TH-TMO 2004

Let f be a function such that $f(0) = 0, f(1) = 1$ , and $f(n) = 2f(n-1)- f(n- 2) + (-1)^n(2n - 4)$ for all integers $n \ge 2$ . Find f(n) in terms of $n$ .

Let $f : R \to R$ satisfy $f(x + f(y)) = 2x + 4y + 2547$ for all reals $x, y$ . Compute $f(0)$ .

Compute the sum $$\sum_{k=0}^{n}\frac{(2n)!}{k!^2(n-k)!^2}.$$

Find the number of ways to select three distinct numbers from ${1, 2, . . . , 3n}$ with a sum divisible by $3$ .

Find the number of positive integer solutions to $(x_1 + x_2 + x_3)(y_1 + y_2 + y_3 + y_4) = 91$

Let $n$ be a positive integer and define $A_n = \{1, 2, ..., n\}$ . How many functions $f : A_n \to A_n$ are there such that for all $x, y \in A_n$ , if $x < y$ then $f(x) \ge f(y)$ ?

Compute the remainder when $29^{30 }+ 31^{28} + 28! \cdot 30!$ is divided by $29 \cdot 31$ .

Compute gcd $(5^{2547} - 1, 5^{2004} - 1)$ .

Find the largest positive integer $n \le 2004$ such that $3^{3n+3} - 27$ is divisible by $169$ .

What are last three digits of $2^{2^{2004}}$ ?

Compute the remainder when $1^{2547} + 2^{2547} +...+ 2547^{2547}$ is divided by $25$ .

Find positive reals $a, b, c$ which maximizes the value of $abc$ subject to the constraint that $b(a^2 + 2) + c(a + 2) = 12$ .

Find positive reals $a, b, c$ which maximizes the value of $a+ 2b+ 3c$ subject to the constraint that $9a^2 + 4b^2 + c^2 = 91$

Two pillars of height $a$ and $b$ are erected perpendicular to the ground. On each pillar, a straight cable is placed connecting the top of the pillar to the base of the other pillar; the two lines of cable intersect at a point above ground. What is the height of this point?

The ratio between the circumradius and the inradius of a given triangle is $7 : 2$ . If the length of two sides of the triangle are $3$ and $7$ , and the length of the remaining side is also an integer, what is the length of the remaining side?