Arithmetic
Let K be the median value in the set of odd composite numbers less than 32. Let J be the smallest odd composite number greater than 10(23)+32. Compute the number of factors of P= K*J
12
A change purse contains only dimes and quarters worth a total of $3.80. Given that there are 20 coins in the purse, compute the number of quarters in the purse.
12
Let N = AB +CD be a positive integer where A, B, C, and D are distinct integers chosen from the set {−1, 2, −3, 4}. Compute the minimum possible value of N.
10
Given that x2 + 2x + 4 = 0, compute the value of x6
64
Sam is driving at 46 feet per second. The diameter of each of Sam’s wheels is 23 inches. Compute, to the nearest whole number, the number of minutes it takes a wheel to turn through 4800 revolutions.
10
Let the set S = {3, 5, 7, 11, 13, 16, 17, 23, 25, 29}. Compute the least number of distinct numbers in S that add to 50.
3
Brady’s age is B years old. Brady has three children, and their ages add to B years old. Also, N years ago, Brady’s age was three times the sum of his children’s ages then. Compute B/N .
4
For positive values of x, compute:
(2x+1*3x+2*6x+3)/(9x+1*4x-1)
1728
Factor 9x2 + 4y2 − 9z2 − 12xy into the product of two trinomials with integer coefficients. For each polynomial, let the coefficient of x be positive
(3x − 2y + 3z)(3x − 2y − 3z)
In right triangle BUG, cos G = 2/5 . Compute sin G + tan G.
7(√21)/10
When written as a decimal, 20/19 = 1.052631578947368421... with (052631578947368421) repeating after the decimal point. Compute the sum of the first 2019 digits after the decimal point in the decimal expansion of 20/19 .
9079
Jenna and Jimmy always run their training runs along the same course, starting at the same point and ending at the same point. Jimmy runs 2 mph faster than Jenna. If Jimmy starts 30 minutes after Jenna, they will both reach the finish line at the same time, 2 hours after Jimmy starts. Compute the length of the training run in miles.
20
Compute √ ( [(√3+√15)2/2 ]* [1 − (√5)/3])
2
Express a2 − ac − 4b2 − 2bc as the product of two polynomials with integer coefficients. For each polynomial, let the coefficient of a be 1.
(a − 2b − c)(a + 2b)
A car is traveling at 5π meters per second. The wheel radius is 330 millimeters. If the car travels for 22 minutes, compute the number of revolutions the wheel makes in that time.
10,000
How many natural numbers between 244 and 870 are divisible by 7 or 11?
139
Three real numbers multiply to 16848. The second number is three times the first. The third number is three more than the second number. Compute the sum of the three numbers.
87
Compute the value of (20192020 + 2019−2020)2 − (20192020 − 2019−2020)2
4
Factor as the product of two trinomials each of whose leading coefficients is 1: x3 + ax2 + 7x2 + 3ax + 13x + a + 4.
(x2 + 3x + 1)(x + 4 + a)
Suppose that triangle GBM is drawn with m∠G = 30◦ , m∠B = 105◦ , and BM = 6√2. Compute the area of triangle GBM
18 + 18√3
The number 20202020 + 20212021 + 2020 + 2021 is written as a decimal number. Compute the remainder when this decimal number is divided by 100.
62
PJ is thinking of four positive integers a, b, c, and d, with a < b < c < d. His friend Ari asks for the four numbers. PJ says, “I won’t tell you the numbers, but I will give you their six pairwise sums: 18, 15, 13, 12, 10, and 7. From this information, Ari was able to compute the four numbers. Compute a2 + b2 +c2 + d2.
193
Compute the value of √(33 − 20√2) + √(17 + 12√2)
8
Compute (2 · 20202 − 673 · 2020 − 675)/4080399 as a fraction in simplest form.
5/3
In triangle GBM, BL ⊥ GM and BI is an angle bisector for ∠GBM. Given that m∠BGM = 60◦ and m∠BMG = 45◦ , compute cot(∠BIG).
√ 2 − 2 + √ 6 − √ 3