ASMO Math 2022 — Grade 9 — School Round

It is given that $\sqrt{x}-\sqrt{y}=20$, where x and y are real numbers. Determine the maximum possible value of $x-5y$.

Find the largest prime number less than 5000 of the form $a^n-1$, where a and n are positive integers, and n is greater than 1.

A square piece of graph paper of side length 30 mm contains 900 smallest squares each of side length 1 mm each. Its four rectangular corners, denoted by A, B, C, D in clockwise order, are cut away from the square piece of graph paper. The resultant graph paper, which has the shape of a cross, is shown in the figure below. Let N denote the total number of rectangles, excluding all the squares which are contained in the resultant graph paper. Determine the value of $N/10$.

The numbers $1/1,1/2,\ldots,1/2010$ are written on a blackboard. A student chooses any two of the numbers, say p, q, erases them and then writes down $p+q+pq$. She continues to do this until only one number is left on the blackboard. What is this number?

Determine the remainder when $2^{55}+1$ is divided by 33.

Determine the sum of all the 5-digit integers which are not multiples of 11 and whose digits are 1, 3, 4, 7, 9.

Let a, b and c be integers with $a>b>c>0$. If b and c are relatively prime, $b+c$ is a multiple of a, and $a+c$ is a multiple of b, find the value of $abc$.

Determine the sum of all positive integers p such that the expression $(x-p)(x-13)+4$ can be expressed in the form $(x+q)(x+r)$ for distinct integers q and r.

It is given that $x=1/(2-\sqrt{3})$. Determine the value of $x^6-2\sqrt{3}x^5-x^4+x^3-4x^2+2x-\sqrt{3}$.

Figure below shows an equilateral triangle PRS. The area of triangle PRS is half that of the triangle PQR. Determine the angle PRQ in degrees.

Determine the number of positive divisors of $(2008^3+3\times2008\times2009+1)^2$.

Suppose that ABC is a triangle and D is a point on side AB with $AD=BD=CD$. If $\angle BCA=m^\circ$, then determine the value of m.

Let $m_1,m_2,\ldots$ be a sequence of rational numbers such that $m_1=2$ and for $n\geq1$, $m_{n+1}=(1+m_n)/(1-m_n)$. Find $300m_{2008}$.

If $2010!=P\times10^b$, where P is an integer not divisible by 10, determine b.

If p, q and r are positive real numbers such that $pq+p+q=qr+q+r=rp+r+p=35$, determine $(p+1)(q+1)(r+1)$.

Let k be a real number such that $k^2-15k+1=0$. Determine $k^4+1/k^4$.

Determine the number of ways that 2010 can be written as a sum of one or more positive integers in non-decreasing order such that the difference between the last term and the first term is at most 1.

Determine the value of the expression $10\sqrt{10\sqrt[3]{10\sqrt[4]{10\sqrt[5]{\cdots}}}}$.

Find the number of ways of tiling a $4\times9$ rectangle by tiles of size $1\times2$.

Suppose $a\sqrt{a}+b\sqrt{b}=183$ and $b\sqrt{a}+a\sqrt{b}=182$. If a and b are real numbers, find $(9/5)(a+b)$.