Problem 31

A two-digit integer n has the property that the sum of n and its digits is equal to 63. Find n.

Problem 30

Compare 65²³ and 255¹⁷.

Problem 29

Factorise P(x) =  2x⁷ + 5x⁴ + 3.

Problem 24

In a hall, there are 1098 seats. There are two aisles. Each row has the same number of seats. In each row, the number of seats between the aisles is 31. The rest of the seats are equally divided between the aisles. Find the number of rows.

Problem 17

Prove that for any m, n,  m² - n², 2mn,  and m² + n² form a Pythagorean triple.

Problem 15

Prove that if n is an odd number, then n, 0.5 (n² - 1) and 0.5 (n² + 1) form a Pythagorean triple.

Problem 12

Prove that for all natural numbers n, the formula

 is true.

Problem 11

Prove that for all natural numbers n, the formula

 is true.

  

Problem 10

 Prove that for all natural numbers n, the formula 1 + 3 + 5 +…+ (2n - 1) =  n² is true.

Problem 6

A vending machine has $13.25 in it. There are 10 coins total and the machine only accepts 2-dollar, 1-dollar and 50-cent coins. How many of each coin are in the machine?

Problem 4

The recursive formula for a sequence {u_n} is written in the form:

u₁ = 2, 
u_n+1 = 3u_n, n ≥ 1

Write the general explicit formula.