How many integers between 100 and 1000 contain no digits other than 3, 4 or 5?

In circle O, the lengths of chords AB and BC are equal and m\(\angle\)ABC = 90 degrees. Given that circle O has a radius of 3 meters, what is the length of arc ABC? Express your answer in terms of π.

If \(\sqrt{x}-\sqrt{y}=10\) and \(\sqrt{x}+\sqrt{y}=14\), what is the value of x + y?

Francisco is born at 1:00 a.m. on a Tuesday and gets married exactly 2¹⁸ hours later. On what day of the week does Francisco get married?

Penner has a deck of 40 cards composed of four suits (red, blue, green, and yellow) and cards numbered 1 through 10 in each suit. Tell secretly chooses a card. Penner then chooses the following 4 cards from the deck: Red-2, Blue-3, Green-5 and Yellow-7. For each card Penner chooses, Tell says “yes” if his secret card is of the same color or shares a common factor greater than 1 with Penner’s card. Otherwise Tell says “no.” Tell says “no,” “yes,” “no,” and “yes,” respectively, in response to Penner’s cards. With this information, what is the best possible probability Penner has of guessing Tell’s secret card? Express your answer as a common fraction. 

How many pairs of positive integers a and b exist such that a²-b² = 144?

If f(x) = ax² + bx + c, with f(0) = 4, f(2) = 2 and f(4) – f(3) = 4, what is the value of f(1)?

If \(\dfrac{x^2+8x+15}{x+5}\) = 4.01, then what is the value of x? Express your answer as a decimal to the nearest hundredth. 

If p, q and r are prime numbers such that pq + r = 73, what is the least possible value of p + q + r? 

A set S contains some, but not all, of the positive integers from 3 to 7. Some statements describing S are given below. The statement numbered n is true if the number n is in S and false if n is not in S. What is the product of the numbers that are in S?

3. The sum of the numbers in S is odd.

4. The sum of the numbers in S is less than 15.

5. S contains exactly one composite number.

6. S contains exactly one prime number.

7. The product of the numbers in S is odd.

Suppose that A & B = k × A^m × Bⁿ , where k, m and n are constants. Suppose that 5 & 3 = 18, 10 & 3 = 72 and 5 & 6 = 36. What is the value of 10 & 6?

Prove that: \(n^4-10n^2+9⋮384\) with all even natural n.

Prove that: \(n^3+6n^2+8n⋮48\) with all odd natural n.

Prove that: \(\left(n^2+n-1\right)^2-1⋮24\left(\forall n\in Z\right)\)

Prove that: \(n^3+3n^2+2n⋮6\left(\forall n\in Z\right)\)

Prove that: \(A=\left(n+1\right)^4+n^4+1⋮\) a square number \(\ne1\) with \(\forall n\in Z\)+

Given a,b,c,d are 4 consecutive positive integers. Prove that: \(abcd+1\) is a square number

Find x;y such that: \(x^x\) has y numbers and \(y^y\) has x numbers

Prove that: \(\dfrac{3}{5}< \dfrac{1}{2004}+\dfrac{1}{2005}+...+\dfrac{1}{4006}< \dfrac{3}{4}\)

Prove that: \(abc\ge\left(b+c-a\right)\left(a+c-b\right)\left(a+b-c\right)\)