ASMO Math 2022 — Grade 10 — School Round

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 $(9/5)(a+b)$.

Evaluate $\frac{100^3(100-1)}{100^2-1}\times\frac{(100-1)(100+1)}{100^2}$.

Solve $\sum_{k=2}^{n}\frac{1}{\sqrt{2k}+\sqrt{2k+2}}=1$.

A cubic function is defined as $f(x)=ax^3+bx^2+cx$, where a, b and c are real numbers. It is known that $f(2)=26$, $f(5)=290$ and $f(1)=6$. Find $a+b+c$.

You are given the following rational value of x such that $x=1/(3-\sqrt{8})$. Solve the following equation with the given value of x: $\frac{x}{x^2-1}(x^2-6x+1)-(x^2-2\sqrt{8}x)+4x^2-4x$.

A manager who works in a retail shop often records number of customers purchased item(s) on weekdays and he showed the following sample record. Monday: $x+3$; Tuesday: $5x+2$; Wednesday: $3x+2$; Thursday: $4x+9$; Friday: $7x+4$. It is known that the mean number of customers purchased item(s) is 16. Find the sum of minimum, median and maximum number of customers.

Let $x$ be an angle satisfying

$$\cos x=\sin x+1,\qquad \sin x\lt0.$$

Evaluate

$$\frac{\sin^2 x+\sin x-2}{\sin x-1}\times\frac{\sin^3 x+16\sin^2 x-5\sin x+5}{7\sin^2 x+2\sin x}.$$

Hint: Use $\sin^2 x+\cos^2 x=1$ to determine $\sin x$. You do not need to find $x$.

Compute $\frac{1}{\log_2(2\sqrt{3})}+\frac{1}{\log_8(2\sqrt{3})}+\frac{1}{\log_3(2\sqrt{3})}+\frac{1}{\log_3(2\sqrt{3})}$.

Find all possible ordered pairs $(x,y)$ of positive integers satisfying $\sqrt{2x}\sqrt{4xy}-\sqrt{2x}\sqrt{320}+\sqrt{2y}\sqrt{4xy}-\sqrt{2y}\sqrt{320}+\sqrt{320}\sqrt{4xy}-320=0$.

It is known that $X\times Y=4^5+2^{16}$ where X and Y are positive integers. Evaluate $X/4+(Y+5)/10$.

The following figure shows both shaded and non-shaded polygons. The unit is in meter (m). The polygon ABCDEFG represents a rectangle with right angle triangles BFE, BCE and ECD and trapezoid ABFG. The given lengths are 3m, 4m and 5m for GF, FE and FB respectively. The angle of BEC is 45 degrees. Find the area of shaded regions.