HKIMO 2020 Grade 8 National Final (Heat) Set 01

How many integers $x$ satisfy $10>\frac{1-3x}{4}>13$?

Find the average of the sequence $2350,2353,2356,2360,2363,2366,\ldots,3026,3030,3033,3036$.

Given $A:B=2:3$, $B:C=1:5$, $C:D=5:4$ and $A+B+C+D=2400$, find D.

There are 90 balls. Peter and Mary take turns removing 4 to 8 balls. Whoever takes the last ball wins. Peter moves first and may add balls before the game. What is the least number he must add to guarantee victory?

A box contains 18 red socks, 22 white socks and 12 blue socks. What is the least number to draw at random to guarantee three pairs of white socks and four pairs of non-white socks?

Editorial convention: each pair consists of two socks of the same colour.

Given $x>0$ and $x=\cfrac{15}{2+\cfrac{15}{2+\cfrac{15}{2+\cdots}}}$, find x.

Factorize $x^2+6y^2-5xy+4x-8y$.

Find y if $|3x+2y-12|+|y-x+9|=0$.

Given $a>3$ and that $ax^2+24x+8c=0$ has real roots, find the greatest possible integer c.

Given $f(x)=ax^2+bx+c$, $f(-2)=8$, $f(2)=24$ and $f(-4)=16$, find $f(4)$.

A four-digit number has greatest common divisor 263 with 4734. Find its largest possible value.

Find the remainder when $92^{25}$ is divided by 7.

If $8x\equiv7\pmod{15}$, find the largest two-digit integer x.

Find the sum of all positive divisors of 2040.

Find the units digit of $A=1+8+27+64+\cdots+8000$.

A triangle has side lengths 13, 14 and 15. Find its area.

Line L passes through $A=(10,6)$ and $B=(18,-4)$. Find its x-intercept.

A cone is formed from a circle and a semicircle. Its total surface area is $7392\ \mathrm{cm}^2$. Find the radius of the semicircle, using $\pi=22/7$.

Right triangles satisfy $\triangle ABC\sim\triangle DEF$, with legs $AB=15$ and $AC=20$. Find $DE/EF$.

An interior angle of a regular n-sided polygon is $(6n+42)^\circ$. Find the greatest possible n.