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 .
A fair coin is tossed repeatedly until either heads comes up three times in a row or tails comes up three times in a row. What is the probability that the coin will be tossed more than times? Express your answer as a common fraction.
Consider the equilateral triangle with sides of length cm. A point in the interior of is said to be “special” if it is a distance of cm from one side of the triangle and a distance of cm from another side. Consider the convex polygon whose vertices consist of the special points. What is the area of this polygon? Express your answer as a decimal to the nearest tenth.
The three coin denominations used in Coinistan have values 7 cents, 12 cents and 23 cents. Fareed and Krzysztof notice that their two Coinistan coin collections have the same number of coins and the same total value, but not the same number of 7-cent coins. What is the smallest possible value of Krzysztof’s collection?
How many four-digit integers , with , have the property that the three two-digit integers form an increasing arithmetic sequence? One such number is , where and .
Four lines are drawn through the figure shown (see problem statement in video). What is the maximum number of non-overlapping regions created inside the figure?