What is the remainder when 3²⁰²⁰ is divided by 73?

(a)  

What is the largest integer k such that k + 1 divides

k ²⁰²⁰ + 2k ²⁰¹⁹ + 3k ²⁰¹⁸ + · · · + 2020k + 2021 ?

(a)  

Suppose f is a second-degree polynomial for which f(2) = 1, f(4) = 2, and f(8) = 3. Find the sum of the roots of f.

(a)  

A triangle has side lengths 7, 11, 14. Find the length of its in radius.

(a)  

Suppose that (1 + sec θ)(1 + csc θ) = 6. Determine the value of (1 + tan θ)(1 + cot θ)

(a)  

Determine the number of ordered quadruples (a, b, c, d) of positive integers such that abcd = 216.

(a)  

A 10 × 1 rectangular pavement is to be covered by tiles which are either green or yellow, each of width 1 and of varying integer lengths from 1 to 10. Suppose you have an unlimited supply of tiles for each color and for each of the varying lengths. How many distinct tilings of the rectangle are there, if at least one green and one yellow tile should be used, and adjacent tiles should have different colors?

(a)  

Let P = (3¹ + 1)(3² + 1)(3³ + 1). . .(3²⁰²⁰ + 1). Find the largest value of the integer n such that 2ⁿ divides P.

(a)  

Find the 2020th term of the following sequence:

1, 1, 3, 1, 3, 5, 1, 3, 5, 7, 1, 3, 5, 7, 9, 1, 3, 5, 7, 9, 11, . . .

(a)  

How many infinite arithmetic sequences of positive integers are there which contain the numbers 3 and 39?

(a)  

What is the sum of all four-digit numbers that each use the digits 2, 4, 6, and 8 exactly once?

(a)  

What is the probability that a rectangle with perimeter 36 cm has area greater than 36 cm² ?

(a)  

Let a and b be real numbers that satisfy the equations

 $\frac{a}{b}+\frac{b}{a}=\frac{5}{2}\ \ \ and\ \ a-b=\frac{3}{2}.$  

(a)  

How many permutations of the string “000011112222” contain the substring “2020”?

(a)  

Kyle secretly selects a subset of {1, 2, 3, 4}. Albert also secretly selects a subset of {1, 2, 3, 4}. What is the probability that their chosen subsets have at least one element in common?

(a)  

A 20 × 19 rectangle is plotted on the Cartesian plane with one corner at the origin and with sides parallel to the coordinate axes. How many unit squares do the two diagonals of this rectangle pass through?

(a)  

In a race with six runners, A finished between B and C, B finished between C and D, and D finished between E and F. If each sequence of winners in the race is equally likely to occur, what is the probability that F placed last?

(a)  

Given triangle ABC, let D be a point on side AB and E be a point on side AC. Let F be the intersection of BE and CD. If  $\Delta$  DBF has an area of 4,  $\Delta$  BFC has an area of 6, and  $\Delta$  FCE has an area of 5, find the area of quadrilateral ADFE.

(a)  

If a 3 + b 3 + c 3 = 3abc = 6 and a 2 + b 2 + c 2 = 8, find the value of  $\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}.$  

(a)  

Compute the sum of all possible distinct values of m + n if m and n are positive integers such that lcm(m, n) + gcd(m, n) = 2(m + n) + 11.

(a)  

In convex pentagon ABCDE, AB = BC, CD = DE, ∠ABC = 100^◦ , ∠CDE = 80^◦ , and BD² = 100/sin100^◦ . Find the area of the pentagon.

(a)