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We are asked to find the largest sum of calendar dates for seven consecutive Fridays in any given year.

Mike wrote a list of 6 positive integers on his paper. The first two are chosen randomly. Each of the remaining integers is the sum of the two previous integers. We are asked to find the ratio of the fifth integer to the sum of all 6 integers.

Evaluate $\frac{1 + 2 + 3 + 4 + 5 + 6 + 7}{8 + 9 + 10 + 11 + 12 + 13 + 14}$.

Chad has $100$ cookies that he wants to distribute among four friends. Two of them, Jeff and Qiao, are rivals; neither wants the other to receive more cookies than they do. The other two, Jim and Townley, don't care about how many cookies they receive. In how many ways can Chad distribute all $100$ cookies to his four friends so that everyone is satisfied? (Some of his four friends may receive zero cookies.)

Define a sequence of positive integers $s_1, s_2, . . . , s_{10}$ to be $terrible$ if the following conditions are satisfied for any pair of positive integers $i$ and $j$ satisfying $1 \le i < j \le 10$: - $s_i > s_j$ - $j - i + 1$ divides the quantity $s_i + s_{i+1} + \cdots + s_j$ Determine the minimum possible value of $s_1 + s_2 + \cdots + s_{10}$ over all terrible sequences.

Let sequences {$a_n$} and {$b_n$} satisfy: $a_n=a_{n-1}\cos{\theta} - b_{n-1}\sin{\theta}$ and $b_n=a_{n-1}\sin{\theta}+b_{n-1}\cos{\theta}$. If $a_1=1$ and $b_1=\tan{\theta}$, where $\theta$ is a known real number, find the general formula for {$a_n$} and {$b_n$}.

If $x$ and $y$ are positive integer solutions to the equation $x^2 - 2y^2 = 1$, then $6\mid xy$.

Simplify $\displaystyle\frac{2^2-2}{2^2+2}\cdot\displaystyle\frac{3^2-3}{3^2+3}\cdots\displaystyle\frac{10^2-2}{10^2+10}$

Compute $\Large(\sqrt{6+4\sqrt{2}} + \sqrt{6-4\sqrt{2}}\Large)^2$

The Fibonacci numbers are defined by $F_1=1, F_2=1$, and $F_n=F_{n-1} + F_{n-2}$ for $n=3, 4, \cdots$. Find and prove a formula for the sum of the first $n$ Fibonacci numbers, i.e. $F_1 + F_2 + \cdots +F_n$.

Let {$a_n$} be a sequence with $a_1=1$. If for any $n > 1$, $a_n$ equals one plus twice of the sum of all the previous terms, express $a_n$ in terms of $n$.

How many among the first $1000$ Fibonacci numbers are multiples of $11$?

Let $F(1)=1, F(2)=1, F(n+2)= F(n+1)+F(n)$ be the Fibonacci sequence. Prove if $i | j$, then $F(i) | F(j)$. Or described in another way, every $k^{th}$ element is a multiple of $F(k)$.

Let $a$, $b$, and $c$ form a geometric sequence. Can the last two digits of $N=a^3+b^3+c^3-3abc$ be 20?

Prove that $5x^2\pm 4$ is a perfect square if and only if $x$ is a term in the Fibonacci sequence.

Find the maximal value of $m^2+n^2$ if $m$ and $n$ are integers between $1$ and $1981$ satisfying $(n^2-mn-m^2)^2=1$.

Sequence $ \{a_{n}\}$ is defined by $ a_{1}= 2007,\, a_{n+1}=\frac{a_{n}^{2}}{a_{n}+1}$ for $ n \ge 1.$ Prove that $ [a_{n}] =2007-n$ for $ 0 \le n \le 1004,$ where $ [x]$ denotes the largest integer no larger than $ x.$

What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1$, $k_2$, $\dots$ , $k_n$ for which $$k_1^2 + k_2^2 + \cdots +k_n^2 = 2002$$

A sequence of three real numbers forms an arithmetic progression with a first term of 9. If 2 is added to the second term and 20 is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression?

Let $a_1,a_2,\cdots$, be a sequence with the following properties. (i) $a_1=1$, and (ii) $a_{2n}=n\cdot a_n$ for any positive integer $n$. What is the value of $a_{2^{100}}$?

Find the sum of the reciprocals of all perfect squares whose prime factorization contains only 3, 5, and 7, i.e. $$\frac{1}{9}+\frac{1}{25}+\frac{1}{49}+\frac{1}{9}+\frac{1}{81}+\frac{1}{225}+\frac{1}{441}+\frac{1}{625}+\cdots$$

Find the general formula of the sequence defined as $a_1=6$ and $a_n=\frac{1}{2}a_{n-1}+4$.

There is a sequence with $a(2) = 0$, $a(3) = 1$ and $a(n) = a(\lfloor{\frac{n}{2}}\rfloor)+a(\lceil{\frac{n}{2}}\rceil)$ for $n\ge 4$. Find $a(2014)$.

Given that $x_{n+2} =\frac{20x_{n+1}}{14x_n}$, $x_0 = 25$, $x_1 = 11$, it follows that $$\sum_{n=0}^{\infty}\frac{x_{3n}}{2^n}=\frac{p}{q}$$ for some positive integers $p, q$ with $GCD(p, q) = 1$. Find $p + q$.

The sum of $n$ consecutive positive integers is 100. What is the greatest possible value of $n$?