Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$. Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$. [i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$. Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.

Let $D$ and $E$ be points on sides $AB$ and $AC$ of a triangle $ABC$ such that $DB = BC = CE$. The segments $BE$ and $CD$ intersect at point $P$. Prove that the incenter of triangle $ABC$ lies on the circles circumscribed around the triangles $BDP$ and $CEP$.

There are $m$ pieces of candy held in $n$ trays($n,m\ge 4$). An [i]operation[/i] is defined as follow: take out one piece of candy from any two trays respectively, then put them in a third tray. Determine, with proof, if we can move all candies to a single tray by finite [i]operations[/i].

Let $T$ be a finite set of squarefree integers. (a) Show that there exists an integer polynomial $P(x)$ such that the set of squarefree numbers in the range of $P(n)$ across all $n \in \mathbb{Z}$ is exactly $T$. (b) Suppose that $T$ is allowed to be infinite. Is it still true that for all choices of $T$, such an integer polynomial $P(x)$ exists? [i]Allen Wang[/i]

Point Sets $A=\{(x,y)|(x-3)^2+(y-4)^2\leq\left( \frac{5}{2}\right)^2\},B=\{(x,y)|(x-4)^2+(y-5)^2>\left( \frac{5}{2}\right)^2\}$, then the number of integral points in $A\cap B$ is________.

Find the integer $n$ such that $6!\times7!=n!$. [i]2017 CCA Math Bonanza Individual Round #1[/i]

Prove that there exists a positive integer $ n$ such that for every $ x\ge0$ the inequality $ (x\minus{}1)(x^{2005}\minus{}2005x^{n\plus{}1}\plus{}2005x^n\minus{}1)\ge0$ holds.

Find the limit $$\lim_{n\to\infty}\left(\prod_{k=1}^n\frac k{k+n}\right)^{e^{\frac{1999}n}-1}.$$

Suppose that $ a + b = 20$, $b + c = 22$, and $c + a = 2022$. Compute $\frac{a-b}{c-a}$ .

Solve the following equation in $\mathbb{Z}$: \[3^{2a + 1}b^2 + 1 = 2^c\]

Let $p$ be an odd prime number and $M$ a set derived from $\frac{p^2 + 1}{2}$ square numbers. Investigate whether $p$ elements can be selected from this set whose arithmetic mean is an integer. (Walther Janous)

Prove that for every positive integer $n$ coprime to $10$ there exists a multiple of $n$ that does not contain the digit $1$ in its decimal representation.

Find all positive integer pairs $(m, n)$ such that $m- n$ is a positive prime number and $mn$ is a perfect square. Justify your answer.

Paul can write polynomial $(x+1)^n$, expand and simplify it, and after that change every coefficient by its reciprocal. For example if $n=3$ Paul gets $(x+1)^3=x^3+3x^2+3x+1$ and then $x^3+\frac13x^2+\frac13x+1$. Prove that Paul can choose $n$ for which the sum of Paul’s polynomial coefficients is less than $2.022$.

Prove that if the product of $n$ real and positive numbers is equal to $1$, its sum is greater than or equal to $n$.

In obtuse triangle $ABC$ with $AB = 7$, $BC = 20$, and $C A = 15$, let point $D$ be the foot of the altitude from $C$ to line $AB$. Evaluate $[ACD]+[BCD]$. (Note that $[XY Z]$ means the area of triangle $XY Z$.) [i]Proposed by Jonathan Liu[/i]

Let $(F_n)_{n\in\mathbb{N}}$ be the Fibonacci sequence difined as $F_0=F_1=1, F_{n+2}=F_{n+1}+F_n, \forall n\in\mathbb{N}$. Show that for every nonnegative integer $r$ there is a term in the Fibonacci sequence that is divided by $r$.

Suppose that we have $n$ events $A_1,\dots, A_n,$ each of which has probability at least $1-a$ of occuring, where $a<1/4.$ Further suppose that $A_i$ and $A_j$ are mutually independent if $|i-j|>1.$ Assume as known that the recurrence $u_{k+1}=u_k-au_{k-1}, u_0=1, u_1=1-a,$ defines positive real numb $u_k$ for $k=0,1,\dots.$ Show that the probability of all of $A_1,\dots, A_n$ occuring is at least $u_n.$

What is the radius of the circle passing through the center of the square $ABCD$ with side length $1$, its corner $A$, and midpoint of its side $[BC]$? $ \textbf{(A)}\ \dfrac {\sqrt 3}4 \qquad\textbf{(B)}\ \dfrac {\sqrt 5}4 \qquad\textbf{(C)}\ \sqrt 2 \qquad\textbf{(D)}\ \sqrt 3 \qquad\textbf{(E)}\ \dfrac {\sqrt {10}}4 $

Consider a flat field on which there exist a valley in the form of an infinite strip with arbitrary width $\omega$. There exist a polyhedron of diameter $d$(Diameter in a polyhedron is the maximum distance from the points on the polyhedron) is in one side and a pit of diameter $d$ on the other side of the valley. We want to roll the polyhedron and put it into the pit such that the polyhedron and the field always meet each other in one point at least while rolling (If the polyhedron and the field meet each other in one point at least then the polyhedron would not fall into the valley). For crossing over the bridge, we have built a rectangular bridge with a width of $\frac{d}{10}$ over the bridge. Prove that we can always put the polyhedron into the pit considering the mentioned conditions. (You will earn a good score if you prove the decision for $\omega = 0$).

Let $ABCD$ be a convex quadrilateral such that $\angle A, \angle B, \angle C$ are acute. $AB$ and $CD$ meet at $E$ and $BC,DA$ meet at $F$. Let $K,L,M,N$ be the midpoints of $AB,BC,CD,DA$ repectively. $KM$ meets $BC,DA$ at $X$ and $Y$, and $LN$ meets $AB,CD$ at $Z$ and $W$. Prove that the line passing $E$ and the midpoint of $ZW$ is parallel to the line passing $F$ and the midpoint of $XY$.

Let $\vartriangle ABC$ be an equilateral triangle with side length $1$ and let $\Gamma$ the circle tangent to $AB$ and $AC$ at $B$ and $C$, respectively. Let $P$ be on side $AB$ and $Q$ be on side $AC$ so that $PQ // BC$, and the circle through $A, P$, and $Q$ is tangent to $\Gamma$ . If the area of $\vartriangle APQ$ can be written in the form $\frac{\sqrt{a}}{b}$ for positive integers $a$ and $b$, where $a$ is not divisible by the square of any prime, find $a + b$.

Prove that for each positive integer $n$ there are at most two pairs $(a, b)$ of positive integers with following two properties: (i) $a^2 + b = n$, (ii) $a+b$ is a power of two, i.e. there is an integer $k \ge 0$ such that $a+b = 2^k$.

A wooden cube has edges of length $3$ meters. Square holes, of side one meter, centered in each face are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. The entire surface area including the inside, in square meters, is $\textbf {(A) } 54 \qquad \textbf {(B) } 72 \qquad \textbf {(C) } 76 \qquad \textbf {(D) } 84\qquad \textbf {(E) } 86$

Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$. Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$