For any integers $1\leq n\leq m\leq5$, how many hyperbolas does the equation $\rho=\frac{1}{1-\text{C}_m^n \cos\theta}$ represent? Note: $\text{C}_m^n=\frac{m!}{n!(m-n)!}$. $\text{(A)}15\qquad\text{(B)}10\qquad\text{(C)}7\qquad\text{(D)}6$

Mr. Schwartz has been hired to paint a row of 7 houses. Each house must be painted red, blue, or green. However, to make it aesthetically pleasing, he doesn't want any three consecutive houses to be the same color. Find the number of ways he can fulfill his task. [i]Proposed by Daniel Monroe[/i]

Show that for any $a,b,c$ positive reals, \[ (a+b)(a+c) \geq 2 \sqrt{abc(a+b+c)} \]

The square $0\le x\le 1$, $0\le y\le 1$ is drawn in the plane $Oxy$. A grasshopper sitting at a point $M$ with noninteger coordinates outside this square jumps to a new point which is symmetrical to $M$ with respect to the leftmost (from the grasshopper’s point of view) vertex of the square. Prove that no matter how many times the grasshopper jumps, it will never reach the distance more than $10 d$ from the center $C$ of the square, where $d$ is the distance between the initial position $M$ and the center $C$. (A Kanel)

Let $P$ be the set of positive integers that are prime numbers. Find the number of subsets of $P$ that have the property that the sum of their elements is $34$ such as $\{3, 31\}$.

Determine the smallest integer of the form $\frac{ \overline{AB}}{B}$ .where $A$ and $B$ are three-digit positive integers and $\overline{AB}$ denotes the six-digit number that is form by writing the numbers $A$ and $B$ consecutively.

Let $a,b,c,d$ be positive integers satisfying \[\frac{ab}{a+b}+\frac{cd}{c+d}=\frac{(a+b)(c+d)}{a+b+c+d}.\] Determine all possible values of $a+b+c+d$.

Find the remainder when \[9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}\] is divided by $ 1000$.

Let $P$ be the set of prime numbers. Consider a subset $M$ of $P$ with at least three elements. We assume that, for each non empty and finite subset $A$ of $M$, with $A \neq M$, the prime divisors of the integer $( \prod_{p \in A} ) - 1$ belong to $M$. Prove that $M = P$.

Let $x,y$ be distinct real numbers such that $\frac{x^n-y^n}{x-y}$ is an integer for $4$ consecutive positive integer $n$. Prove that $\frac{x^n-y^n}{x-y}$ is an integer for all positive integers $n$.

An entry in a grid is called a [i]saddle [/i] point if it is the largest number in its row and the smallest number in its column. Suppose that each cell in a $ 3 \times 3$ grid is filled with a real number, each chosen independently and uniformly at random from the interval $[0, 1]$. Compute the probability that this grid has at least one saddle point.

A positive integer $n$ is called[i] bad [/i]if it cannot be expressed as the product of two distinct positive integers greater than $1$. Find the number of bad positive integers less than $100. $ [i]Proposed by Michael Ren[/i]

Find all integers $k$, such that there exists an integer sequence ${\{a_n\}}$ satisfies two conditions below (1) For all positive integers $n$,$a_{n+1}={a_n}^3+ka_n+1$ (2) $|a_n| \leq M$ holds for some real $M$

Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could [i]not[/i] be the total value of the four coins, in cents? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 45 \qquad \textbf{(E)}\ 55$

Alphonse and Beryl play a game starting with a blank blackboard. Alphonse goes first and the two players alternate turns. On Alphonse's first turn, he writes the integer $10^{2011}$ on the blackboard. On each subsequent turn, each player can do exactly one of the following two things: [b](i)[/b] replace any number $x$ that is currently on the blackboard with two integers a and b greater than $1$ such that $x = ab,$ or [b](ii)[/b] erase one or two copies of a number $y$ that appears at least twice on the blackboard. Thus, there may be many numbers on the board at any time. The first player who cannot do either of these things loses. Determine which player has a winning strategy and explain the strategy.

Let $S = \{1, 22, 333, \dots , 999999999\}$. For how many pairs of integers $(a, b)$ where $a, b \in S$ and $a < b$ is it the case that $a$ divides $b$?

The side lengths $a,b,c$ of a triangle $ABC$ are positive integers. Let:\\ \[T_{n}=(a+b+c)^{2n}-(a-b+c)^{2n}-(a+b-c)^{2n}+(a-b-c)^{2n}\] for any positive integer $n$. If $\frac{T_{2}}{2T_{1}}=2023$ and $a>b>c$ , determine all possible perimeters of the triangle $ABC$.

Determine all polynomials P(x) with real coefficients such that [(x + 1)P(x − 1) − (x − 1)P(x)] is a constant polynomial.

Let $A=(a_{jk})$ be a $10\times 10$ array of positive real numbers such that the sum of numbers in row as well as in each column is $1$. Show that there exists $j<k$ and $l<m$ such that \[a_{jl}a_{km}+a_{jm}a_{kl}\ge \frac{1}{50}\]

Given is a natural number $n\geqslant 3$. Find the smallest $k{}$ for which the following statement is true: for any $n{}$-gon and any two points inside it there is a broken line with $k{}$ segments connecting these points, lying entirely inside the $n{}$-gon. [i]Proposed by L. Emelyanov[/i]

A closed broken self-intersecting line is drawn in the plane. Each of the links of this line is intersected exactly once and no three links intersect at the same point. Further, there are no self-intersections at the vertices and no two links have a common segment. Can it happen that every point of self-intersection divides both links in halves?

Define that $x*y=ax+by+cxy$ for all real numbers $x,y$, where $a,b,c$ are uncertain. It is known that $1*2=3,2*3=4$. If there exists a real number $d$, for any real number $x$, $x*d=x$, then $d=$________.

We are given a semicircle $k$ with center $O$ and diameter $AB$. Let $C$ be a point on $k$ such that $CO \bot AB$. The bisector of $\angle ABC$ intersects $k$ at point $D$. Let $E$ be a point on $AB$ such that $DE \bot AB$ and let $F$ be the midpoint of $CB$. Prove that the quadrilateral $EFCD$ is cyclic.

Perpendicular bisectors of the sides $BC$ and $AC$ of an acute-angled triangle $ABC$ intersect lines $AC$ and $BC$ at points $M$ and $N$. Let point $C$ move along the circumscribed circle of triangle $ABC$, remaining in the same half-plane relative to $AB$ (while points $A$ and $B$ are fixed). Prove that line $MN$ touches a fixed circle.

Let $ a\in (1,\infty) $ and a countinuous function $ f:[0,\infty)\longrightarrow\mathbb{R} $ having the property: $$ \lim_{x\to \infty} xf(x)\in\mathbb{R} . $$ [b]a)[/b] Show that the integral $ \int_1^{\infty} \frac{f(x)}{x}dx $ and the limit $ \lim_{t\to\infty} t\int_{1}^a f\left( x^t \right) dx $ both exist, are finite and equal. [b]b)[/b] Calculate $ \lim_{t\to \infty} t\int_1^a \frac{dx}{1+x^t} . $