Let $n\ge  2$ be a positive integer and $p $ a prime such that $n|p-1$ and $p | n^3-1$. Show $ 4p-3$ is a square.

(a) Is $2005^{2004}$ the sum of two perfect squares? (b) Is $2004^{2005}$ the sum of two perfect squares?

Consider the following operation on positive real numbers written on a blackboard: Choose a number $ r$ written on the blackboard, erase that number, and then write a pair of positive real numbers $ a$ and $ b$ satisfying the condition $ 2 r^2 \equal{} ab$ on the board. Assume that you start out with just one positive real number $ r$ on the blackboard, and apply this operation $ k^2 \minus{} 1$ times to end up with $ k^2$ positive real numbers, not necessarily distinct. Show that there exists a number on the board which does not exceed kr.

In rectangular coordinate system, give two points $M(-1,2),N(1,4)$, $P$ is a moving point on $x$-axis, when $\angle MPN$ takes its maximum value, the $x$-axis of $P$ is________.

In a trapezoid $ABCD$, the internal bisector of angle $A$ intersects the base $BC$(or its extension) at the point $E$. Inscribed in the triangle $ABE$ is a circle touching the side $AB$ at $M$ and side $BE$ at the point $P$. Find the angle $DAE$ in degrees, if $AB:MP=2$.

Consider the rectangular parallelepiped $ABCDA'B'C'D' $ as such the measure of the dihedral angle formed by the planes $(A'BD)$ and $(C'BD)$ is $90^o$ and the measure of the dihedral angle formed by the planes $(AB'C)$ and $(D'B'C)$ is $60^o$. Determine and measure the dihedral angle formed by the planes $(BC'D)$ and $(A'C'D)$.

A triangle $ABC$ is inscribed in a circle $S$. Let $A_0$ and $C_0$ be the midpoints of the arcs $BC$ and $AB$ on $S$, not containing the opposite vertex, respectively. The circle $S_1$ centered at $A_0$ is tangent to $BC$, and the circle $S_2$ centered at $C_0$ is tangent to $AB$. Prove that the incenter $I$ of $\triangle ABC$ lies on a common tangent to $S_1$ and $S_2$.

What is the sum of all distinct values of $x$ that satisfy $x^4-x^3-7x^2+13x-6=0$? [i]2018 CCA Math Bonanza Lightning Round #1.4[/i]

There are $10{}$ apples, each with a with a weight which is no more than $100{}$ g. There is a weighing scale with two plates which shows the difference between the weights on the plates. Prove that 1) It is possible to put some (more than one) apples on the plates of the scale such that the difference between the weights on the plates will be less than $1$ g. 2) It is possible to put an equal amount (more than one) of apples on each plate of the scale such that the difference between the weights on the plates will be less than $2$ g.

Find all natural numbers $k$ for which there exists a set $M$ of ten real numbers such that there are exactly $k$ pairwise non-congruent triangles whose side lengths are three (not necessarily distinct) elements of $M$.

Let $ABC$ be an acute scalene triangle inscribed in circle $\Omega$. Circle $\omega$, centered at $O$, passes through $B$ and $C$ and intersects sides $AB$ and $AC$ at $E$ and $D$, respectively. Point $P$ lies on major arc $BAC$ of $\Omega$. Prove that lines $BD, CE, OP$ are concurrent if and only if triangles $PBD$ and $PCE$ have the same incenter.

In $\triangle ABC$ shown in the figure, $AB=7$, $BC=8$, $CA=9$, and $\overline{AH}$ is an altitude. Points $D$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $\overline{BD}$ and $\overline{CE}$ are angle bisectors, intersecting $\overline{AH}$ at $Q$ and $P$, respectively. What is $PQ$? [asy]draw((0,0)--(7,0)); draw((0,0)--(33/7,7.66651)); draw((33/7,7.66651)--(7,0)); draw((11/5,7*7.66651/15)--(7,0)); draw((63/17,0)--(33/7,7.66651)); draw((0,0)--(45/7,7.66651/4)); dot((0,0)); label("A",(0,0),SW); dot((7,0)); label("B",(7,0),SE); dot((33/7,7.66651)); label("C",(33/7,7.66651),N); dot((11/5,7*7.66651/15)); label("D",(11/5+.2,7*7.66651/15-.25),S); dot((63/17,0)); label("E",(63/17,0),NE); dot((45/7,7.66651/4)); label("H",(44/7,7.66651/4),NW); dot((27/7,3*7.66651/20)); label("P",(27/7,3*7.66651/20),NW); dot((5,7*7.66651/36)); label("Q",(5,7*7.66651/36),N); label("9",(33/14,7.66651/2),NW); label("8",(41/7,7.66651/2),NE); label("7",(3.5,0),S);[/asy] $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ \frac{5}{8}\sqrt{3} \qquad \textbf{(C)}\ \frac{4}{5}\sqrt{2} \qquad \textbf{(D)}\ \frac{8}{15}\sqrt{5} \qquad \textbf{(E)}\ \frac{6}{5}$

Prove that the arithmetic mean $a$ of $x_1,\ldots ,x_n$ satisfies \[ (x_1-a)^2+\ldots +(x_n-a)^2\le \frac{1}{2}(|x_1-a|+\ldots +|x_n-a|)^2\]

Suppose that integers are given $m <n $. Consider a spreadsheet of size $n \times n $, whose cells arbitrarily record all integers from $1 $ to ${{n} ^ {2}} $. Each row of the table is colored in yellow $m$ the largest elements. Similarly, the blue colors the $m$ of the largest elements in each column. Find the smallest number of cells that are colored yellow and blue at a time

There are 2018 points marked on a sphere. A zebra wants to paint each point white or black and, perhaps, connect some pairs of points of different colors with a segment. Find the residue modulo 5 of the number of ways to do this.

Cosider ellipse $\epsilon$ with two foci $A$ and $B$ such that the lengths of it's major axis and minor axis are $2a$ and $2b$ respectively. From a point $T$ outside of the ellipse, we draw two tangent lines $TP$ and $TQ$ to the ellipse $\epsilon$. Prove that \[\frac{TP}{TQ}\ge \frac{b}{a}.\] [i]Proposed by Morteza Saghafian[/i]