The Void Cube is a $3\times3\times3$ cube with no center pieces just holes which can be looked through, as shown in the diagram. Consider the cross-section of the cube passing through the vertices $A,B$ and $C$. Draw a picture of this cross section. (Shade the solid part of the triangle $ABC$ in black and the hollow part in white.)
There are 314 coins in 21 open boxes. In each move, you can take 1 coin from each of any two boxes and put them into a third box. In the final move, you take all the coins from one box. What is the maximum number of coins you can get?
A right triangle with hypotenuse $a$ stands on its shorter side $c=1$. We rotate it sequentially three times as it is shown below. The diagram labels $a>b>c$.
If $TS$ is a common tangent of the three arcs, find $a$.
Answer format: enter an exact expression; sqrt(n) denotes a square root.
Let $P(x)$ be a polynomial of degree 3 and $x_1,x_2,x_3$ are the solutions of $P(x)=0$. Let
$$\frac{P(1/3)-P(-1/3)}{P(0)}=8,\qquad\frac{P(1/4)-P(-1/4)}{P(0)}=9$$
and $x_1+x_2+x_3=35$. Find the value of
$$\frac{x_2+x_3}{x_1}+\frac{x_1+x_3}{x_2}+\frac{x_1+x_2}{x_3}.$$
The quadrilateral $ABCD$ is inscribed in a circle with center $O$. Connect $AC$ and $BD$ intersecting at $K$. $O_1$ is the circumcenter of triangle $ABK$ and $O_2$ is the circumcenter of triangle $CDK$. A line $l$ through $K$ intersect the two circumcircles at $E$ and $F$ respectively, and the circumcircle of $ABCD$ at $G$ and point $H$. Prove that $EG=FH$.
Suppose $x,y$ and $z$ are all non-negative real numbers. Let $k$ and $m$ be the minimum possible values of $\frac{x^2+y^2+z^2+1}{xy+yz+z}$ and $\frac{x^2+y^2+z^2+1}{xy+y+z}$ respectively. Find $km+k+m$.
Use three non-overlapping copies of a L-tetromino to construct a symmetric figure. Each L-tetromino must have a common point with at least one other L-tetromino.
Find a solution in which the copies may be rotated but not reflected.
How many ways are there of placing a single L-pentomino on the $8\times8$ chessboard so that it completely covers some five small squares of the chessboard?
