Team Selection Test

# Team Selection Test ## Easy Problems (1 point each) 1. A rectangular pool has diagonal $17$ $units$ and area $120$ $units^2$. What is the perimeter of the pool? 2. What is the least positive integer $n$ such that $2020!$ is not a multiple of $7^n$? 3. For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral? * a square * a rectangle that is not a square * a rhombus that is not a square * a parallelogram that is not a rectangle or a rhombus * an isosceles trapezoid that is not a parallelogram 4. Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2:1$ ? ## Medium Problems (2 points each) 1. If $x^2 + y^2 = 47xy$, then $log(k(x+y)) = \frac{1}{2}(log(x)+log(y))$. Find the value of $k$. 2. What are the last two digits of $2002^{2002^{2002}}$ 3. On December 9, 2004, Tracy McGrady scored 13 points in 33 seconds to beat the San Antonio Spurs. Given that McGrady never misses and that each shot made counts for 2, 3, or 4 points, how many shot sequences could McGrady have taken to achieve such a feat assuming that order matters? 4. Consider the set of all fractions $\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by $1$, the value of the fraction is increased by $10\%$? ## Difficult Problems (4 points each) 1. Let $S$ be the set of positive integer divisors of $20^9$. Three numbers are chosen independently and at random with replacement from the set S and labeled $a_1$, $a_2$,and $a_3$ in the order they are chosen. The probability that both $a_1$ is a multiple of $a_2$ and $a_2$ is a multiple of $a_3$ is $\frac{m}{n}$, where m and n are relatively prime positive integers. Find m. 2. Three spheres are centered at the vertices of a triangle in the horizontal plane and are tangent to each other. The triangle formed by the uppermost points of the spheres has side lengths $10$, $26$, and $2\sqrt{145}$. What is the area of the triangle whose vertices are at the centers of the spheres?