$ABCDEF$ is a hexagon inscribed in a circle such that the measure of $\angle{ACE}$ is $90^{\circ}$. What is the average of the measures, in degrees, of $\angle{ABC}$ and $\angle{CDE}$? [i]2018 CCA Math Bonanza Lightning Round #1.3[/i]

For what real values of $k$ do $1988x^2 + kx + 8891$ and $8891x^2 + kx + 1988$ have a common zero?

Pratyya and Payel have a number each, $n$ and $m$ respectively, where $n>m.$ Everyday, Pratyya multiplies his number by $2$ and then subtracts $2$ from it, and Payel multiplies his number by $2$ and then add $2$ to it. In other words, on the first day their numbers will be $(2n-2)$ and $(2m+2)$ respectively. Find minimum integer $x$ with proof such that if $n-m\geq x,$ then Pratyya's number will be larger than Payel's number everyday.

Let $A$ be the set of positive real numbers less than $1$ that have a periodic decimal expansion with a period of ten different digits. Find a positive integer $n$ greater than $1$ and less than $10^{10}$ such that $na-a$ is a positive integer for all $a$. of set $A$.

A graph has $n$ points and $\frac{n(n-1)}{2}$ edges. Each edge is colored with one of $k$ colors so that there are no closed monochrome paths. What is the largest possible value of $n$ (given $k$)?

In triangle $ABC$, points $D$, $E$, and $F$ are on sides $BC$, $CA$, and $AB$ respectively, such that $BF = BD = CD = CE = 5$ and $AE - AF = 3$. Let $I$ be the incenter of $ABC$. The circumcircles of $BFI$ and $CEI$ intersect at $X \neq I$. Find the length of $DX$.

In a school with $101$ students, each student has at least one friend among the other students. Show that for every integer $1<n<101$, a group of $n$ students can be selected from this school in such a way that each selected student has at least one friend among the other selected students.

All the students in an algebra class took a $ 100$-point test. Five students scored $ 100$, each student scored at least $ 60$, and the mean score was $ 76$. What is the smallest possible number of students in the class? $ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ 11 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 13 \qquad \textbf{(E)}\ 14$

Find all finite and non-empty sets $A$ of functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that for all $f_1, f_2 \in A$, there exists $g \in A$ such that for all $x, y \in \mathbb{R}$ $$f_1 \left(f_2 (y)-x\right)+2x=g(x+y)$$

Define $m(n)$ to be the greatest proper natural divisor of $n\in \mathbb{N}$. Find all $n \in \mathbb{N} $ such that $n+m(n) $ is a power of $10$. [i]N. Agakhanov[/i]

Let $f(x,y)$ be a function from ordered pairs of positive integers to real numbers such that \[ f(1,x) = f(x,1) = \frac{1}{x} \quad\text{and}\quad f(x+1,y+1)f(x,y)-f(x,y+1)f(x+1,y) = 1 \] for all ordered pairs of positive integers $(x,y)$. If $f(100,100) = \frac{m}{n}$ for two relatively prime positive integers $m,n$, compute $m+n$. [i]David Yang[/i]

Given real positive a,b , find the larget real c such that $c\leq max(ax+\frac{1}{ax},bx+\frac{1}{bx})$ for all positive ral x. There is a solution here,,,, http://www.kalva.demon.co.uk/irish/soln/sol039.html but im wondering if there is a better one . Thank you.

Two equal hard triangles are given. One of their angles is equal to $ \alpha$ (these angles are marked). Dispose these triangles on the plane in such a way that the angle formed by some three vertices would be equal to $ \alpha / 2$. [i](No instruments are allowed, even a pencil.)[/i] (E. Bakayev, A. Zaslavsky)

Let $ \alpha$ be the positive root of the equation $ x^{2} \equal{} 1991x \plus{} 1$. For natural numbers $ m$ and $ n$ define \[ m*n \equal{} mn \plus{} \lfloor\alpha m \rfloor \lfloor \alpha n\rfloor. \] Prove that for all natural numbers $ p$, $ q$, and $ r$, \[ (p*q)*r \equal{} p*(q*r). \]

Octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle where $A_1A_2=A_2A_3=A_3A_4=A_6A_7=13$ and $A_4A_5=A_5A_6=A_7A_8=A_8A_1=5. $ The sum of all possible areas of $A_1A_2A_3A_4A_5A_6A_7A_8$ can be expressed as $a+b\sqrt{c}$ where $\gcd{a,b}=1$ and $c$ is squarefree. Find $abc.$ [asy] label("$A_1$",(5,0),E); label("$A_2$",(2.92, -4.05),SE); label("$A_3$",(-2.92,-4.05),SW); label("$A_4$",(-5,0),W); label("$A_5$",(-4.5,2.179),NW); label("$A_6$",(-3,4), NW); label("$A_7$",(3,4), NE); label("$A_8$",(4.5,2.179),NE); draw((5,0)--(2.9289,-4.05235)); draw((2.92898,-4.05325)--(-2.92,-4.05)); draw((-2.92,-4.05)--(-5,0)); draw((-5,0)--(-4.5, 2.179)); draw((-4.5, 2.179)--(-3,4)); draw((-3,4)--(3,4)); draw((3,4)--(4.5,2.179)); draw((4.5,2.179)--(5,0)); dot((0,0)); draw(circle((0,0),5)); [/asy]

Suppose $ABCD$ is a cyclic quadrilateral with $\angle ABC = \angle ADC = 90^{\circ}$. Let $E$ and $F$ be the feet of perpendiculars from $A$ and $C$ to $BD$ respectively. Prove that $BE = DF$.

[b]2.[/b] Let $S$ be a given finite set of hyperplanes in $\mathbb{R}^n$, and let $O$ be a point. Show that there exists a compact set $K \subseteq \mathbb{R}^n$ containing $O$ such that the orthogonal projection of any point of $K$ onto any hyperplane in $S$ is also in $K$. ([b]G.37[/b]) [Gy. Pap]

Figure $ABCD$ is a trapezoid with $AB || DC, AB = 5, BC = 3 \sqrt 2, \measuredangle BCD = 45^\circ$, and $\measuredangle CDA = 60^\circ$. The length of $DC$ is $\textbf{(A) }7 + \frac{2}{3} \sqrt{3}\qquad \textbf{(B) }8\qquad \textbf{(C) }9 \frac{1}{2}\qquad \textbf{(D) }8 + \sqrt 3\qquad \textbf{(E) }8 + 3 \sqrt 3$

Suppose a function $ f: \mathbb R\to \mathbb R$ satisfies $ f(f(x)) \equal{} \minus{} x$ for all $ x\in \mathbb R$. Prove that $ f$ has infinitely many points of discontinuity.

How many ordered triples $(x, y, z)$ of positive integers satisfy $\text{lcm}(x, y) = 72$, $\text{lcm}(x, z)= 600$, and $\text{lcm}(y, z) = 900$? $\textbf{(A) } 15 \qquad\textbf{(B) } 16 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 27 \qquad\textbf{(E) } 64$

It is given $4$ circles in a plane and every one of them touches the other three as shown: [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC82L2FkYWQ5NThhMWRiMjAwZjYxOWFhYmE1M2YzZDU5YWI2N2IyYzk2LnBuZw==&rn=a3J1Z292aS5wbmc=[/img] Biggest circle has radius $2$, and every one of the medium has $1$. Find out the radius of fourth circle.

There is a ruler and a "rusty" compass, with which you can construct a circle of radius $R$. The point $K$ is from the line $\ell$ at a distance greater than $R$. How to use this ruler and this compass to draw a line passing through the point $K$ and perpendicular to line $\ell$? (Misha Sidorenko, Katya Sidorenko, Rodion Osokin)

Let $ABC$ be a triangle . A circle through $B$ and $C$ crosses sides $AB$ and $AC$ at $P$ and $Q$, respectively. Points $X$ and $Y$ on segments $BQ$ and $CP$, respectively, satisfy $\angle ABY=\angle AXP$ and $ACX=\angle AYQ$. Prove that $XY$ and $BC$ are parallel.

The following number is the product of the divisors of $n$. $$46, 656, 000, 000$$ What is $n$?

If the inequality \[ ((x+y)^2+4)((x+y)^2-2)\geq A\cdot (x-y)^2 \] is hold for every real numbers $x,y$ such that $xy=1$, what is the largest value of $A$? $ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 14 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 20 $