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Arithmetic
Equation
Sequences
Number Theory
Mystery (double pts)
100

If a @ b = a2 - b2 and a # b = (a - b)2, then what is the value of (5 @ 3) # 6?

100

100

If 7x + 12 = 5x - 78, what is the value of x?

-45

100
The 19th term of an arithmetic sequence is 234 and the common difference is 8. What is the value of the first term?

90

100

What is the units digit of 5123456789?

5

100

How many ways are there to choose 8 balls from 12 identical balls?

1

200

What is the value of 100 - 101 + 102 - 103 + ... + 200 - 201?

51

200
Find the product of x and y if 5x + 3y = -7 and x + 3y = 9. Express your answer as a common fraction.

52/3

200

In a geometric sequence, the first term is 27 and the common difference is 2/3. What is the smallest integer of the sequence?

8

200
What is the remainder when 20252 is divided by 8?

1

200

If 10 ⋅ 10! + 9 ⋅ 9! + 8 ⋅ 8! + ... + 2 ⋅ 2! + 1 ⋅ 1! can be written as a! - b, what is the sum of a and b?

12
300

What is the square root of 13 + 23 + 33 + ... + 93?

45

300

There are a total of 67 animals in a zoo of only birds and donkeys. All the birds have two legs except for a disabled one-leg bird, and all donkeys have four legs except two disabled two-leg donkeys. If there are a total of 201 legs, what fraction of disabled donkeys are there? Express your answer as a common fraction.

1/18

300

What is the sum of the arithmetic sequence 4 + 11 + 18 + 25 + ... + 354?

9129

300

Convert the number 12345 to base 10.

194

300
What is the sum of roots (solutions) to the equation |x2 - 12x + 34| = 2?

18

400

What is the value of 12 + 22 + 32 + ... + 102?

385

400
Find the value of x+2y+3z if 3x-2y+11z = 12 and -x + 2y - 4z = -3.

6

400

What is the sum of the infinite sequence 1/2 + 1/16 + 1/128 + 1/1024 + ...?

4/7

400

For any positive integer M, the notation M denotes the product of the integers 1 through M. What is the largest integer n for which 5n is a factor of the sum 98! + 99! + 100! ?

26

400

What is the largest area of a circle that can be inscribed in a right triangle with side lengths 7, 24, and 25?

 3π

500

If the value of (3+2)(32+22)(34+24)(38+28) = ab - cb, what is the value of a + b + c?

21

500

What is the value of x+y+z if:

2x + 5y + 2z = -38

3x - 2y + 4z = 17

-6x + y - 7z = -12

-7

500

Suppose that {an} is an arithmetic sequence with a1 + a2 + a3 + ... + a100 = 100 and a101 + a102 + a103 + ... + a200 = 200. What is the value of a2 - a1?

0.01

500

If a four-digit number 2X9Y = 2X ⋅ 9Y, then what is the value of the product of X and Y?

10

500

What is the sum of all the values of three-digit N such that N is divisible by 11 and N/11 is equal to the sum of the squares of the digits of N?

1353