Determine the largest integer $n$ such that there exist monic quadratic polynomials $p_1(x)$, $p_2(x)$, $p_3(x)$ with integer coefficients so that for all integers $ i \in [1, n]$ there exists some $j \in [1, 3]$ and $m \in Z$ such that $p_j (m) = i$.

Let $a,b,c$ be real numbers satisfying: \[ab-a=b+119\] \[bc-b=c+59\] \[ca-c=a+71\] Determine all possible values of $a+b+c$.

Prove that for all $x_1, x_2,\ldots , x_n \in \mathbb R$ the following inequality holds: \[\sum_{n \geq i >j \geq 1} \cos^2(x_i - x_j ) \geq \frac{n(n-2)}{4}\]

Colin has $900$ Choco Pies. He realizes that for some integer values of $n \le 900$, if he eats n pies a day, he will be able to eat the same number of pies every day until he runs out. How many possible values of $n$ are there?

Positive integers $a,b,c,x,y,z$ satisfy: $a^2+b^2=c^2$, $x^2+y^2=z^2$ and $|x-a| \leq 1$ , $|y-b| \leq 1$. Prove that sets $\{a,b\}$ and $\{x,y\}$ are equal.

Determine all possible values of $A^3+B^3+C^3-3ABC$ where $A$, $B$, and $C$ are nonnegative integers.

In the Cartesian plane $Oxy$ we consider the points ${A_1}\left( {40,1} \right), {A_2}\left( {40,2} \right), \ldots , {A_{40}}\left( {40,40} \right)$ as well as the segments $O{A_1},O{A_2},\ldots,O{A_{40}}$. A point of the Cartesian plane $Oxy$ is called "good", if its coordinates are integers and it is internal of one segment $O{A_i}, i=1,2,3,\ldots,40$. Additionally, one of the segments $O{A_1},O{A_2},\ldots,O{A_{40}}$ is called "good" if it contains a "good" point. Find the number of "good" segments and "good" points.

Prove that for $\forall$ $k\geq 2$, $k\in \mathbb{N}$ there exist a natural number that could be presented as a sum of two, three … $k$ cubes of natural numbers.

Let $ABC$ be a scalene triangle and $D$ be the feet of altitude from $A$ to $BC$. Let $I_1$, $I_2$ be incenters of triangles $ABD$ and $ACD$ respectively, and let $H_1$, $H_2$ be orthocenters of triangles $ABI_1$ and $ACI_2$ respectively. The circles $(AI_1H_1)$ and $(AI_2H_2)$ meet again at $X$. The lines $AH_1$ and $XI_1$ meet at $Y$, and the lines $AH_2$ and $XI_2$ meet at $Z$. Suppose the external common tangents of circles $(BI_1H_1)$ and $(CI_2H_2)$ meet at $U$. Prove that $UY=UZ$. [i]Proposed by Ivan Chan Kai Chin[/i]

Find all pairs $(p,q)$ of prime numbers which $p>q$ and $$\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}$$ is an integer.

$ \lim_{n\to\infty } \left( (1+1/n)^{-n}\sum_{i=0}^n\frac{1}{i!} \right)^{2n} $

If $ AD$ is the altitude, $ BE$ the angle bisector, and $ CF$ the median of a triangle $ ABC$, prove that $ AD,BE,$ and $ CF$ are concurrent if and only if: $ a^2(a\minus{}c)\equal{}(b^2\minus{}c^2)(a\plus{}c),$ where $ a,b,c$ are the lengths of the sides $ BC,CA,AB$, respectively.

Find the set $S$ of primes such that $p \in S$ if and only if there exists an integer $x$ such that $x^{2010} + x^{2009} + \cdots + 1 \equiv p^{2010} \pmod{p^{2011}}$. [i]Brian Hamrick.[/i]

Let be a natural number $ k, $ and a matrix $ M\in\mathcal{M}_k(\mathbb{R}) $ having the property that $$ \det\left( I-\frac{1}{n^2}\cdot A^2 \right) +1\ge\det \left( I -\frac{1}{n}\cdot A \right) +\det \left( I +\frac{1}{n}\cdot A \right) , $$ for all natural numbers $ n. $ Prove that the trace of $ A $ is $ 0. $ [i]Nelu Chichirim[/i]

A disc is divided into $30$ segments which are labelled by $50,100,150,\ldots,1500$ in some order. Show that there always exist three successive segments, the sum of whose labels is at least $2350$.

Compute the number of integers $1 \leq n \leq 1024$ such that the sequence $\lceil n \rceil$, $\lceil n/2 \rceil$, $\lceil n/4 \rceil$, $\lceil n/8 \rceil$, $\ldots$ does not contain any multiple of $5$. [i]Proposed by Sean Li[/i]

Let $a_1,a_2$ - two naturals, and $1<b_1<a_1,1<b_2<a_2$ and $b_1|a_1,b_2|a_2$. Prove that $a_1b_1+a_2b_2-1$ is not divided by $a_1a_2$

2. Let $A$ be a set of $2020$ distinct real numbers. Call a number [i]scarily epic[/i] if it can be expressed as the product of two (not necessarily distinct) numbers from $A$. What is the minimum possible number of distinct [i]scarily epic[/i] numbers? [i]Proposed by Monkey_king1[/i]

Let $n\geq 2$. Find all matrices $A\in\mathcal{M}_n(\mathbb{C})$ such that $$\text{rank}(A^2)+\text{rank}(B^2)\geq 2\text{rank}(AB),$$ for all $B\in\mathcal{M}_n(\mathbb{C})$. [i]Cristi Săvescu[/i]

Let $n$ be a positive integer, and let $x_1, x_2, \dots, x_n$ be distinct positive integers with $x_1 = 1$. Construct an $n \times 3$ table where the entries of the $k$-th row are $x_k, 2x_k, 3x_k$ for $k = 1, 2, \dots, n$. Now follow a procedure where, in each step, two identical entries are removed from the table. This continues until there are no more identical entries in the table. [list=a] [*] Prove that at least three entries remain at the end of the procedure. [*] Prove that there are infinitely many possible choices for $n$ and $x_1, x_2, \dots, x_n$ such that only three entries remain. [/list]

It is election year in PUMACland, and for the presidential election there are $27$ people voting for either Vraj Patel or Vedant Shah. Each voter selects a candidate uniformly at random, and their ballots are labeled $1$ through $27.$ The election takes place as a series of rounds. In each round, the surviving ballots are sorted by label and separated into consecutive groups of three. From each group, the person with the most votes wins, and exactly one of the ballots bearing the winner’s name is allowed to proceed to the next round. This procedure continues until a single ballot remains, and the person whose name is on the ballot wins. Alice, Bob, and Carol submitted ballots numbered $1, 15,$ and $27,$ respectively. Suppose that Alice, Bob, and Carol had all flipped their votes. If the probability that the outcome of the election would have changed is $\tfrac{a}{b}$ for relatively prime positive integers $a, b,$ find $a + b.$

Given a convex polygon $ A_1A_2 \ldots A_n$ with area $ S$ and a point $ M$ in the same plane, determine the area of polygon $ M_1M_2 \ldots M_n,$ where $ M_i$ is the image of $ M$ under rotation $ R^{\alpha}_{A_i}$ around $ A_i$ by $ \alpha_i, i \equal{} 1, 2, \ldots, n.$

Find the least positive integer $n$ with the property: Among arbitrarily $n$ selected consecutive positive integers, all smaller than $2018$, there is at least one that is divisible by its sum of digits .

The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller? $\textbf{(A) }\dfrac54\qquad\textbf{(B) }\dfrac32\qquad\textbf{(C) }\dfrac95\qquad\textbf{(D) }2\qquad\textbf{(E) }\dfrac52$

Let $A$ and $B$ be two points inside a given circle $k$. Prove that there exist (infinitely many) circles through $A$ and $B$ which lie entirely in $k$.