Find all integers of the form $\frac{m-6n}{m+2n}$ where $m,n$ are nonzero rational numbers satisfying $m^3=(27n^2+1)(m+2n)$.

In triangle $ABC$, let $X$ and $Y$ be the midpoints of $AB$ and $AC$, respectively. On segment $BC$, there is a point $D$, different from its midpoint, such that $\angle{XDY}=\angle{BAC}$. Prove that $AD\perp BC$.

For every $k= 1,2, \ldots$ let $s_k$ be the number of pairs $(x,y)$ satisfying the equation $kx + (k+1)y = 1001 - k$ with $x$, $y$ non-negative integers. Find $s_1 + s_2 + \cdots + s_{200}$.

In triangle $ ABC$ points $ E$ and $ F$ lie on sides $ AC$ and $ BC$ such that segments $ AE$ and $ BF$ have equal length, and circles formed by $ A,C,F$ and by $ B,C,E,$ respectively, intersect at point $ C$ and another point $ D.$ Prove that that the line $ CD$ bisects $ \angle ACB.$

In $1960$ only $5\%$ of the working adults in Carlin City worked at home. By $1970$ the "at-home" work force increased to $8\%$. In $1980$ there were approximately $15\%$ working at home, and in $1990$ there were $30\%$. The graph that best illustrates this is [asy] unitsize(13); draw((0,4)--(0,0)--(7,0)); draw((0,1)--(.2,1)); draw((0,2)--(.2,2)); draw((0,3)--(.2,3)); draw((2,0)--(2,.2)); draw((4,0)--(4,.2)); draw((6,0)--(6,.2)); for (int a = 1; a < 4; ++a) { for (int b = 1; b < 4; ++b) { draw((2*a,b-.1)--(2*a,b+.1)); draw((2*a-.1,b)--(2*a+.1,b)); } } label("1960",(0,0),S); label("1970",(2,0),S); label("1980",(4,0),S); label("1990",(6,0),S); label("10",(0,1),W); label("20",(0,2),W); label("30",(0,3),W); label("$\%$",(0,4),N); draw((12,4)--(12,0)--(19,0)); draw((12,1)--(12.2,1)); draw((12,2)--(12.2,2)); draw((12,3)--(12.2,3)); draw((14,0)--(14,.2)); draw((16,0)--(16,.2)); draw((18,0)--(18,.2)); for (int a = 1; a < 4; ++a) { for (int b = 1; b < 4; ++b) { draw((2*a+12,b-.1)--(2*a+12,b+.1)); draw((2*a+11.9,b)--(2*a+12.1,b)); } } label("1960",(12,0),S); label("1970",(14,0),S); label("1980",(16,0),S); label("1990",(18,0),S); label("10",(12,1),W); label("20",(12,2),W); label("30",(12,3),W); label("$\%$",(12,4),N); draw((0,12)--(0,8)--(7,8)); draw((0,9)--(.2,9)); draw((0,10)--(.2,10)); draw((0,11)--(.2,11)); draw((2,8)--(2,8.2)); draw((4,8)--(4,8.2)); draw((6,8)--(6,8.2)); for (int a = 1; a < 4; ++a) { for (int b = 1; b < 4; ++b) { draw((2*a,b+7.9)--(2*a,b+8.1)); draw((2*a-.1,b+8)--(2*a+.1,b+8)); } } label("1960",(0,8),S); label("1970",(2,8),S); label("1980",(4,8),S); label("1990",(6,8),S); label("10",(0,9),W); label("20",(0,10),W); label("30",(0,11),W); label("$\%$",(0,12),N); draw((12,12)--(12,8)--(19,8)); draw((12,9)--(12.2,9)); draw((12,10)--(12.2,10)); draw((12,11)--(12.2,11)); draw((14,8)--(14,8.2)); draw((16,8)--(16,8.2)); draw((18,8)--(18,8.2)); for (int a = 1; a < 4; ++a) { for (int b = 1; b < 4; ++b) { draw((2*a+12,b+7.9)--(2*a+12,b+8.1)); draw((2*a+11.9,b+8)--(2*a+12.1,b+8)); } } label("1960",(12,8),S); label("1970",(14,8),S); label("1980",(16,8),S); label("1990",(18,8),S); label("10",(12,9),W); label("20",(12,10),W); label("30",(12,11),W); label("$\%$",(12,12),N); draw((24,12)--(24,8)--(31,8)); draw((24,9)--(24.2,9)); draw((24,10)--(24.2,10)); draw((24,11)--(24.2,11)); draw((26,8)--(26,8.2)); draw((28,8)--(28,8.2)); draw((30,8)--(30,8.2)); for (int a = 1; a < 4; ++a) { for (int b = 1; b < 4; ++b) { draw((2*a+24,b+7.9)--(2*a+24,b+8.1)); draw((2*a+23.9,b+8)--(2*a+24.1,b+8)); } } label("1960",(24,8),S); label("1970",(26,8),S); label("1980",(28,8),S); label("1990",(30,8),S); label("10",(24,9),W); label("20",(24,10),W); label("30",(24,11),W); label("$\%$",(24,12),N); draw((0,9)--(2,9.25)--(4,10)--(6,11)); draw((12,8.5)--(14,9)--(16,10)--(18,10.5)); draw((24,8.5)--(26,8.8)--(28,10.5)--(30,11)); draw((0,0.5)--(2,1)--(4,2.8)--(6,3)); draw((12,0.5)--(14,.8)--(16,1.5)--(18,3)); label("(A)",(-1,12),W); label("(B)",(11,12),W); label("(C)",(23,12),W); label("(D)",(-1,4),W); label("(E)",(11,4),W);[/asy]

Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[ f(f(x))+yf(xy+1) = f(x-f(y)) + xf(y)^2. \]for all real numbers $x$ and $y$.

Let $S=\{(x,y,z)\in \mathbb{Z}^3\}$ the set of points with integer coordinates in the space. Gugu has infinitely many solid spheres. All with radii $\ge (\frac{\pi}2)^3$. Is it possible for Gugu to cover all points of $S$ with his spheres?

Eight people are posing together in a straight line for a photo. Alice and Bob must stand next to each other, and Claire and Derek must stand next to each other. How many different ways can the eight people pose for their photo?

Two circles are given for which there is a family of quadrilaterals circumscribed around the first circle and inscribed in the second. Let's denote by $a, b, c$ and $d{}$ the consecutive lengths of the sides of one of these quadrilaterals. Prove that the sum \[\frac{a}{c}+\frac{c}{a}+\frac{b}{d}+\frac{d}{b}\]does not depend on the choice of the quadrilateral. [i]Proposed by I. Weinstein[/i]

Across all polynomials $P$ such that $P(n)$ is an integer for all integers $n$, determine, with proof, all possible values of $P(i)$, where $i^2=-1$.

Solve the system of equations: $\left\{ \begin{aligned} x^4+y^2-xy^3-\frac{9}{8}x = 0 \\ y^4+x^2-yx^3-\frac{9}{8}y=0 \end{aligned} \right.$

The equation $x^3y+xy^3+xy=0$ represents $\textbf{(A)}~\text{a circle}$ $\textbf{(B)}~\text{a circle and a pair of straight lines}$ $\textbf{(C)}~\text{a rectangular hyperbola}$ $\textbf{(D)}~\text{a pair of straight lines}$

For a convex polygon (i.e. all angles less than $180^\circ$) call a diagonal [i]bisector[/i] if its bisects both area and perimeter of the polygon. What is the maximum number of bisector diagonals for a convex pentagon? [i]Proposed by Morteza Saghafian[/i]

$13.$ At a party, each man danced with exactly four women and each woman danced with exactly three men. Nine men attended the party. How many women attended the party $?$

Determine all real numbers $a, b, c, d$ for which $ab+cd=6$ $ac+bd=3$ $ad+bc=2$ $a+b+c+d=6$

Let $ABC$ be an acute-angled triangle and variable $D \in [BC]$ . Let's denote by $E, F$ the feet of the perpendiculars from $D$ to $AB$, $AC$ respectively . a) Show that $$\frac{4S^2}{b^2+c^2}\le DE^2 + DF^2\le max \{h_B^2 + h_C^2 \}.$$ b) Proved that, if $D_0 \in [BC]$ is the point where the minimum of the sum $DE^2 + DF^2$ is achieved, then $D_0$ is the leg of the symmetrical median of $A$ facing the bisector of angle $A$. c) Specify the position, of $D \in [BC]$ for which the maximum of the sum $DE^2 + DF^2$ is achieved. (The area of the triangle $ABC$ was denoted by $S$ and $h_b, h_c$ are the lengths of the altitudes from $B$ and $C$ respectively)

$x$ is a base-$10$ number such that when the digits of $x$ are interpreted as a base-$20$ number, the resulting number is twice the value as when they are interpreted as a base-$13$ number. Find the sum of all possible values of $x$.

Solve in the reals the following system. $$ \left\{ \begin{matrix} \log_2|x|\cdot\log_2|y| =3/2 \\x^2+y^2=12 \end{matrix} \right. $$ [i]Gheorghe Marchitan[/i]

Let $S$ be the set of all ordered triples $\left(a,b,c\right)$ of positive integers such that $\left(b-c\right)^2+\left(c-a\right)^2+\left(a-b\right)^2=2018$ and $a+b+c\leq M$ for some positive integer $M$. Given that $\displaystyle\sum_{\left(a,b,c\right)\in S}a=k$, what is \[\displaystyle\sum_{\left(a,b,c\right)\in S}a\left(a^2-bc\right)\] in terms of $k$? [i]2018 CCA Math Bonanza Lightning Round #4.1[/i]

Let $A$ be a real $n\times n$ matrix satisfying $$A+A^{\text T}=I,$$where $A^{\text T}$ denotes the transpose of $A$ and $I$ the $n\times n$ identity matrix. Show that $\det A>0$.

For a sequence $ \left( x_n \right)_{n\ge 1} $ of real numbers that are at least $ 1, $ prove that the series $ \sum_{i=1}^{\infty } \frac{1}{x_i} $ converges if and only if the series $ \sum_{i=1}^{\infty } \frac{1}{1+x_i} $ converges if and only if the series $ \sum_{i=1}^{\infty } \frac{1}{\lfloor x_i\rfloor } $ converges.

What is the maximum number of acute interior angles a non-overlapping planar $2017$-gon can have?

On a right triangle of paper, two points $A$ and $B$ have been painted. You have scissors and you have the right to make cuts (on paper) as follows: cut through a height of the given triangle. In doing so, remove, without the respective altitude, one of the two triangles and continue the process. Prove that after a finite number of cuts you can separate points $A$ and $B$ leaving one of them outside the remaining triangles.

Six people, with distinct weights, want to form a triangular position where there are three people in the bottom row, two in the middle row, and one in the top row, and each person in the top two rows must weigh less than both of their supports. How many distinct formations are there?

Let $x$, $y$, $z$ be positive real numbers satisfying the simultaneous equations \begin{align*}x(y^2+yz+z^2)&=3y+10z\\y(z^2+zx+x^2)&=21z+24x\\z(x^2+xy+y^2)&=7x+28y.\end{align*} Find $xy+yz+zx$.