Which of the following polynomials does not divide $x^{60} - 1$? $ \textbf{a)}\ x^2+x+1 \qquad\textbf{b)}\ x^4-1 \qquad\textbf{c)}\ x^5-1 \qquad\textbf{d)}\ x^{15}-1 \qquad\textbf{e)}\ \text{None of above} $

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

$a, b, c, d$ are positive integers, and $r=1-\frac{a}{b}-\frac{c}{d}$. And, $a+c \le 1982, r \ge 0$. Prove that $r>\frac{1}{1983^3}$.

For any permutation $ f : \{ 1, 2, \cdots , n \} \to \{1, 2, \cdots , n \} $, and define \[ A = \{ i | i > f(i) \} \] \[ B = \{ (i, j) | i<j \le f(j) < f(i) \ or \ f(j) < f(i) < i < j \} \] \[ C = \{ (i, j) | i<j \le f(i) < f(j) \ or \ f(i) < f(j) < i < j \} \] \[ D = \{ (i, j) | i< j \ and \ f(i) > f(j)\} \] Prove that $ |A| + 2|B| + |C| = |D| $.

Prove that each positive integer is equal to a difference of two positive integers with the same number of the prime divisors.

One right pyramid has a base that is a regular hexagon with side length $1$, and the height of the pyramid is $8$. Two other right pyramids have bases that are regular hexagons with side length $4$, and the heights of those pyramids are both $7$. The three pyramids sit on a plane so that their bases are adjacent to each other and meet at a single common vertex. A sphere with radius $4$ rests above the plane supported by these three pyramids. The distance that the center of the sphere is from the plane can be written as $\frac{p\sqrt{q}}{r}$ , where $p, q$, and $r$ are relatively prime positive integers, and $q$ is not divisible by the square of any prime. Find $p+q+r$.

Meghal is playing a game with $2016$ rounds $1, 2, ..., 201$6. In round $n$, two rectangular double-sided mirrors are arranged such that they share a common edge and the angle between the faces is $\frac{2\pi}{n+2}$. Meghal shoots a laser at these mirrors and her score for the round is the number of points on the two mirrors at which the laser beam touches a mirror. What is the maximum possible score Meghal could have after she finishes the game?

Let $x \neq y$ be positive reals satisfying $x^3+2013y=y^3+2013x$, and let $M = \left( \sqrt{3}+1 \right)x + 2y$. Determine the maximum possible value of $M^2$. [i]Proposed by Varun Mohan[/i]

For $1\leq n\leq 2016$, how many integers $n$ satisfying the condition: the reminder divided by $20$ is smaller than the one divided by $16$.

Find the area of a triangle with side lengths $\sqrt{13},\sqrt{29},$ and $\sqrt{34}.$ The area can be expressed as $\frac{m}{n}$ for $m,n$ relatively prime positive integers, then find $m+n.$ [i]Proposed by Kaylee Ji[/i]

The integer n is a number expressed as the sum of an even number of different positive integers less than or equal to 2000. 1+2+ · · · +2000 Find all of the following positive integers that cannot be the value of n.

Consider the polynomials $P(x)=x^4-3x^3+x-3,\,\,\,\,Q(x)=x^2-2x-3 \,\,\,\, R(x)=-x^2-5x+a$ i) Find $a \in $R such that polynomial $R(x)$ is dividide by $x-2$ ii) Factor polynomials $P(x),Q(x)$ iii) Prove that exrpession $-x^2+x+\frac{P(x)}{Q(x)}+15$ is a perfect square.

This line graph represents the price of a trading card during the first $6$ months of $1993$. [asy] unitsize(18); for (int a = 0; a <= 6; ++a) { draw((4*a,0)--(4*a,10)); } for (int a = 0; a <= 5; ++a) { draw((0,2*a)--(24,2*a)); } draw((0,5)--(4,4)--(8,8)--(12,3)--(16,9)--(20,6)--(24,2),linewidth(1.5)); label("$Jan$",(2,0),S); label("$Feb$",(6,0),S); label("$Mar$",(10,0),S); label("$Apr$",(14,0),S); label("$May$",(18,0),S); label("$Jun$",(22,0),S); label("$\textbf{1993 PRICES FOR A TRADING CARD}$",(12,10),N); label("$\begin{tabular}{c}\textbf{P} \\ \textbf{R} \\ \textbf{I} \\ \textbf{C} \\ \textbf{E} \end{tabular}$",(-2,5),W); label("$1$",(0,2),W); label("$2$",(0,4),W); label("$3$",(0,6),W); label("$4$",(0,8),W); label("$5$",(0,10),W); [/asy] The greatest monthly drop in price occurred during $\text{(A)}\ \text{January} \qquad \text{(B)}\ \text{March} \qquad \text{(C)}\ \text{April} \qquad \text{(D)}\ \text{May} \qquad \text{(E)}\ \text{June}$

Jeffrey stands on a straight horizontal bridge that measures $20000$ meters across. He wishes to place a pole vertically at the center of the bridge so that the sum of the distances from the top of the pole to the two ends of the bridge is $20001$ meters. To the nearest meter, how long of a pole does Jeffrey need?

Jane tells you that she is thinking of a three-digit number that is greater than $500$ that has exactly $20$ positive divisors. If Jane tells you the sum of the positive divisors of her number, you would not be able to figure out her number. If, instead, Jane had told you the sum of the \textit{prime} divisors of her number, then you also would not have been able to figure out her number. What is Jane's number? (Note: the sum of the prime divisors of $12$ is $2 + 3 = 5$, not $2 + 2 + 3 = 7$.)

Given a trihedral angle Sxyz. A plane (P) not through S cuts Sx,Sy,Sz respectively at A,B,C. On the plane (P), outside triangle ABC, construct triangles DAB,EBC,FCA which are confruent to the triangles SAB,SBC,SCA respectively. Let (T) be the sphere lying inside Sxyz, but not inside the tetrahedron SABC, toucheing the planes containing the faces of SABC. Prove that (T) touches the plane (P) at the circumcenter of triangle DEF.

Prove that for positive real numbers $a, b, c$ the following inequality holds: \begin{align*} \frac{a}{9bc+1}+\frac{b}{9ca+1}+\frac{c}{9ab+1}\geq \frac{a+b+c}{1+(a+b+c)^2} \end{align*} When does equality occur?

How many positive integer solutions does the following system of equations have? $$\begin{cases}\sqrt{2020}(\sqrt{a}+\sqrt{b})=\sqrt{(c+2020)(d+2020)}\\\sqrt{2020}(\sqrt{b}+\sqrt{c})=\sqrt{(d+2020)(a+2020)}\\\sqrt{2020}(\sqrt{c}+\sqrt{d})=\sqrt{(a+2020)(b+2020)}\\\sqrt{2020}(\sqrt{d}+\sqrt{a})=\sqrt{(b+2020)(c+2020)}\\ \end{cases}$$

Find the number of ordered triples of sets $(T_1, T_2, T_3)$ such that 1. each of $T_1, T_2$, and $T_3$ is a subset of $\{1, 2, 3, 4\}$, 2. $T_1 \subseteq T_2 \cup T_3$, 3. $T_2 \subseteq T_1 \cup T_3$, and 4. $T_3\subseteq T_1 \cup T_2$.

Consider triangle $ABC$ with $AB = AC$ and $\angle A = 40 ^o$. The points $S$ and $T$ are on the sides $AB$ and $BC$, respectively, so that $\angle BAT = \angle BCS= 10 ^o$. The lines $AT$ and $CS$ intersect at point $P$. Prove that $BT = 2PT$.

Find all triplets of integers $(a, b, c)$, that verify the equation $$|a+3|+b^2+4\cdot c^2-14\cdot b-12\cdot c+55=0.$$

For a positive integer $n$, let $f(n)$ be the number obtained by writing $n$ in binary and replacing every 0 with 1 and vice versa. For example, $n=23$ is 10111 in binary, so $f(n)$ is 1000 in binary, therefore $f(23) =8$. Prove that \[\sum_{k=1}^n f(k) \leq \frac{n^2}{4}.\] When does equality hold? (Proposed by Stephan Wagner, Stellenbosch University)

Nine distinct light bulbs are placed in a circle, each of which is off. Determine the number of ways to turn on some of the light bulbs in the circle such that no four consecutive bulbs are all off.

Suppose that the rational numbers $a, b$ and $c$ are the roots of the equation $x^3+ax^2 + bx + c = 0$. Find all such rational numbers $a, b$ and $c$. Justify your answer

Jesse has ten squares, which are labeled $1, 2, \dots, 10$. In how many ways can he color each square either red, green, yellow, or blue such that for all $1 \le i < j \le 10$, if $i$ divides $j$, then the $i$-th and $j$-th squares have different colors?