Problem Statement A sequence of integers $a_1, a_2, a_3, \ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \ge 3$. What is the sum of the first 2001 terms of this sequence if t...

Problem Statement Determine the value of $ab$ if $\log_8a+\log_4b^2=5$ and $\log_8b+\log_4a^2=7$. Problem Link Solution $$\log_8a+\log_4b^2=5 \text{(1)}$$ $$\log_8b+\log_4a^2=7 \text{(2)}...

Problem Statement Find the value of $10\cot(\cot^{-1}3+\cot^{-1}7+\cot^{-1}13+\cot^{-1}21).$ Problem Link Solution $$\cot^{-1}(x) = \tan^{-1}(\dfrac{1}{x})$$ $$\tan^{-1}(\tan(\cot^{-1}(x)...

Problem Statement Let $N$ be the number of consecutive $0$’s at the right end of the decimal representation of the product $1!2!3!4!\cdots99!100!.$ Find the remainder when $N$ is divided by $100...

Problem Statement Suppose that $a$, $b$, $c$, and $d$ are positive integers that satisfy the equations [ab + cd = 38] [ac + bd = 34] [ad + bc = 43] What is the value of $a$ + $b$ + $c$ + $d$...

Problem Statement An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability t...

Problem Statement Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement...

Problem Statement The nine horizontal and nine vertical lines on an $8 \times 8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ wh...

Problem Statement How many even integers between 4000 and 7000 have four different digits? Problem Link Solution $$\text{Case 1: Number starts with 5}$$ $$\text{There are 5 options for th...

Problem Statement The numbers $1447$, $1005$ and $1231$ have something in common: each is a $4$-digit number beginning with $1$ that has exactly two identical digits. How many such numbers are t...