TH-TMO 2005

A die is thrown six times. How many ways are there for the six rolls to sum to $21$ ?

How many ways are there to express $2548$ as a sum of at least two positive integers, where two sums that differ in order are considered different?

For each subset $T$ of $S = \{1, 2, ... , 7\}$ , the result $r(T)$ of T is computed as follows: the elements of $T$ are written, largest to smallest, and alternating signs $(+, -)$ starting with $+$ are put in front of each number. The value of the resulting expression is $r(T)$ . (For example, for $T =\{2, 4, 7\}$ , we have $r(T) = +7 - 4 + 2 = 5$ .) Compute the sum of $r(T)$ as $T$ ranges over all subsets of $S$ .

Compute gcd $\left( \frac{135^{90}-45^{90}}{90^2} , 90^2 \right)$

What is the remainder when $\sum_{k=1}^{2005}k^{2005\cdot 2^{2005}}$ is divided by $2^{2005}$ ?

Find the smallest positive integer $x$ such that $2^{254}$ divides $x^{2005} + 1$ .

Find the number of even integers n such that $0 \le n \le 100$ and $5 | n^2 \cdot 2^{{2n}^2}+ 1$ .

Find all odd integers $k$ for which there exists a positive integer $m$ satisfying the equation $k + (k + 5) + (k + 10) + ... + (k + 5(m - 1)) = 1372$ .

A function $f : Z \to Z$ is given so that $f(m + n) = f(m) + f(n) + 2mn - 2548$ for all positive integers $m, n$ . Given that $f(2548) = -2548$ , find the value of $f(2)$ .

A function $f : R \to R$ satisfy the functional equation $f(x + 2y) + 2f(y - 2x) = 3x -4y + 6$ for all reals $x, y$ . Compute $f(2548)$ .

Compute the sum of roots of $(2 - x)^{2005} + x^{2005} = 0$ .

For $a, b \ge 0$ we define $a * b = \frac{a+b+1}{ab+12}$ . Compute $0*(1*(2*(... (2003*(2004*2005))...)))$ .

Compute the sum $$\sum_{k=0}^{1273}\frac{1}{1 + tan^{2548}\left(\frac{k\pi}{2548}\right)}$$

Let $P(x)$ be a monic polynomial of degree $4$ such that for $k = 1, 2, 3$ , the remainder when $P(x)$ is divided by $x - k$ is equal to $k$ . Find the value of $P(4) + P(0)$ .

Let $a, b, c, d > 0$ satisfy $36a + 4b + 4c + 3d = 25$ . What is the maximum possible value of $ab^{1/2}c^{1/3}d^{1/4}$ ?

Compute the minimum value of $cos(a-b) + cos(b-c) + cos(c-a)$ as $a,b,c$ ranges over the real numbers.