Consider the polynomial \[P(x)=x^3+3x^2+6x+10.\] Let its three roots be $a$, $b$, $c$. Define $Q(x)$ to be the monic cubic polynomial with roots $ab$, $bc$, $ca$. Compute $|Q(1)|$. [i]Proposed by Nathan Xiong[/i]

Let $x(t)$ be a solution to the differential equation \[\left(x+x^\prime\right)^2+x\cdot x^{\prime\prime}=\cos t\] with $x(0)=x^\prime(0)=\sqrt{\frac{2}{5}}$. Compute $x\left(\dfrac{\pi}{4}\right)$.

Solve the system of simultaneous equations \[\begin{cases}a^3+3ab^2+3ac^2-6abc=1\\ b^3+3ba^2+3bc^2-6abc=1\\c^3+3ca^2+3cb^2-6abc=1\end{cases}\] in real numbers.

Find the real solution(s) to the equation $ (x \plus{} y)^2 \equal{} (x \plus{} 1)(y \minus{} 1)$.

Given are $n$ pairwise intersecting convex $k$-gons on the plane. Any of them can be transferred to any other by a homothety with a positive coefficient. Prove that there is a point in a plane belonging to at least $1 +\frac{n-1}{2k}$ of these $k$-gons.

Convex equiangular hexagon $ABCDEF$ has $AB=CD=EF=1$ and $BC = DE = FA = 4$. Congruent and pairwise externally tangent circles $\gamma_1$, $\gamma_2$, and $\gamma_3$ are drawn such that $\gamma_1$ is tangent to side $\overline{AB}$ and side $\overline{BC}$, $\gamma_2$ is tangent to side $\overline{CD}$ and side $\overline{DE}$, and $\gamma_3$ is tangent to side $\overline{EF}$ and side $\overline{FA}$. Then the area of $\gamma_1$ can be expressed as $\frac{m\pi}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Sean Li[/i]

Suppose $x_1,x_2,x_3,\dotsc$ is a strictly decreasing sequence of positive real numbers such that the series $x_1+x_2+x_3+\cdots$ diverges. Is it necessary true that the series $\sum_{n=2}^{\infty}{\min \left\{ x_n,\frac{1}{n\log (n)}\right\} }$ diverges?

Suppose that \[ \frac {2x}{3} \minus{} \frac {x}{6} \]is an integer. Which of the following statements must be true about $ x$? $ \textbf{(A)}\ \text{It is negative.} \qquad \textbf{(B)}\ \text{It is even, but not necessarily a multiple of }3\text{.}$ $ \textbf{(C)}\ \text{It is a multiple of }3\text{, but not necessarily even.}$ $ \textbf{(D)}\ \text{It is a multiple of }6\text{, but not necessarily a multiple of }12\text{.}$ $ \textbf{(E)}\ \text{It is a multiple of }12\text{.}$

For how many positive integers $n\le100$ is it true that $10n$ has exactly three times as many positive divisors as $n$ has?

There are $11$ integers (not necessarily distinct) written on the board. Can it turn out that the product of any five of them is greater than the product of the other six?

Cynthia loves Pokemon and she wants to catch them all. In Victory Road, there are a total of $80$ Pokemon. Cynthia wants to catch as many of them as possible. However, she cannot catch any two Pokemon that are enemies with each other. After exploring around for a while, she makes the following two observations: 1. Every Pokemon in Victory Road is enemies with exactly two other Pokemon. 2. Due to her inability to catch Pokemon that are enemies with one another, the maximum number of the Pokemon she can catch is equal to $n$. What is the sum of all possible values of $n$?

While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers $a$, $b$, and $c$, and recalled that their product is $24$, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than $25$ with fewer than $6$ divisors. Help James by computing $a+b+c$.

Let $n$ be a positive integer. Prove that at least one of the integers $[2^n \cdot \sqrt2]$, $[2^{n+1} \cdot \sqrt2]$, $...$, $[2^{2n} \cdot \sqrt2]$ is even, where $[a]$ denotes the integer part of $a$.

Integers are written in the cells of a table $2010 \times 2010$. Adding $1$ to all the numbers in a row or in a column is called a [i]move[/i]. We say that a table is [i]equilibrium[/i] if one can obtain after finitely many moves a table in which all the numbers are equal. [list=a] [*]Find the largest positive integer $n$, for which there exists an [i]equilibrium[/i] table containing the numbers $2^0, 2^1, \ldots , 2^n$. [*] For this $n$, find the maximal number that may be contained in such a table. [/list]

Let $a, b$ be positive integers such that $a^2 + ab + 1$ a multiple of $b^2 + ab + 1$. Prove that $a = b$.

$$(b+c)x^2+(a+c)x+(a+b)=0$$ has not real roots. Prove that $$4ac-b^2 \leq 3a(a+b+c)$$

Let $(P, P')$ and $(Q, Q')$ be two pairs of points isogonally conjugated with respect to a triangle $ABC$, and $R$ be the common point of lines $PQ$ and $P'Q'$. Prove that the pedal circles of points $P$, $Q$, and $R$ are coaxial.

Consider all sets $A$ of one hundred different natural numbers with the property that any three elements $a,b,c \in A$ (not necessarily different) are the sides of a non-obtuse triangle. Denote by $S(A)$ the sum of the perimeters of all such triangles. Compute the smallest possible value of $S(A)$.

In an infinite sequence $a_1, a_2, a_3, \cdots$, the number $a_1$ equals $1$, and each $a_n, n > 1$, is obtained from $a_{n-1}$ as follows: [list]- if the greatest odd divisor of $n$ has residue $1$ modulo $4$, then $a_n = a_{n-1} + 1,$ - and if this residue equals $3$, then $a_n = a_{n-1} - 1.$[/list] Prove that in this sequence [b](a) [/b] the number $1$ occurs infinitely many times; [b](b)[/b] each positive integer occurs infinitely many times. (The initial terms of this sequence are $1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, \cdots$ )

Prove that one can find 100 distinct pairs of integers such that every digit of each number is no less than 6 and the product of the numbers in each pair is also a number with all its digits being no less than 6. [i](4 points)[/i]

Let $f(x)=\sin x+2\cos x+3\tan x$, using radian measure for the variable $x$. In what interval does the smallest positive value of $x$ for which $f(x)=0$ lie? $\textbf{(A) } (0,1) \qquad \textbf{(B) } (1,2) \qquad \textbf{(C) } (2,3) \qquad \textbf{(D) } (3,4) \qquad \textbf{(E) } (4,5)$

Let $ABCDEF$ be a convex hexagon such that all of its vertices are on a circle. Prove that $AD$, $BE$ and $CF$ are concurrent if and only if $\frac {AB}{BC}\cdot\frac {CD}{DE}\cdot\frac {EF}{FA}= 1$.

A sequence $(a_1, a_2,\ldots, a_k)$ consisting of pairwise distinct squares of an $n\times n$ chessboard is called a [i]cycle[/i] if $k\geq 4$ and squares $a_i$ and $a_{i+1}$ have a common side for all $i=1,2,\ldots, k$, where $a_{k+1}=a_1$. Subset $X$ of this chessboard's squares is [i]mischievous[/i] if each cycle on it contains at least one square in $X$. Determine all real numbers $C$ with the following property: for each integer $n\geq 2$, on an $n\times n$ chessboard there exists a mischievous subset consisting of at most $Cn^2$ squares.

During the mathematics Olympiad, students solved three problems. Each task was evaluated with an integer number of points from $0$ to $7$. There is at most one problem for each pair of students, for which they got after the same number of points. Determine the maximum number of students could participate in the Olympics.

Find the distance of the centre of gravity of a uniform $100$-dimensional solid hemisphere of radius $1$ from the centre of the sphere with $10\%$ relative error.