There are $90$ cards and two different digits are written on each one: $01$, $02$, $03$, $04$, $05$, $06$, $07$, $08$, $09$, $10$, $12$, and so on up to $98$. A set of cards is [i]correct [/i]if it does not contain any cards whose first digit is the same as the second digit of another card in the set. We call the [i]value [/i]of a set of cards the sum of the numbers written on each card. For example, the four cards $04$, $35$, $78$ and $98$ form a correct set and their value is $215$, since$ 04+35+78+98=215$. Find a correct set that has the largest possible value. Explain why it is impossible to achieve a correct set of higher value.

A four-digit number $n$ is said to be [i]literally 1434[/i] if, when every digit is replaced by its remainder when divided by $5$, the result is $1434$. For example, $1984$ is [i]literally 1434[/i] because $1$ mod $5$ is $1$, $9$ mod $5$ is $4$, $8$ mod $5$ is $3$, and $4$ mod $5$ is $4$. Find the sum of all four-digit positive integers that are [i]literally 1434[/i]. [i]Proposed by Evin Liang[/i] [hide=Solution] [i]Solution.[/i] $\boxed{67384}$ The possible numbers are $\overline{abcd}$ where $a$ is $1$ or $6$, $b$ is $4$ or $9$, $c$ is $3$ or $8$, and $d$ is $4$ or $9$. There are $16$ such numbers and the average is $\dfrac{8423}{2}$, so the total in this case is $\boxed{67384}$. [/hide]

The absolute value of every number in the sequence $\{a_n\}$ is smaller than 2005, and \[a_{n+6}=a_{n+4}+a_{n+2}-a_n.\] holds for all positive integers n. Prove that $\{a_n\}$ is periodic. Incredibly, this was probably the most difficult problem of our independent study problems in the 1st TST (excluding the final exam).

Let $ABC$ be an acute scalene triangle. Choose a circle $\omega$ passing through $B$ and $C$ which intersects the segments $AB$ and $AC$ at the interior points $D$ and $E$, respectively. The lines $BE$ and $CD$ intersects at $F$. Let $G$ be a point on the circumcircle of $ABF$ such that $GB$ is tangent to $\omega$ and let $H$ be a point on the circumcircle of $ACF$ such that $HC$ is tangent to $\omega$. Prove that there exists a point $T\neq A$, independent of the choice of $\omega$, such that the circumcircle of triangle $AGH$ passes through $T$.

$a_1a_2a_3$ and $a_3a_2a_1$ are two three-digit decimal numbers, with $a_1$ and $a_3$ different non-zero digits. Squares of these numbers are five-digit numbers $b_1b_2b_3b_4b_5$ and $b_5b_4b_3b_2b_1$ respectively. Find all such three-digit numbers.

Find all natural numbers $x,y$ such that $$x^5=y^5+10y^2+20y+1.$$

Let $ABC$ be an acute triangle. Suppose that circle $\Gamma_1$ has it's center on the side $AC$ and is tangent to the sides $AB$ and $BC$, and circle $\Gamma_2$ has it's center on the side $AB$ and is tangent to the sides $AC$ and $BC$. The circles $\Gamma_1$ and $ \Gamma_2$ intersect at two points $P$ and $Q$. Show that if $A, P, Q$ are collinear, then $AB = AC$.

In the cells of the table $ 100 \times100 $ are placed in pairs different numbers. Every minute each of the numbers changes to the largest of the numbers in the adjacent cells on the side. Can after $4$ hours all the numbers in the table be the same?

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

The two diagonals of a rhombus have lengths with ratio $3 : 4$ and sum $56$. What is the perimeter of the rhombus?

Let $P(x)=x^2-3x-9$. A real number $x$ is chosen at random from the interval $5\leq x \leq 15$. The probability that $\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor )}$ is equal to $\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}-d}{e}$, where $a,b,c,d$ and $e$ are positive integers and none of $a,b,$ or $c$ is divisible by the square of a prime. Find $a+b+c+d+e$.

Six ants are placed on the vertices of a regular hexagon with an area of $12$. At each point in time, each ant looks at the next ant in the hexagon (in counterclockwise order), and measures the distance, $s$, to the next ant. Each ant then proceeds towards the next ant at a speed of $\frac{s}{100}$ units per year. After T years, the ants’ new positions are the vertices of a new hexagon with an area of $4$. T is of the form $a \ln b$, where $b$ is square-free. Find $a + b$.

For $\forall$ $m\in \mathbb{N}$ with $\pi (m)$ we denote the number of prime numbers that are no bigger than $m$. Find all pairs of natural numbers $(a,b)$ for which there exist polynomials $P,Q\in \mathbb{Z}[x]$ so that for $\forall$ $n\in \mathbb{N}$ the following equation is true: $\frac{\pi (an)}{\pi (bn)} =\frac{P(n)}{Q(n)}$.

For a given chord $MN$ of a circle discussed the triangle $ABC$, whose base is the diameter $AB$ of this circle,which do not intersect the $MN$, and the sides $AC$ and $BC$ pass through the ends of $M$ and $N$ of the chord $MN$. Prove that the heights of all such triangles $ABC$ drawn from the vertex $C$ to the side $AB$, intersect at one point.

Two people, $A$ and $B$, play the following game: $A$ start choosing a positive integrer number and then, each player in it's turn, say a number due to the following rule: If the last number said was odd, the player add $7$ to this number; If the last number said was even, the player divide it by $2$. The winner is the player that repeats the first number said. Find all numbers that $A$ can choose in order to win. Justify your answer.

Prove that if the odd prime $p$ divides $a^{b}-1$, where $a$ and $b$ are positive integers, then $p$ appears to the same power in the prime factorization of $b(a^{d}-1)$, where $d=\gcd(b,p-1)$.

In an exam every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions in this exam.

There exists a certain right triangle with the smallest area in the $2$D coordinate plane such that all of its vertices have integer coordinates but none of its sides are parallel to the $x$- or $y$-axis. Additionally, all of its sides have distinct, integer lengths. What is the area of this triangle?

In a triangle $ABC$, let $O$ be the circumcenter, $H$ the ortho­center, and $M$ the midpoint of the segment $AH$. The perpendicular at $M$ onto $OM$ intersects lines $AB$ and $AC$ at $P$ and $Q$, respectively. Prove that $MP = MQ$.

We write the digits of $1995$ in the following way: $199511999955111999999555......$ 1. Determine how many digits we have to write such that the sum of the written digits is $2880$. 2.Which digit is in position number $1995$?

The circle inscribed in the triangle $A_1A_2A_3$ is tangent to its sides $A_1A_2, A_2A_3, A_3A_1$ at points $T_1, T_2, T_3$, respectively. Denote by $M_1, M_2, M_3$ the midpoints of the segments $A_2A_3, A_3A_1, A_1A_2$, respectively. Prove that the perpendiculars through the points $M_1, M_2, M_3$ to the lines $T_2T_3, T_3T_1, T_1T_2$ meet at one point.

Kelvin the frog currently sits at $(0,0)$ in the coordinate plane. If Kelvin is at $(x,y),$ either he can walk to any of $(x,y + 1),$ $(x + 1,y),$ or $(x + 1,y + 1),$ or he can jump to any of $(x,y + 2), (x + 2,y),$ or $(x+1,y+1).$ Walking and jumping from $(x,y)$ to $(x+1,y+1)$ are considered distinct actions. Compute the number of ways Kelvin can reach $(6,8).$

Suppose $x,y$ are nonnegative real numbers such that $x + y \le 1$. Prove that $8xy \le 5x(1 - x) + 5y(1 - y)$ and determine the cases of equality.

The following diagram shows a square where each side has four dots that divide the side into three equal segments. The shaded region has area 105. Find the area of the original square. [center][img]https://snag.gy/r60Y7k.jpg[/img][/center]

Is it true that for $\forall$ prime number $p$, there exist non-constant polynomials $P$ and $Q$ with $P,Q\in \mathbb{Z} [x]$ for which the remainder modulo $p$ of the coefficient in front of $x^n$ in the product $PQ$ is 1 for $n=0$ and $n=4$; $p-2$ for $n=2$ and is 0 for all other $n\geq 0$?