Problems

If $y+4 = (x-2)^2, x+4 = (y-2)^2$ , and $x \neq y$ , what is the value of $x^2+y^2$ ?

A line that passes through the origin intersects both the line $x=1$ and the line $y=1+\frac{\sqrt{3}}{3}x$ . The three lines create an equilateral triangle. What is the perimeter of the triangle?

The isosceles right triangle $ABC$ has right angle at $C$ and area $12.5$ . The rays trisecting $\angle ACB$ intersect $AB$ at $D$ and $E$ . What is the area of $\bigtriangleup CDE$ ?

Erin the ant starts at a given corner of a cube and crawls along exactly $7$ edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point. How many paths are there meeting these conditions?

In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$ . What is $FG+JH+CD$ ?

In rectangle $ABCD$ , $AB=1$ , $BC=2$ , and points $E$ , $F$ , and $G$ are midpoints of $\overline{BC}$ , $\overline{CD}$ , and $\overline{AD}$ , respectively. Point $H$ is the midpoint of $\overline{GE}$ . What is the area of the shaded region?

Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?

Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\triangle ABC$ ?

Two concentric circles have radii $1$ and $2$ . Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?

Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?

The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is bad if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$ . Arrangements that differ only by a rotation or a reflection are considered the same. How …

In a small pond there are eleven lily pads in a row labeled $0$ through $10$ . A frog is sitting on pad $1$ . When the frog is on pad $N$ , $0<N<10$ , it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$ . Each jump is independent of the previous jumps. If the …

A triangle with vertices $(6, 5)$ , $(8, -3)$ , and $(9, 1)$ is reflected about the line $x=8$ to create a second triangle. What is the area of the union of the two triangles?

A unit square is rotated $45^\circ$ about its center. What is the area of the region swept out by the interior of the square?

All 20 diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect?

The sums of three whole numbers taken in pairs are $12$ , $17$ , and $19$ . What is the middle number?

The closed curve in the figure is made up of $9$ congruent circular arcs each of length $\frac{2\pi}{3}$ , where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side $2$ . What is the area enclosed by the curve?

A $3 \times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90\,^{\circ}$ clockwise about its center, and every white square in a position formerly occupied by a black …

Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?

Real numbers $x$ , $y$ , and $z$ are chosen independently and at random from the interval $[0,n]$ for some positive integer $n$ . The probability that no two of $x$ , $y$ , and $z$ are within 1 unit of each other is greater than $\frac {1}{2}$ . What is the smallest possible value of $n$ ?

What is the sum of all integer solutions to $1<(x-2)^2<25$ ?

Two equilateral triangles are contained in a square whose side length is $2\sqrt 3$ . The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. What is the area of the rhombus?

A bug travels from A to B along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there?

Square $EFGH$ has one vertex on each side of square $ABCD$ . Point $E$ is on $\overline{AB}$ with $AE=7\cdot EB$ . What is the ratio of the area of $EFGH$ to the area of $ABCD$ ? {{}}

Two points on the circumference of a circle of radius $r$ are selected independently and at random. From each point a chord of length $r$ is drawn in a clockwise direction. What is the probability that the two chords intersect?