The Problems
♣ Problem #12 (November 14) pdf version
Each of the letters in the sum below stands for a different digit, 0 through 9 (same letters represent the same digit of course). What should each letter represent so that the sum is correct?
♣ Problem #11 (November 7) pdf version
Find all positive integers `m` starting with 7 such that if the first digit of `m` is removed then the resulting number is `m/15`.
♣ Problem #10 (October 31) pdf version
A marker is fixed in the center of a toilet paper roll and the roll placed on
the tray of a marker board. As the toilet paper is unrolled from left to
right, the marker traces a decreasing curve on the board (see the figure
below). Will the curve be concave up, concave down, or a straight line?
♣ Problem #9 (October 24) pdf version
Five chinchillas are lined up in a row. The weight of each one is an integer value, strictly between 300 and 400 grams. For any chinchilla (except the one on the far right), the sum of his weight and half the weight of his neighbor on the right is exactly 584 grams. Find the weights of the chinchillas.
♣ Problem #8 (October 17) pdf version
Find the line with positive slope that is tangent to both of the curves `y = x^2 + 4` and `y = -x^2 + 4x - 8`.
♣ Problem #7 (October 10) pdf version
Four circles, each of radius `1`, are placed with their centers at points `a`, `b`, `c` and `d`. Find the area of the shaded region.
♣ Problem #6 (September 30) pdf version Solution
Tavi has written the number `8^{2005}` in expanded form on the walls of his room (in very tiny numbers). He adds up the digits to get a new number, adds up the digits of that number to get another one, and so on until he ends up with just a single digit.
For example: if Tavi started with `94583` then he would get `29` then `11` then `2`.
What number will Tavi have at the end of his long additions starting with `8^{2005}`?
♣ Problem #5 (September 23) pdf version Solution
Albert and Bertha are playing a game. On the table between them are nine cards, numbered from $1$ to $9$ (face up, so all numbers are visible). They take turns removing one card from the table. The winner is the first player to remove cards summing to $15$.
For example, if Albert removes the $5$ card, Bertha removes the $6$ card, Albert removes the $7$ card and then Bertha removes the $9$ card then Bertha will win (since $9 + 6 = 15$).
If Albert goes first, what move should he make to guarantee that he will win the game (no matter what Bertha does)?
♣ Problem #4 (September 16) pdf version Solution
Find an example of a function `f(x)` that satisfies all of the following conditions simultaneously:
- `f` is differentiable everywhere
- `f'(0) = 1`
- for every non-vertical line `L`, the graph of `y = f(x)` intersects `L` infinitely many times
♣ Problem #3 (September 9) pdf version Solution
You are given twelve hexagonal tiles with lines and arrows drawn on them, as shown below. The tiles may be placed together according to the following rules:
- tiles may be rotated but not flipped,
- tiles must be placed edge-to-edge,
- the tiles must form an unbroken line,
- directions of the arrows on connected lines must match.
Examples:
♣ Problem #2 (September 2) pdf version Solution
Nine squares are placed in a rectangle, as shown below (their proportions are not exact). If the smallest square has an area of four square inches, what is the area of the rectangle?
♣ Problem #1 (August 26) pdf version Solution
Find all solutions $(n,k)$ to the equation $1! + 2! + 3! + \cdots + n! = k^2$ where $n$ and $k$ are positive integers.