Barbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbara’s turn, she must remove 2 or 4 coins, unless only one coin remains, in which case she loses her turn. When it is Jenna’s turn, she must remove 1 or 3 coins. A coin flip determines who goes first. Whoever removes the last coin wins the game. Assume both players use their best strategy. Who will win when the game starts with 2013 coins and when the game starts with 2014 coins?
(Note: I have the wrong problem number written in the video whiteboard.)
Find the number of integers such that the equation has distinct solutions.
Find the number of pairs of positive integers with such that there exists a real number satisfying .
Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences.
Find the number of ways identical coins can be separated into three nonempty piles so that there are fewer coins in the first pile than in the second pile and fewer coins in the second pile than the third pile.
Circles and intersect at points and . Line is tangent to and at and , respectively, with line closer to point than to . Circle passes through and intersecting again at and intersecting again at . The three points and are collinear, and . Find .
Centered at each lattice point in the coordinate plane are a circle with radius 1/10 and a square with sides of length 1/5 whose sides are parallel to the coordinate axes. The line segment from (0,0) and (1001, 429) intersects m of the squares and n of the circles. Find m+n.
This one is the classic jumping frog with state diagrams, this time in the coordinate plane between a fence and river.
Find the least positive integer such that is a product of at least four not necessarily distinct primes.
A strictly increasing sequence of positive integers has the property that for every positive integer , the subsequence is geometric and the subsequence is arithmetic. Suppose that . Find .