HKIMO 2019 Grade 8 National Final (Heat) Set 02

How many integers x satisfy $8>\frac{2-3x}{4}>11$?

Find the average of $1550,1553,1556,1560,1563,1566,\ldots,2136,2140,2143,2146$.

There are 70 balls. Peter and Mary alternately take 4 to 8 balls. The person who takes the last ball wins. Peter moves first and may add balls before play. What is the least number he must add to guarantee victory?

Editorial reading of the printed last-ball rule: every move must take 4–8 balls. If 1–3 remain, no legal move is possible and neither player has taken the last ball; being unable to move is not itself a win for the other player.

Given $A:B=2:3$, $B:C=5:1$, $C:D=7:2$ and $A+B+C+D=2020$, find D.

A box contains 17 red socks, 32 white socks and 16 blue socks. What is the least number to draw to guarantee four white pairs and five non-white pairs?

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

Given $x>0$ and $x=2+\cfrac{24}{2+\cfrac{24}{2+\cfrac{24}{2+\cdots}}}$, find x.

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

Given $f(x)=ax^2+bx+c$, $f(-4)=7$, $f(4)=23$ and $f(-1)=6$, find $f(1)$.

Given $a>3$ and real roots of $ax^2+12x-2c=0$, find the least possible integer c.

Find y if $|3x+2y-5|+|y-2x+8|=0$.

A four-digit number has greatest common divisor 261 with 3132. Find the greatest possible number.

If $7x\equiv6\pmod{13}$, find the largest two-digit integer x.

Find the remainder when $52^{25}$ is divided by 7.

Find the sum of all positive divisors of 616.

Find the units digit of $A=3+10+29+\cdots+(3^9+8)+(3^{10}+9)$.

A triangle has side lengths 12, 13, 7 and area $n\sqrt3$. Find n.

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

Right triangles satisfy $\triangle ABC\sim\triangle DEF$, with legs $AB=15$, $AC=36$. Find $DE/EF$.

The hypotenuse of a triangle is 24. Find the largest possible area.

An interior angle of a regular n-sided polygon equals $(8n+54)^\circ$. Find the greatest possible n.