2021 AIME I #4

June 11, 2022 mathproblemsolvingskillsLeave a comment

Find the number of ways $66$ 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.

2016 AIME I #15

June 3, 2022 mathproblemsolvingskillsLeave a comment

Circles $\omega_1$ and $\omega_2$ intersect at points $X$ and $Y$ . Line $l$ is tangent to $\omega_1$ and $\omega_2$ at $A$ and $B$ , respectively, with line $AB$ closer to point $X$ than to $Y$ . Circle $\omega$ passes through $A$ and $B$ intersecting $\omega_1$ again at $D \neq A$ and intersecting $\omega_2$ again at $C \neq B$ . The three points $C, Y,$ and $D$ are collinear, $XC = 67, XY = 47,$ and $XD = 37$ . Find $AB^2$ .

2016 AIME I #14

May 26, 2022 mathproblemsolvingskillsLeave a comment

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.

2016 AIME I #13

May 20, 2022 mathproblemsolvingskillsLeave a comment

This one is the classic jumping frog with state diagrams, this time in the coordinate plane between a fence and river.

2016 AIME I #12

May 13, 2022 mathproblemsolvingskillsLeave a comment

Find the least positive integer $m$ such that $m^2-m+11$ is a product of at least four not necessarily distinct primes.

2016 AIME I #10

May 7, 2022 mathproblemsolvingskillsLeave a comment

A strictly increasing sequence of positive integers $a_1, a_2, a_3, \dots$ has the property that for every positive integer $k$ , the subsequence $a_{2k-1}, a_{2k}, a_{2k+1}$ is geometric and the subsequence $a_{2k}, a_{2k+1}, a_{2k+2}$ is arithmetic. Suppose that $a_{13} = 2016$ . Find $a_1$ .

2022 MathCounts State Team Round #9

May 3, 2022 mathproblemsolvingskillsLeave a comment

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 $10$ times? Express your answer as a common fraction.

2022 MathCounts State Team Round #10

April 30, 2022 mathproblemsolvingskillsLeave a comment

Consider the equilateral triangle $ABC$ with sides of length $8 \sqrt{3}$ cm. A point in the interior of $ABC$ is said to be “special” if it is a distance of $3$ cm from one side of the triangle and a distance of $7$ 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.

2021 MathCounts National Sprint #28 (60 second solve!)

April 21, 2022 mathproblemsolvingskillsLeave a comment

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?

2016 AMC 10 B #24 (video solution)

April 17, 2022 mathproblemsolvingskillsLeave a comment

How many four-digit integers $abcd$ , with $a \neq 0$ , have the property that the three two-digit integers $ab < bc < cd$ form an increasing arithmetic sequence? One such number is $4692$ , where $a=4, b=6, c=9,$ and $d=2$ .