HKIMO 2019 Grade 7 National Final (Heat) Set 02

Find the average of $2001,2003,2006,2008,2011,2013,\ldots,2991,2993,2996,2998$.

A cage contains chickens and rabbits. There are seven times as many chickens as rabbits, and the chickens have 120 more legs in total than the rabbits. How many rabbits are there?

How many integers $x$ satisfy $-3\le\frac{(x-7)\times2+5}{3}<5$?

$A:B=1:3$, $B:C=4:3$, $C:D=4:5$, and $A+B+C+D=870$. Find $A$.

A box contains 18 red socks, 7 white socks and 16 blue socks. What is the minimum number drawn at random that guarantees three pairs, one pair of each colour?

Find $x$ if $(18-2x)-(45-5x)=0$.

Real numbers $a,b$ satisfy $a+b=11$ and $ab=8$. Find $a^3b+ab^3$.

For positive $a\ne2$, $\frac a3+\frac1{2-a}=\frac{5a}3$. Find the sum of all possible values of $a$.

Factorise $x^2-4x-45$.

Find $x$ if $|3x-2y+16|+|5-y|=0$.

Find the sum of all positive divisors of 420.

Find the largest three-digit integer $x$ such that $x\equiv10\pmod{11}$.

Find the last digit of $3^{888}+5^{999}+9^{1110}$.

How many positive fractions less than 1, in lowest terms, have denominator 3234?

A three-digit number leaves remainders 4, 7 and 14 on division by 6, 13 and 16, respectively. Find it.

$M(11,8)$ is the midpoint of $A(x,4)$ and $B(-5,y)$. Find $x+y$.

A solid cuboid is formed from 972 unit cubes. Find its minimum total surface area.

A right triangle has hypotenuse 18. Find its maximum area.

In the figure, $ABD$ is equilateral, $ACE$ is right-angled, and $\angle CAD=\angle DAE$. Find $AE/AC$. The figure is illustrative.

Unit cubes are stacked as shown. The first three levels contain 10 cubes in total. Find the total volume for six levels.