Find the least non-negative residue of $70! \pmod{5183}$.

Show that it is impossible to cover an $8\times 8$ square using fifteen $4\times 1$ rectangles and one $2\times 2$ square.

It is possible to cover a $6\times 6$ grid using one L-shaped piece made of 3 grids and eleven $3\times 1$ smaller grid ?

Two players, $A$ and $B$, take turns naming positive integers, with $A$ playing first. No player may name an integer that can be expressed as a linear combination, with positive integer coefficients, of previously named integers. The player who names 1 loses. Show that no matter how A and B play, the game will always end.

$\textbf{Heaps of Beans}$

A game starts with four heaps of beans, containing $3$, $4$, $5$ and $6$ beans, respectively. The two players move alternately. A move consists of taking either one bean from a heap, provided at least two beans are left behind in that heap, or a complete heap of two or three beans. The player who takes the last bean wins. Does the first or second player have a winning strategy?

A positive integer $n$ is said to be good if there exists a perfect square whose sum of digits in base $10$ is equal to $n$. For instance, $13$ is good because $7^2 = 49$ and $4 + 9 = 13$. How many good numbers are among $1, 2, 3, \cdots , 2007$?

Prove that the sum $$\sqrt{1001^2 + 1} + \sqrt{1002^2 + 1} + \cdots + \sqrt{2000^2 + 1}$$ is irrational.

Let $n$ be an even positive integer, and let $p(x)$ be an $n$-degree polynomial such that $p(-k) = p(k)$ for $k = 1, 2, \dots , n$. Prove that there is a polynomial $q(x)$ such that $p(x) = q(x^2)$.

Let $a, b, c$ be distinct integers. Can the polynomial $(x - a)(x - b)(x - c) - 1$ be factored into the product of two polynomials with integer coefficients?

Let $P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ be a polynomial with integral coefficients. Suppose that there exist four distinct integers $a, b, c, d$ with $P(a) = P(b) = P(c) = P(d) = 5$. Prove that there is no integer $k$ satisfying $P(k) = 8$.

Show that $(1 + x + \cdots + x^n)^2 - x^n$ is the product of two polynomials.

Determine all polynomials such that $P(0) = 0$ and $P(x^2 + 1) = P(x)^2 + 1$.

Find the largest integer not exceeding $1 + \frac{1}{\sqrt{2}}+ \frac{1}{\sqrt{3}} + \cdots + + \frac{1}{\sqrt{10000}}$

Let $n$ be an integer greater than or equal to 3. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.

A highly valued secretive recipe is kept in a safe with multiple locks. Keys to these locks are distributed among 9 members of the managing committee. The established rule requires that at least 6 members must present in order to open the safe. What is the minimal number of locks required? Correspondingly, how many keys are required?

Let $a_n=\binom{200}{n}(\sqrt[3]{6})^{200-n}\left(\frac{1}{\sqrt{2}}\right)^n$, where $n=1$, $2$, $\cdots$, $95$. Find the number of integer terms in $\{a_n\}$.

Let sequence $\{a_n\}$ satisfy the condition: $a_1=\frac{\pi}{6}$ and $a_{n+1}=\arctan(\sec a_n)$, where $n\in Z^+$. There exists a positive integer $m$ such that $\sin{a_1}\cdot\sin{a_2}\cdots\sin{a_m}=\frac{1}{100}$. Find $m$.

As shown in diagram below, find the degree measure of $\angle{ADB}$.

An infinite number of equilateral triangles are constructed as shown on the right. Each inner triangle is inscribed in its immediate outsider and is shifted by a constant angle $\beta$. If the area of the biggest triangle equals to the sum of areas of all the other triangles, find the value of $\beta$ in terms of degrees.

For $-1 < r < 1$, let $S(r)$ denote the sum of the geometric series $$12+12r+12r^2+12r^3+\cdots .$$Let $a$ between $-1$ and $1$ satisfy $S(a)S(-a)=2016$. Find $S(a)+S(-a)$.

Find the least positive integer $m$ such that $m^2 - m + 11$ is a product of at least four not necessarily distinct primes.

(Weitzenbock's Inequality) Let $a, b, c$, and $S$ be a triangle's three sides' lengths and its area, respectively. Show that $$a^2 + b^2 + c^2 \ge 4\sqrt{3}\cdot S$$

Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another.

In $\triangle{ABC}$, $AE$ and $AF$ trisects $\angle{A}$, $BF$ and $BD$ trisects $\angle{B}$, $CD$ and $CE$ trisects $\angle{C}$. Show that $\triangle{DEF}$ is equilateral.

As shown, in $\triangle{ABC}$, $AB=AC$, $\angle{A} = 20^\circ$, $\angle{ABE} = 30^\circ$, and $\angle{ACD}=20^\circ$. Find the measurement of $\angle{CDE}$.

PolynomialAndEquation
Root
AlgebraFundamentalTheorem
RationalRootTheorem
CeilingAndFloor
BasicSequence
LinearRecursion
SpecialSequence
FunctionProperty
PlaneGeometry
Triangle
CevaTheorem
BinomialExpansion
NumberTheoryBasic
SquareNumber
MOD
EulerFermatTheorem
CRT
Trigonometry
TrigInTriangle
LawOfSines
LawOfCosines
TrigTransformation
Inequality
Induction
ProofByContradiction
ColoringMethod
LogicalAndReasoning
TwoStateProblem
Modeling
Invariant
BrainTeaser
ConstructionMethod