Let $m, n$ be positive integers. Show that the polynomial $$f(x)=x^m(x^2-100)^n-11$$ cannot be expressed as a product of two non-constant polynomials with integral coefficients.

Let the $ABCD$ be a quadrilateral without parallel sides, inscribed in a circle. Let $P$ and $Q$ be the intersection points between the lines containing the quadrilateral opposite sides. Show that the bisectors to the angles at $P$ and $Q$ are parallel to the bisectors of the angles at the intersection point of the diagonals of the quadrilateral.

Big Al, the ape, ate 100 bananas from May 1 through May 5. Each day he ate six more bananas than on the previous day. How many bananas did Big Al eat on May 5? $ \textbf{(A)}\ 20\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 34 $

Show that the number \[\sqrt[n]{\sqrt{2019} + \sqrt{2018}} + \sqrt[n]{\sqrt{2019} - \sqrt{2018}}\] is irrational for any $n\ge 2$.

In $\vartriangle ABC$, the incircle is tangent to the sides $BC, CA, AB$ at $D, E, F$ respectively. Let $P$ and $Q$ be the midpoints of $DF$ and $DE$ respectively. Lines $P C$ and $DE$ intersect at $R$, and lines $BQ$ and$ DF$ intersect at $S$. Prove that a) Points $B, C, P, Q$ lie on a circle. b) Points $P, Q, R, S$ lie on a circle.

Each point on the plane is colored either red, green, or blue. Prove that there exists an isosceles triangle whose vertices all have the same color.

Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ in points $A,C,E$ and $B,D,F$ respectively, in that order from left to right. Prove that the length of the projection of the segment $CD$ onto the $x$-axis equals the sum of the lengths of the projections of $AB$ and $EF$.

Let $\omega_{1}$ be a circle with centre $O$. $P$ is a point on $\omega_{1}$. $\omega_{2}$ is a circle with centre $P$, with radius smaller than $\omega_{1}$. $\omega_{1}$ meets $\omega_{2}$ at points $T$ and $Q$. Let $TR$ be a diameter of $\omega_{2}$. Draw another two circles with $RQ$ as the radius, $R$ and $P$ as the centres. These two circles meet at point $M$, with $M$ and $Q$ lie on the same side of $PR$. A circle with centre $M$ and radius $MR$ intersects $\omega_{2}$ at $R$ and $N$. Prove that a circle with centre $T$ and radius $TN$ passes through $O$.

Inside the segment $AC$, an arbitrary point $B$ is selected and circles with diameters $AB$ and $BC$ are constructed. Points $M$ and $L$ are chosen on the circles (in one half-plane with respect to $AC$), respectively, so that $\angle MBA = \angle LBC$. Points $K$ and $F$ are marked, respectively, on rays $BM$ and $BL$ so that $BK = BC$ and $BF = AB$. Prove that points $M, K, F$ and $L$ lie on the same circle.

In the following diagram, $O$ is the center of the circle. If three angles $\alpha, \beta$ and $\gamma$ be equal, find $\alpha.$ [asy] unitsize(40); import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0.2,1); pen ffttww = rgb(1,0.2,0.4); pen qqwuqq = rgb(0,0.39,0); draw(circle((0,0),2.33),ttttff+linewidth(2.8pt)); draw((-1.95,-1.27)--(0.64,2.24),ffttww+linewidth(2pt)); draw((0.64,2.24)--(1.67,-1.63),ffttww+linewidth(2pt)); draw((-1.95,-1.27)--(1.06,0.67),ffttww+linewidth(2pt)); draw((1.67,-1.63)--(-0.6,0.56),ffttww+linewidth(2pt)); draw((-0.6,0.56)--(1.06,0.67),ffttww+linewidth(2pt)); pair parametricplot0_cus(real t){ return (0.6*cos(t)+0.64,0.6*sin(t)+2.24); } draw(graph(parametricplot0_cus,-2.2073069497794027,-1.3111498158746024)--(0.64,2.24)--cycle,qqwuqq); pair parametricplot1_cus(real t){ return (0.6*cos(t)+-0.6,0.6*sin(t)+0.56); } draw(graph(parametricplot1_cus,0.06654165390165974,0.9342857038103908)--(-0.6,0.56)--cycle,qqwuqq); pair parametricplot2_cus(real t){ return (0.6*cos(t)+-0.6,0.6*sin(t)+0.56); } draw(graph(parametricplot2_cus,-0.766242589858673,0.06654165390165967)--(-0.6,0.56)--cycle,qqwuqq); dot((0,0),ds); label("$O$", (-0.2,-0.38), NE*lsf); dot((0.64,2.24),ds); label("$A$", (0.72,2.36), NE*lsf); dot((-1.95,-1.27),ds); label("$B$", (-2.2,-1.58), NE*lsf); dot((1.67,-1.63),ds); label("$C$", (1.78,-1.96), NE*lsf); dot((1.06,0.67),ds); label("$E$", (1.14,0.78), NE*lsf); dot((-0.6,0.56),ds); label("$D$", (-0.92,0.7), NE*lsf); label("$\alpha$", (0.48,1.38),NE*lsf); label("$\beta$", (-0.02,0.94),NE*lsf); label("$\gamma$", (0.04,0.22),NE*lsf); clip((-8.84,-9.24)--(-8.84,8)--(11.64,8)--(11.64,-9.24)--cycle); [/asy]

$a,b,c,x,y,z$ are positive real numbers and $bz+cy=a$, $az+cx=b$, $ay+bx=c$. Find the least value of following function $f(x,y,z)=\frac{x^2}{1+x}+\frac{y^2}{1+y}+\frac{z^2}{1+z}$

Let $\triangle ABC$ be a triangle with $AB=AC$. Let $D$ be a point on the segment $BC$ such that $BD= 48 \frac{1}{61}$ and $DC=61$. Let $E$ be a point on $AD$ such that $CE$ is perpendicular to $AD$ and $DE=11$. Find $AE$.

Evaluate \[\int_0^a \sqrt{2ax-x^2}\ dx \ (a>0)\]

Let $E$ and $F$ be the midpoints of the lines $AB$ and $DA$ of a square $ABCD$, respectively and let $G$ be the intersection of $DE$ with $CF$. Find the aspect ratio of sidelengths of the triangle $EGC$, $| EG | : | GC | : | CE |$.

Let be $f: (0,+\infty)\rightarrow \mathbb{R}$ monoton-decreasing . If $f(2a^2+a+1)<f(3a^2-4a+1)$ find interval of $a$ .

Let $S$ be a set of $n$ points in the plane such that for any two points $(a, b), (c, d) \in S$, we have that $| a - c | \cdot | b - d | \ge 1$. Show that [list] [*] (a) If $S = \{ Q_1, Q_2, Q_3\}$ such that for any point $Q_i$ in $S$, this point doesn't lie in the axis-aligned rectangle with corners as the other two points, show that the area of $\triangle Q_1Q_2Q_3$ is at least $\frac{\sqrt{5}}{2}$. [*] (b) If all points in $S$ lie in an axis-aligned square with sidelength $a$, then $|S| \le \frac{a^2}{\sqrt{5}} + 2a + 1$. [/list]