Given a positive integer $n$, we say that a polynomial $P$ with real coefficients is $n$-pretty if the equation $P(\lfloor x \rfloor)=\lfloor P(x) \rfloor$ has exactly $n$ real solutions. Show that for each positive integer $n$ [list=a] [*] there exists an n-pretty polynomial; [*] any $n$-pretty polynomial has a degree of at least $\tfrac{2n+1}{3}$. [/list] ([i]Remark.[/i] For a real number $x$, we denote by $\lfloor x \rfloor$ the largest integer smaller than or equal to $x$.)

Find the value of $(52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2}$.

In a triangle with sides of lengths $a,b,$ and $c,$ $(a+b+c)(a+b-c) = 3ab.$ The measure of the angle opposite the side length $c$ is $\displaystyle \text{(A)} \ 15^\circ \qquad \text{(B)} \ 30^\circ \qquad \text{(C)} \ 45^\circ \qquad \text{(D)} \ 60^\circ \qquad \text{(E)} \ 150^\circ$

Suppose there exists a group with exactly $n$ subgroups of index 2. Prove that there exists a finite abelian group $G$ that has exactly $n$ subgroups of index 2.

The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd? $\textbf{(A) }1/21\qquad\textbf{(B) }1/14\qquad\textbf{(C) }5/63\qquad\textbf{(D) }2/21\qquad\textbf{(E) } 1/7$

Solve in integers the following equation: \[y=2x^2+5xy+3y^2\]

9. A sequence of nonzero complex numbers $a_1, a_2, \dots, a_{2020}$ satisfies $a_3=a_2^2+2a_1a_2$ and $$\frac{a_{n+2}}{a_{n+1}}-\frac{a_{n+1}}{a_{n}}=a_n+a_{n+1},$$ for all $2018\ge n\ge 2$. Given $a_2-a_{2020}=2025$, how many integers $0\le a_1\le 2020$ are there, such that $a_1+a_2+\cdots+a_{2019}$ is a real number? [i]Proposed by winnertakeover[/i]

Find all real numbers $x,y$ such that $x^3-y^3=7(x-y)$ and $x^3+y^3=5(x+y)$.

Let $f(x)$ be a polynomial of degree $3$ with real coefficients satisfying $|f(x)| = 12$ for $x = 1, 2, 3, 5, 6, 7$. Find $|f(0)|$.

There are $N$ monsters, each with a positive weight. On each step, two of the monsters are merged into one, whose weight is the sum of weights for the two original monsters. At the end, all monsters will be merged into one giant monster. During this process, if at any mergence, one of the two monsters has a weight greater than $2.020$ times the other monster's weight, we will call this mergence [b]dangerous[/b]. The dangerous level of a sequence of mergences is the number of dangerous mergence throughout its process. Prove that, no matter how the weights being distributed among the monsters, "for every step, merge the lightest two monsters" is always one of the merging sequences that obtain the minimum possible dangerous level. [i]Proposed by houkai[/i]

A triangle $ ABC $ is given, and a segment $ PQ=t $ on $ BC $ such that $ P $ is between $ B $ and $ Q $ and $ Q $ is between $ P $ and $ C $. Let $ PP_1 || AB $, $ P_1 $ is on $ AC $, and $ PP_2 || AC $, $ P_2 $ is on $ AB $. Points $ Q_1 $ and $ Q_2 $ аrе defined similar. Prove that the sum of the areas of $ PQQ_1P_1 $ and $ PQQ_2P_2 $ does not depend from the position of $ PQ $ on $ BC $.

An integer $ n $ is called good if $ |n| $ is not a square of an integer. Find all integers $m$ with the following property: $m$ can be represented in infinite ways as a sum of three disctinct good numbers, the product of which is the square of an odd integer.

Estimate the product of all the nonzero digits in the decimal expansion of $2020!$. If your estimate is $E$ and the answer is $A$, your score for this problem will be \[\max\Big(0, \Big\lfloor 15-0.02\cdot\Big\lvert \log_{10}\Big(\frac{A}{E}\Big)\Big\rvert \Big\rfloor\Big).\] [i]Proposed by Alex Li[/i]

In the future, MIT has attracted so many students that its buildings have become skyscrapers. Ben and Jerry decide to go ziplining together. Ben starts at the top of the Green Building, and ziplines to the bottom of the Stata Center. After waiting $a$ seconds, Jerry starts at the top of the Stata Center, and ziplines to the bottom of the Green Building. The Green Building is $160$ meters tall, the Stata Center is $90$ meters tall, and the two buildings are $120$ meters apart. Furthermore, both zipline at $10$ meters per second. Given that Ben and Jerry meet at the point where the two ziplines cross, compute $100a$.

Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$. Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$. [i]Author: Farzan Barekat, Canada[/i]

Prove that for any real numbers $a,b,c,d \geq \frac{1}{3}$ the following inequality holds: $$\sqrt{\frac{a^6}{b^4+c^3}+\frac{b^6}{c^4+d^3}+\frac{c^6}{d^4+a^3}+\frac{d^6}{a^4+b^3}}\geq \frac{a+b+c+d}{4}$$ [i]D. Zmiaikou[/i]

Prove that if in the following fraction we have $n$ radicals in the numerator and $n - 1$ in the denominator, then $$\frac{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}>\frac14$$

In a room are several people, some of which always lie and all others always tell the truth. Their ages are pairwise distinct. Each person says one of the following phrases: "In this room, there is an equal number of truth-sayers older than me and of liars younger than me" or "In this room, there is an equal number of truth-sayers younger than me and of liars older than me" What is the maximum possible number of truth-sayers in the room? Find an example in which this maximum is achieved and prove a higher number is impossible.

We consider sports tournaments with $n \ge 4$ participating teams and where every pair of teams plays against one another at most one time. We call such a tournament [i]balanced [/i] if any four participating teams play exactly three matches between themselves. So, not all teams play against one another. Determine the largest value of $n$ for which a balanced tournament with $n$ teams exists.

All the angles of the hexagon $ABCDEF$ are equal. Prove that \[AB-DE=EF-BC=CD-FA \]

The sequence $(x_n)$ is defined as follows: $$x_1=2,\, x_{n+1}=\sqrt{x_n+8}-\sqrt{x_n+3}$$ for all $n\geq 1$. a. Prove that $(x_n)$ has a finite limit and find that limit. b. For every $n\geq 1$, prove that $$n\leq x_1+x_2+\dots +x_n\leq n+1.$$

For the positive integers $a, b$, $lcm(a, b) = \gcd(a, b) = p^2q^4$, where $p$ and $q$ are prime numbers. Find $lcm(ap, bq)$. Here lcm and gcd represent the least common multiple and the greatest common divisor respectively.

Let $\Delta ABC$ be a right triangle with $\angle ACB=90^\circ$. The points $P$ and $Q$ on the side $BC$ and $R$ and $S$ on the side $CA$ are such that $\angle BAP=\angle PAQ=\angle QAC$ and $\angle ABS=\angle SBR=\angle RBC$. If $AP\cap BS=T$, prove that $120^\circ<\angle RTB<150^\circ$.

Let a positive integer $n$ be $\textit{nice}$ if there exists a positive integer $m$ such that \[ n^3 < 5mn < n^3 +100. \] Find the number of [i]nice[/i] positive integers. [i]Proposed by Akshaj[/i]

Let $I=3\sqrt2\int^x_0\frac{\sqrt{1+\cos t}}{17-8\cos t}dt$. If $0<x<\pi$ and $\tan I=\frac2{\sqrt3}$, what is $x$?