that there exist real numbers $x,y$ satisfying $f(x+y)<f(xy)$

Let $m$ be a positive integer. Define the sequence $\{a_{n}\}_{n \ge 0}$ by \[a_{0}=0, \; a_{1}=m, \; a_{n+1}=m^{2}a_{n}-a_{n-1}.\] Prove that an ordered pair $(a, b)$ of non-negative integers, with $a \le b$, gives a solution to the equation \[\frac{a^{2}+b^{2}}{ab+1}= m^{2}\] if and only if $(a, b)$ is of the form $(a_{n}, a_{n+1})$ for some $n \ge 0$.

[b]1A.[/b] The iTest, by virtue of being the first national internet-based high school math competition, saves a lot of paper every year. The quantity of trees saved (“$a$”) is determined by the following formula: $a = x^2 + 3x + 9$, where $x$ is the number of participating students in the competition. If $x$ is the correct answer from short answer [hide=problem 22]x=20[/hide], then find $a$. [i](1 point)[/i] [b]1B.[/b] Let $q$ be the sum of the digits of $a$. If $q = b! - (b-1)! + (b-2)! - (b-3)!$, find $b$. [i](2 points)[/i] [b]1C.[/b] Find the number of the following statements that are false: [i] (4 points)[/i] 1. $q$ is the first prime number resulting from the sum of cubes of distinct fractions, where both the numerator and denominator are primes. 2. $q$ is composite. 3. $q$ is composite and is the sum of the first four prime numbers and $1$. 4. $q$ is the smallest prime equal to the difference of cubes of two consecutive primes. 5. $q$ is not the smallest prime equal to the product of twin primes plus their arithmetic mean. 6. The sum of $q$ consecutive Fibonacci numbers, starting from the $q^{th}$ Fibonacci number, is prime. 7. $q$ is the largest prime factor of $1bbb$. 8. $q$ is the $8^{th}$ largest prime number. 9. $a$ is composite. 10. $a + q + b = q^2$. 11. The decimal expansion of $q^q$ begins with $q$. 12. $q$ is the smallest prime equal to the sum of three distinct primes. 13. $q^5 + q^2 + q^1 + q^3 + q^5 + q^6 + q^4 + q^0 = 52135640$. 14. $q$ is not the smallest prime such that $q$ and $q^2$ have the same sum of their digits. 15. $q$ is the smallest prime such that $q$ = (the product of its digits + the sum of its digits). [hide=ANSWER KEY]1A. 469 1B. 4 1C. 6[/hide]

Every day, Pinky the flamingo eats either $1$ or $2$ shrimp, each with equal probability. Once Pinky has consumed $10$ or more shrimp in total, its skin will turn pink. Once Pinky has consumed $11$ or more shrimp in total, it will get sick. What is the probability that Pinky does not get sick on the day its skin turns pink?

Integers $a_1, a_2, \ldots, a_n$ satisfy $$1<a_1<a_2<\ldots < a_n < 2a_1.$$ If $m$ is the number of distinct prime factors of $a_1a_2\cdots a_n$, then prove that $$(a_1a_2\cdots a_n)^{m-1}\geq (n!)^m.$$

How many ways can $2^{2012}$ be expressed as the sum of four (not necessarily distinct) positive squares?

We are given a $4\times4$ square, consisting of $16$ squares with side length of $1$. Every $1\times1$ square inside the square has a non-negative integer entry such that the sum of any five squares that can be covered with the figures down below (the figures can be moved and rotated) equals $5$. What is the greatest number of different numbers that can be used to cover the square?

Show that the number \[p_n=\left(\frac{3+\sqrt5}{2}\right)^n+\left(\frac{3-\sqrt5}{2}\right)^n-2\] is a positive integer for any positive integer $n.$ Furthermore, show that the numbers $p_{2n-1}$ and $p_{2n}/5$ are perfect squares $($for any positive integer $n).$

There are n points in the plane with the property that the distance between any two points is at least 1. Prove that for sufficiently large n , the number of pairs of points whose distance is in $[ t_1 , t_1 + 1] \cup [ t_2 , t_2 + 1]$ for some $t_1, t_2$ , is at most $[\frac{n^2}{3}]$ , and the bound is sharp.

For positive integers $n$ and $k$, let $f(n,k)$ be the remainder when $n$ is divided by $k$, and for $n>1$ let $F(n) = \displaystyle\max_{1 \le k \le \frac{n}{2}} f(n,k)$. Find the remainder when $\displaystyle\sum_{n=20}^{100} F(n)$ is divided by $1000$.

Show that if $a$ and $b$ are integer numbers, and $a^2 + b^2 + 9ab$ is divisible by $11$, then $a^2-b^2$ divisible by $11$.

Determine all the triples $\{a, b, c \}$ of positive integers coprime (not necessarily pairwise prime) such that $a + b + c$ simultaneously divides the three numbers $a^{12} + b^{12}+ c^{12}$, $ a^{23} + b^{23} + c^{23} $ and $ a^{11004} + b^{11004} + c^{11004}$