It is possible to cover a $6\times 6$ grid using one L-shaped piece made of 3 grids and eleven $3\times 1$ smaller grid ?
A chocolate bar is made up of a rectangular $m\times n$ grid of small squares. Two players take turns breaking up the bar. On a given turn, a player picks a rectangular piece of chocolate and breaks it into two smaller rectangular pieces, by snapping along one whole line of subdivisions between its squares. The player who makes the last break wins. Does one of the players have a winning strategy for this game?
Two players, $A$ and $B$, take turns naming positive integers, with $A$ playing first. No player may name an integer that can be expressed as a linear combination, with positive integer coefficients, of previously named integers. The player who names 1 loses. Show that no matter how A and B play, the game will always end.
Alan and Barbara play a game in which they take turns filling entries of an initially empty $1024$ by $1024$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all the entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?
A regiment had 48 soldiers but only half of them had uniforms. During inspection, they form a 6 × 8 rectangle, and it was just enough to conceal in its interior everyone without a uniform. Later, some new soldiers joined the regiment, but again only half of them had uniforms. During the next inspection, they used a different rectangular formation, again just enough to conceal in its interior everyone without a uniform. How many new soldiers joined the regiment?
Each of the numbers 2, 3, 4, 5, 6, 7, 8 and 9 is used once to fill in one of the boxes in the equation below to make it correct. Of the three fractions being added, what is the value of the largest one?
Use each of the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9 exactly once to fill in the nine small circles in the Olympic symbol below, so that the numbers inside each large circle is 14.
There are 14 points of intersection in the seven-pointed star in the diagram. Label these points with the numbers
1, 2, 3, $\dots$, 14 such that the sum of the labels of the four points on each line is the same. Give one of solution.
Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?
Let $k$ be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with $(k+1)$ digits. Every time Bernardo writes a number, Silvia erases the last $k$ digits of it. Bernardo then writes the next perfect square, Silvia erases the last $k$ digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let $f(k)$ be the smallest positive integer not written on the board. For example, if $k = 1$, then the numbers that Bernardo writes are $16, 25, 36, 49, 64$, and the numbers showing on the board after Silvia erases are $1, 2, 3, 4,$ and $6$, and thus $f(1) = 5$. What is the sum of the digits of $f(2) + f(4)+ f(6) + ... + f(2016)$?
Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number $S$. What is the smallest possible value for the sum of the digits of $S$?
All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3\times3$ array of squares, one number in each square, in such a way that if two numbers of consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18$. What is the number in the center?
Josh writes the numbers $1,2,3,\dots,99,100$. He marks out $1$, skips the next number $(2)$, marks out $3$, and continues skipping and marking out the next number to the end of the list. Then he goes back to the start of his list, marks out the first remaining number $(2)$, skips the next number $(4)$, marks out $6$, skips $8$, marks out $10$, and so on to the end. Josh continues in this manner until only one number remains. What is that number?
There are twelve different mixed numbers that can be created by substituting three of the numbers $1$, $2$, $3$ and $5$ for $a$, $b$ and $c$ in the expression $a\frac{b}{c}$ , where $b < c$. What is the mean of these twelve mixed numbers? Express your answer as a mixed number.
In the list of numbers $1, 2, \cdots, 9999$, the digits $0$ through $9$ are replaced with the letters $A$ through $J$, respectively. For example, the number $501$ is replaced by the string $FAB$ and $8243$ is replaced by the string $ICED$. The resulting list of $9999$ strings is sorted alphabetically. How many strings appear before $CHAI$ in this list?
Billy wrote a sequence of five numbers on the board, each an integer between $0$ and $4$, inclusive. Penny then wrote a sequence of five numbers that measured some statistics about Billy’s sequence. In particular, Penny first wrote down the number of $0$s in Billy’s sequence. Then Penny wrote the number of $1$s in Billy’s sequence, and then the number of $2$s, the number of $3$s, and finally the number of $4$s. It turned out that Penny’s sequence was exactly the same as Billy’s! What was this sequence? Express your answer as an ordered $5$-tuple.
A cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are colored red and $6$ are colored white. If the $3$-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white?
Three members of the Euclid Middle School girls' softball team had the following conversation.
Ashley: I just realized that our uniform numbers are all $2$-digit primes.
Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month.
Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month.
Ashley: And the sum of your two uniform numbers is today's date.
What number does Caitlin wear?
$\textbf{Prisoners' Problem}$
One hundred prisoners will be lined up. Each one will be assigned either a red hat or a blue hat. No one can see the color of his or her own hat. However, each person is able to see the color of the hat worn by every person in front of him or her. That is, for example, the last person in line can see the colors of the hats on $99$ people in front of him or her; and the first person, who is at the front of the line, cannot see the color of any hat.
Beginning with the last person in line, and then moving to the $99^{th}$ person, the $98^{th}$, etc., each will be asked to name the color of his or her own hat. He or she can only answer "red" or "blue". If the color is correctly named, the person lives; if not, the person is shot dead on the spot. Everyone in line is able to hear all the responses but not the gunshots.
Before being lined up, the $100$ prisoners are allowed to discuss a strategy aiming to save as many of them as possible. How many people can be saved if they can agree on a good strategy?
As a more challenging question, what if the hats can have $100$ known different colors instead of $2$?
Take a list of positive integers $1$, $2$, $3$, $\cdots$, $2017$. At each step, pick up two of the numbers on the list, say $a$ and $b$, cross them out and replace them by the single number $(ab+a+b)$. Keep doing this until only a single number is left. What is (are) the possible value(s) of this last number?
Let $a$ and $b$ be five-digit palindromes (without leading zeroes) such that $a < b$ and there are no other
five-digit palindromes strictly between $a$ and $b$. What are all possible values of $b - a$? (A number is a
palindrome if it reads the same forwards and backwards in base $10$.)
How many ways are there to insert $+$’s between the digits of $111111111111111$ (fifteen $1$’s) so that the
result will be a multiple of $30$?
Find the smallest square which can cover $n$ congruent equilateral triangles so that these triangles do not overlap.
$\textbf{Coin Flipping}$
There are $9$ coins on the table, all heads up. In each operation, you can flip any two of them. Is it possible to make all of them heads down after a series of operations? If yes, please list a series of such operations. If no, please explain.
$\textbf{Medalists}$
Five runners, $A$, $B$, $C$, $D$, and $E$, enter the final. The fastest three win a gold, silver, and bronze medal, respectively. The other two get nothing. Who are the three medalists if all of the following statements are false?
- $A$ does not win the gold and $B$ does not get the silver.
- $B$ does not get the bronze and $D$ does not win silver.
- $C$ wins a medal, but $D$ does not.
- $A$ wins a medal, but $C$ does not.
- Both $D$ and $E$ win a medal.