How far is it from the point $(9, 17)$ to its reflection across the line \[3x + 4y = 15?\]

Find the largest constant $K>0$ such that for any $0\le k\le K$ and non-negative reals $a,b,c$ satisfying $a^2+b^2+c^2+kabc=k+3$ we have $a+b+c\le 3$. (Dan Schwarz)

A “large” circular disk is attached to a vertical wall. It rotates clockwise with one revolution per minute. An insect lands on the disk and immediately starts to climb vertically upward with constant speed $\frac{\pi}{3}$ cm per second (relative to the disk). Describe the path of the insect [i](a)[/i] relative to the disk; [i](b)[/i] relative to the wall.

How many real roots does the polynomial $x^5 + x^4 - x^3 - x^2 - 2x - 2$ have? $ \textbf{a)}\ 1 \qquad\textbf{b)}\ 2 \qquad\textbf{c)}\ 3 \qquad\textbf{d)}\ 4 \qquad\textbf{e)}\ \text{None of above} $

On each cell of a $2005\times2005$ chessboard, there is a marker. In each move, we are allowed to remove a marker that is a neighbor to an even number of markers (but at least one). Two markers are considered neighboring if their cells share a vertex. (a) Find the least number $n$ of markers that we can end up with on the chessboard. (b) If we end up with this minimum number $n$ of markers, prove that no two of them will be neighboring.

In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.

$(GBR 5)$ Let us define $u_0 = 0, u_1 = 1$ and for $n\ge 0, u_{n+2} = au_{n+1}+bu_n, a$ and $b$ being positive integers. Express $u_n$ as a polynomial in $a$ and $b.$ Prove the result. Given that $b$ is prime, prove that $b$ divides $a(u_b -1).$

A river flows at speed u. A boat has speed v relative to the water. If its velocity is at an angle $\alpha$ relative the direction of the river, what is its speed relative to the river bank? What $\alpha$ minimises the time taken to cross the river?

For positive integer $n,$ let $a_n=\sum_{k=0}^{[\frac{n}{2}]}\binom{n-2}{k}(-\frac{1}{4})^k.$ Find $a_{1997}.$ (For real $x,$ $[x]$ is defined as largest integer that does not exceeds $x.$)

Equilateral triangles $AMB,BNC,CKA$ are constructed on the exterior of a triangle $ABC$. The perpendiculars from the midpoints of $MN, NK, KM$ to the respective lines $CA, AB, BC$ are constructed. Prove that these three perpendiculars pass through a single point.

Let $ C$ be a circle with center $ M$ in a plane $ V$, and $ P$ be a point not on the circle $ C$. $ (a)$ If $ P$ is fixed, prove that $ AP^2\plus{}BP^2$ is a constant for every diameter $ AB$ of the circle $ C$. $ (b)$ Let $ AB$ be a fixed diameter of $ C$ and $ P$ a point on a fixed sphere $ S$ not intersecting $ V$. Determine the points $ P$ on $ S$ that minimize $ AP^2\plus{}BP^2$.

A spaceborne energy storage device consists of two equal masses connected by a tether and rotating about their center of mass. Additional energy is stored by reeling in the tether; no external forces are applied. Initially the device has kinetic energy $E$ and rotates at angular velocity $\omega$. Energy is added until the device rotates at angular velocity $2\omega$. What is the new kinetic energy of the device? (a) $\sqrt{2}E$ (b) $2E$ (c) $2\sqrt{2}E$ (d) $4E$ (e) $8E$

Given an integer $n\ge 2$, show that there exist two integers $a,b$ which satisfy the following. For all integer $m$, $m^3+am+b$ is not a multiple of $n$.

A right circular cone pointing downward forms an angle of $60^\circ$ at its vertex. Sphere $S$ with radius $1$ is set into the cone so that it is tangent to the side of the cone. Three congruent spheres are placed in the cone on top of S so that they are all tangent to each other, to sphere $S$, and to the side of the cone. The radius of these congruent spheres can be written as $\tfrac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime. Find $a + b + c$. [asy] size(150); real t=0.12; void ball(pair x, real r, real h, bool ww=true) { pair xx=yscale(t)*x+(0,h); path P=circle(xx,r); unfill(P); draw(P); if(ww) draw(ellipse(xx-(0,r/2),0.85*r,t*r)); } pair X=(0,0); real H=17, h=5, R=h/2; draw(H*dir(120)--(0,0)--H*dir(60)); draw(ellipse((0,0.87*H),H/2,t*H/2)); pair Y=(R,h+2*R),C=(0,h); real r; for(int k=0;k<20;++k) { r=-(dir(30)*Y).x; Y-=(sqrt(3)/2*Y.x-r,abs(Y-C)-R-r)/3; } ball(Y.x*dir(90),r,Y.y,false); ball(X,R,h); ball(Y.x*dir(-30),r,Y.y); ball(Y.x*dir(210),r,Y.y);[/asy]

Let $D$ be an interior point of the triangle $ABC$. $CD$ and $AB$ intersect at $D_{c}$, $BD$ and $AC$ intersect at $D_{b}$, $AD$ and $BC$ intersect at $D_{a}$. Prove that there exists a triangle $KLM$ with orthocenter $H$ and the feet of altitudes $H_{k}\in LM, H_{l}\in KM, H_{m}\in KL$, so that $(AD_{c}D) = (KH_{m}H)$ $(BD_{c}D) = (LH_{m}H)$ $(BD_{a}D) = (LH_{k}H)$ $(CD_{a}D) = (MH_{k}H)$ $(CD_{b}D) = (MH_{l}H)$ $(AD_{b}D) = (KH_{l}H)$ where $(PQR)$ denotes the area of the triangle $PQR$

Show that the number $ 7^{100}-3^{100} $ has $ 85 $ digits and find its last $ 4 $ ones.

The inscribed sphere of a given tetrahedron touches all four faces of the tetrahedron at their respective centroids. Prove that the tetrahedron is regular.

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard: [list] [*]In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin. [*]In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are two (not necessarily distinct) numbers from the first line. [*]In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line. [/list] Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.

Let $A_1B_1C_1D_1A_2B_2C_2D_2$ be a unit cube, with $A_1B_1C_1D_1$ and $A_2B_2C_2D_2$ opposite square faces, and let $M$ be the center of face $A_2 B_2 C_2 D_2$. Rectangular pyramid $MA_1B_1C_1D_1$ is cut out of the cube. If the surface area of the remaining solid can be expressed in the form $a + \sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime, find $a+b$. [i]Author: Alex Zhu[/i]

Let $n$ be an integer greater than or equal to $2$. Consider real numbers $a_1, a_2, \dots, a_{2n}$ satisfying the condition \[ |a_k - a_{n+k}| \geqslant 1 \quad \text{for all } 1 \leqslant k \leqslant n. \] Determine the minimum possible value of \[ (a_1 - a_2)^2 + (a_2 - a_3)^2 + \dots + (a_{2n-1} - a_{2n})^2 + (a_{2n} - a_1)^2. \]

On a \(4 \times 4\) board, each cell is colored either black or white such that each row and each column have an even number of black cells. How many ways can the board be colored?

If you pick a random $3$-digit number, what is the probability that its hundreds digit is triple the ones digit?

A set of lines in the plan is called [i]nice [/i]i f every line in the set intersects an odd number of other lines in the set. Determine the smallest integer $k \ge 0$ having the following property: for each $2018$ distinct lines $\ell_1, \ell_2, ..., \ell_{2018}$ in the plane, there exist lines $\ell_{2018+1},\ell_{2018+2}, . . . , \ell_{2018+k}$ such that the lines $\ell_1, \ell_2, ..., \ell_{2018+k}$ are distinct and form a [i]nice [/i] set.

The circle $\omega$ with center at point $I$ inscribed in an triangle $ABC$ ($AB\neq AC$) touches the sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$, respectively. The circumcircles of triangles $ABC$ and $AEF$ intersect secondary at point $K.$ The lines $EF$ and $AK$ intersect at point $X$ and intersects the line $BC$ at points $Y$ and $Z$, respectively. The tangent lines to $\omega$, other than $BC$, passing through points $Y$ and $Z$ touch $\omega$ at points $P$ and $Q$, respectively. Let the lines $AP$ and $KQ$ intersect at the point $R$. Prove that if $M$ is a midpoint of segment $YZ,$ then $IR\perp XM$.