Two mirror walls are placed to form an angle of measure $\alpha$. There is a candle inside the angle. How many reflections of the candle can an observer see?

Two players play the following game on a square of $N \times N$ squares. They color one square in turn so that no two colored squares are on the same diagonal. A player who cannot make a move loses. For what values of $N$ does the first player have a winning strategy?

Let $f(x)=ax^2+bx+c$ be a quadratic function, the graph of which doesn't intersect the x-axis. Prove that $$a(2a+3b+6c)>0.$$

Let $a,b$, be two real numbers. If for any $x>0$ holds that $|a-b|<x$, then prove that $a=b$.

a) Prove that a convex $n$-gon cannot have more than $3$ interior angles acute. b) Prove that a convex $n$-gon that has $3$ interior angles equal to $60^0,$ is equilateral.

A bug is standing at each of the vertices of a regular hexagon $ABCDEF$. At the same time each bug picks one of the vertices of the hexagon, which it is not currently in, and immediately starts moving towards that vertex. Each bug travels in a straight line from the vertex it was in originally to the vertex it picked. All bugs travel at the same speed and are of negligible size. Once a bug arrives at a vertex it picked, it stays there. In how many ways can the bugs move to the vertices so that no two bugs are ever in the same spot at the same time?

Assuming $a,b,c$ are the side-lengths of a triangle, show that \begin{align*} \frac{a^2+b^2-c^2}{ab} + \frac{b^2+c^2-a^2}{bc} + \frac{c^2+a^2-b^2}{ca} > 2. \end{align*} Also show that the inequality does not necessarily hold if you replace $2$ (on the right-hand side) by a bigger by a bigger number.

Doug and Dave shared a pizza with $ 8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $ \$8$, and there was an additional cost of $ \$2$ for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug? $ \textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$

The integers $a$ and $b$ have the property that for every nonnegative integer $n$ the number of $2^n{a}+b$ is the square of an integer. Show that $a=0$.

One hundred empty glasses are arranged in a $10 \times 10$ array. Now we pick $a$ of the rows and pour blue liquid into all glasses in these rows, so that they are half full. The remaining rows are filled halfway with yellow liquid. Afterwards, we pick $b$ of the columns and fill them up with blue liquid. The remaining columns are filled up with yellow liquid. The mixture of blue and yellow liquid turns green. If both halves have the same colour, then that colour remains as it is. [list=a] [*] Determine all possible combinations of values for $a$ and $b$ so that exactly half of the glasses contain green liquid at the end. [*] Is it possible that precisely one quarter of the glasses contain green liquid at the end? [/list]

The points lie on three parallel lines with distances as indicated in the figure $A, B$ and $C$ such that square $ABCD$ is a square. Find the area of this square. [img]https://1.bp.blogspot.com/-xeFvahqPVyM/XzcFfB0-NfI/AAAAAAAAMYA/SV2XU59uBpo_K99ZBY43KSSOKe-veOdFQCLcBGAsYHQ/s0/1998%2BMohr%2Bp3.png[/img]

Alice and Bob play a game in which they take turns choosing integers from 1 to $n$. Before any integers are chosen, Bob selects a goal of "odd" or "even". On the first turn, Alice chooses one of the $n$ integers. On the second turn, Bob chooses one of the remaining integers. They continue alternately choosing one of the integers that has not yet been chosen, until the $n$th turn, which is forced and ends the game. Bob wins if the parity of $\{k$ : the number $k$ was chosen on the $k$th turn $\}$ matches his goal. For which values of $n$ does Bob have a winning strategy?

Find all positive integers $m,n$ such that $m^2+5n$ and $n^2+5m$ are perfect squares.

Determine all triples of real numbers $(a,b,c)$ such that \begin{eqnarray*} xyz &=& 8 \\ x^2y + y^2z + z^2x &=& 73 \\ x(y-z)^2 + y(z-x)^2 + z(x-y)^2 &=& 98 . \end{eqnarray*}

Prove that among any $6$ irrational numbers you can pick three numbers $a$, $b$ and $c$ such that numbers $a+b$, $b+c$ and $c+a$ are irrational

Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\Omega$ such that $AB<CD$. Suppose that $AC$ meets $BD$ at $E$, $AD$ meets $BC$ at $F$, and $\Omega$ meets $(FAE)$, $(FBE)$ at $X$, $Y$, respectively. Prove that if $XY$ is diameter of $\Omega$, then $XY$ is perpendicular to $EF$.

Rolly wishes to secure his dog with an 8-foot rope to a square shed that is 16 feet on each side. His preliminary drawings are shown. Which of these arrangements gives the dog the greater area to roam, and by how many square feet? [asy]defaultpen(linewidth(0.7)); size(7cm); D((0,0)--(16,0)--(16,-16)--(0,-16)--cycle, black); D((16,-8)--(24,-8), black); label('Dog', (24, -8), SE); label('I', (8,-8), (0,0)); MP('8', (16,-4), W); MP('8', (16,-12),W); MP('8', (20,-8), N); label('Rope', (20,-8),S); D((0,-20)--(16,-20)--(16,-36)--(0,-36)--cycle, black); D((16,-24)--(24,-24), black); label("II", (8,-28), (0,0)); MP('4', (16,-22), W); MP('8', (20,-24), N); label("Dog",(24,-24),SE); label("Rope", (20,-24), S); dot((24,-24)^^(24,-8));[/asy] $ \textbf{(A)}\text{ I, by }8\pi\qquad\textbf{(B)}\text{ I, by }6\pi\qquad\textbf{(C)}\text{ II, by }4\pi\qquad\textbf{(D) }\text{II, by }8\pi\qquad\textbf{(E)}\text{ II, by }10\pi $

Each point $A$ in the plane is assigned a real number $f(A).$ It is known that $f(M)=f(A)+f(B)+f(C),$ whenever $M$ is the centroid of $\triangle ABC.$ Prove that $f(A)=0$ for all points $A.$

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

Suppose $Z, Y$ , and $W$ are points on a circle such that lengths $ZY = Y W$. Extend $ZY$ and let $X$ be a point on $ZY$ where $ZY = Y X$. If $XW$ is a tangent of the circle, what is $\angle W XY$ ?

To make three kinds of products $A,B,C$, we have three parts $a,b,c$. A product $A$ is made of two$a$ and two $b$; a product $B$ is made of one $b$ and one $c$; a product $C$ is made of two $a$ and one $c$. We have a few parts. If we make $p$ product$A$, $q$ product$B$, $r$ product$C$, then $2$ part $a$ and $1$ part $b$ are remained. Prove: no matter how we make products, we cannot use up all the parts.

(a) Let $z_1,z_2$ be complex numbers such that $z_1z_2=1$ and $|z_1-z_2|=2$. Let $A,B,M_1,M_2$ denote the points in complex plane corresponding to $-1,1,z_1,z_2$, respectively. Show that $AM_1BM_2$ is a trapezoid and compute the lengths of its non-parallel sides. Specify the particular cases. (b) Let $\mathcal C_1$ and $\mathcal C_2$ be circles in the plane with centers $O_1$ and $O_2$, respectively, and with radius $d\sqrt2$, where $2d=O_1O_2$. Let $P$ and $Q$ be two variable points on $\mathcal C_1$ and $\mathcal C_2$ respectively, both on $O_1O_2$ on on different sides of $O_1O_2$, such that $PQ=2d$. Prove that the locus of midpoints $I$ of segments $PQ$ is the same as the locus of points $M$ with $MO_1\cdot MO_2=m$ for some $m$.

Let $\triangle ABC$ be a nondegenerate isosceles triangle with $AB=AC$, and let $D, E, F$ be the midpoints of $BC, CA, AB$ respectively. $BE$ intersects the circumcircle of $\triangle ABC$ again at $G$, and $H$ is the midpoint of minor arc $BC$. $CF\cap DG=I, BI\cap AC=J$. Prove that $\angle BJH=\angle ADG$ if and only if $\angle BID=\angle GBC$. [i]Proposed by David Stoner[/i]

Have $b, c \in R$ satisfy $b \in (0, 1)$ and $c > 0$, then let $A,B$ denote the points of intersection of the line $y = bx+c$ with $y = |x|$, and let $O$ denote the origin of $R^2$. Let $f(b, c)$ denote the area of triangle $\vartriangle OAB$. Let $k_0 = \frac{1}{2022}$ , and for $n \ge 1$ let $k_n = k^2_{n-1}$. If the sum $\sum^{\infty}_{n=1}f(k_n, k_{n-1})$ can be written as $\frac{p}{q}$ for relatively prime positive integers $p, q$, find the remainder when $p+q$ is divided by 1000.

Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$. [i]Proposed by Lewis Chen[/i]