Suppose that $p$ is an odd prime such that $2p+1$ is also prime. Show that the equation $x^{p}+2y^{p}+5z^{p}=0$ has no solutions in integers other than $(0,0,0)$.

An escalator (moving staircase) of $ n$ uniform steps visible at all times descends at constant speed. Two boys, $ A$ and $ Z$, walk down the escalator steadily as it moves, $ A$ negotiating twice as many escalator steps per minute as $ Z$. $ A$ reaches the bottom after taking $ 27$ steps while $ Z$ reaches the bottom after taking $ 18$ steps. Then $ n$ is: $ \textbf{(A)}\ 63 \qquad \textbf{(B)}\ 54 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 30$

Jane has two bags $X$ and $Y$. Bag $X$ contains 4 red marbles and 5 blue marbles (and nothing else). Bag $Y$ contains 7 red marbles and 6 blue marbles (and nothing else). Jane will choose one of her bags at random (each bag being equally likely). From her chosen bag, she will then select one of the marbles at random (each marble in that bag being equally likely). What is the probability that she will select a red marble?

An equilateral triangle $ A_1 A_2 A_3$ is divided into four smaller equilateral triangles by joining the midpoints $ A_4,A_5,A_6$ of its sides. Let $ A_7,...,A_{15}$ be the midpoints of the sides of these smaller triangles. The $ 15$ points $ A_1,...,A_{15}$ are colored either green or blue. Show that with any such colouring there are always three mutually equidistant points $ A_i,A_j,A_k$ having the same color.

Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed 37th and 64th, respectively. How many schools are in the city? $ \textbf{(A)}\ 22\qquad\textbf{(B)}\ 23\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 25\qquad\textbf{(E)}\ 26$

The $U$-tile is made up of $1 \times 1$ squares and has the following shape: [img]https://cdn.artofproblemsolving.com/attachments/8/7/5795ee33444055794119a99e675ef977add483.png[/img] where there are two vertical rows of a squares, one horizontal row of $b$ squares, and also $a \ge 2$ and $b \ge 3$. Notice that there are different types of tile $U$ . For example, some types of $U$ tiles are as follows: [img]https://cdn.artofproblemsolving.com/attachments/0/3/ca340686403739ffbbbb578d73af76e81a630e.png[/img] Prove that for each integer $n \ge 6$, the board of $n\times n$ can be completely covered with $U$-tiles , with no gaps and no overlapping clicks. Clarifications: The $U$-tiles can be rotated. Any amount can be used in the covering of tiles of each type.

Choose arbitrarily $n$ vertices of a regular $2n-$gon and colour them red. The remaining vertices are coloured blue. We arrange all red-red distances into a nondecreasing sequence and do the same with the blue-blue distances. Prove that the two sequences thus obtained are identical.

If $2$ is a solution (root) of $x^3+hx+10=0$, then $h$ equals: $ \textbf{(A) }10\qquad\textbf{(B) }9 \qquad\textbf{(C) }2\qquad\textbf{(D) }-2\qquad\textbf{(E) }-9 $

For some $k > 0$ the lines $50x + ky = 1240$ and $ky = 8x + 544$ intersect at right angles at the point $(m,n)$. Find $m + n$.

Let $ABC$ be a triangle in which $\angle A = 60^\circ$. Let $BE$ and $CF$ be the bisectors of $\angle B$ and $\angle C$ with $E$ on $AC$ and $F$ on $AB$. Let $M$ be the reflection of $A$ in line $EF$. Prove that $M$ lies on $BC$.

Let $P(x) = x^4 + ax^3 + bx^2 + x$ be a polynomial with four distinct roots that lie on a circle in the complex plane. Prove that $ab\ne 9$.

Let $A$ be an Abelian group with $n$ elements. Prove that there are two subgroups in $\text{GL}(n,\Bbb{C})$, isomorphic to $S_n$, whose intersection is isomorphic to the automorphism group of $A$. [i]Proposed by Zoltán Halasi[/i]

Let $ABCD$ be a parallelogram. Let $G,H$ be the feet of the altitudes from $A$ to $CD$ and $BC$ respectively. If $AD=15$, $AG=12$, and $AH=16$, find the length of $AB$. [i]2019 CCA Math Bonanza Lightning Round #2.4[/i]

$P$ and $Q$ are points on angle bisectors of two adjacent angles. Let $PA$, $PB$, $QC$ and $QD$ be altitudes on the sides of these adjacent angles. Prove that lines $AB$, $CD$ and $PQ$ are concurrent.

Suppose that $(i - 1)^{11}$ is a root of the quadratic $x^2 + Ax + B$ for integers $A$ and $B$, where $i =\sqrt{-1}$. Compute the value of $A + B$.

If every altitude of a tetrahedron is at least $1$, show that the shortest distance between each pair of opposite edges is more than $2$.

a) The real number $a$ and the continuous function $f : [a, \infty) \rightarrow [a, \infty)$ are such that $|f(x)-f(y)| < |x–y|$ for every two different $x, y \in [a, \infty)$. Is it always true that the equation $f(x)=x$ has only one solution in the interval $[a, \infty)$? b) The real numbers $a$ and $b$ and the continuous function $f : [a, b] \rightarrow [a, b]$ are such that $|f(x)-f(y)| < |x–y|$, for every two different $x, y \in [a, b]$. Is it always true that the equation $f(x)=x$ has only one solution in the interval $[a, b]$?

In triangle $ABC$, $M$ is the midpoint of side $BC$, the bisector of angle $BAC$ intersects $BC$ and $(ABC)$ at $K$ and $L$, respectively. If the circle with diameter $[BC]$ is tangent to the external angle bisector of angle $BAC$, prove that this circle is tangent to $(KLM)$ as well.

Plot points $A,B,C$ at coordinates $(0, 0)$, $(0, 1)$, and $(1,1)$ in the plane, respectively. Let $S$ denote the union of the two line segments $AB$ and $BC$. Let $X_1$ be the area swept out when Bobby rotates $S$ counterclockwise $45$ degrees about point $A$. Let $X_2$ be the area swept out when Calvin rotates $S$ clockwise $45$ degrees about point $A$. Find $\frac{X_1+X_2}{2}$ .

A house is in the shape of a triangle, perimeter $P$ metres and area $A$ square metres. The garden consists of all the land within 5 metres of the house. How much land do the garden and house together occupy?

Let $P$ be an interior point of a circle $\Gamma$ centered at $O$ where $P \ne O$. Let $A$ and $B$ be distinct points on $\Gamma$. Lines $AP$ and $BP$ meet $\Gamma$ again at $C$ and $D$, respectively. Let $S$ be any interior point on line segment $PC$. The circumcircle of $\vartriangle ABS$ intersects line segment $PD$ at $T$. The line through $S$ perpendicular to $AC$ intersects $\Gamma$ at $U$ and $V$ . The line through $T$ perpendicular to $BD$ intersects $\Gamma$ at $X$ and $Y$ . Let $M$ and $N$ be the midpoints of $UV$ and $XY$ , respectively. Let $AM$ and $BN$ meet at $Q$. Suppose that $AB$ is not parallel to $CD$. Show that $P, Q$, and $O$ are collinear if and only if $S$ is the midpoint of $PC$.

Let $n\geqslant 2$ be an integer. Prove that for any positive real numbers $a_1, a_2,\ldots, a_n$, \[\frac{1}{2\sqrt{2}}\sum_{i=1}^{n}2^{i}a_i^2 \geqslant\sum_{1 \leqslant i < j \leqslant n}a_i a_j.\][i]Proposed by Andrei Vila[/i]

For positive rational $x$, if $x$ is written in the form $p/q$ with $p, q$ positive relatively prime integers, define $f(x)=p+q$. For example, $f(1)=2$. a) Prove that if $f(x)=f(mx/n)$ for rational $x$ and positive integers $m, n$, then $f(x)$ divides $|m-n|$. b) Let $n$ be a positive integer. If all $x$ which satisfy $f(x)=f(2^nx)$ also satisfy $f(x)=2^n-1$, find all possible values of $n$. [i]Anderson Wang.[/i]

Let $x_0, x_1, \ldots, x_{n-1}, x_n=x_0$ be reals and let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. The numbers $y_i$ for $i=0,1, \ldots, n-1$ are chosen such that $y_i$ is between $x_i$ and $x_{i+1}$. Prove that $\sum_{i=0}^{n-1}(x_{i+1}-x_i)f(y_i)$ can attain both positive and negative values, by varying the $y_i$.

Compute the largest prime factor of $3^{12}+3^9+3^5+1.$