ASMO Math 2022 — Grade 11 — School Round

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.

You are given a trigonometric equation $\cos x=\sin x+1$ where x is an angle of triangle and $\sin x<0$. Compute $\frac{\sin^2x+\sin x-2}{\sin x-1}\times\frac{\sin^3x+16\sin^2x-5\sin x+5}{7\sin^2x+2\sin x}$. Hint: Use $\sin^2x+\cos^2x=1$ to obtain $\sin x$; you are not required to find x but need $\sin x<0$.

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.

Evaluate $\frac{1}{\log_2(4\sqrt{5})}+\frac{1}{\log_5(4\sqrt{5})}+\frac{1}{\log_4(4\sqrt{5})}+\frac{1}{\log_2(4\sqrt{5})}$.

Given

$$z=y^3(4y^2+2y-1)+4y^2+4\sqrt{5}\,y-14$$

and

$$y=\frac{1}{1-\sqrt{5}},$$

find the exact value of $z$.

Find the number of pairs of positive integers $(x,y)$ such that $x^6=y^2+127$.

X and Y are mathematically related such that $X\times Y=4^8+32^2$ where X and Y are positive integers. Evaluate $X/8+Y/5$.

Find the smallest integer greater than $(2+\sqrt{5})^4$.

The equation $(m-20)^3+(m-22)^3=(2m-42)^3$ has three positive integer roots $m_1,m_2,m_3$. Evaluate $m_1m_2m_3$.

Let $f:\mathbb{R}\to\mathbb{R}$ be defined by

$$f(x)=x+2x^2+3x^3+\cdots+100x^{100}.$$

Find the number of positive integers $x\ne1$ for which the integer $f(x)$ is divisible by the integer $x-1$.

You are given $f(x)=\frac{1}{2}[x^2+x-1+|x^2-(x-1)|]$ and $g(x)=7x+20$ for real x. Let $h(x)=f(x)-g(x)$. If $h(x)=0$, express the solutions x in the form $a+b\sqrt{c}$.

Evaluate $\sqrt{3}\,\csc20^\circ-\sec20^\circ$ without resorting to calculator.

The figure below shows a right-angled triangle ABCD partitioned into two right-angled triangles ABD and BDC. The length of AB and BC are 3 units and 4 units respectively. The lengths of AD and DC are x and y, and the height BD is z. Find $x+y+z$.