HKIMO 2020 Grade 7 National Final (Heat) Set 01

Find the average of the sequence $2002,2008,2006,2012,2010,2016,\ldots,2394,2400,2398,2404$.

How many integer solutions $x$ satisfy $-5\le\frac{(x-7)\times 2-9}{3}<3$?

Given $A:B=2:3$, $B:C=3:4$, $C:D=4:5$ and $A+B+C+D=910$, find $A$.

A cage contains chickens and rabbits. There are nine times as many chickens as rabbits. The chickens have 840 more legs in total than the rabbits. How many rabbits are there?

A box contains 26 red socks,8 white socks and 16 blue socks. What is the least number to draw at random to guarantee three pairs of different colours, one pair of each colour?

Find $x$ if $(18-x)-(90-5 x)=0$.

Real numbers $a,b$ satisfy $a+b=12$ and $ab=11$. Find $a^4 b^2+a^2 b^4$.

A positive real number $a\ne 3$ satisfies $\frac a 8+\frac 1{3-a}=\frac{5 a}{8}$. Find the sum of all possible values of $a$.

Factorize $x^2-4 x-32$.

Find $x$ if $|3 x-2 y+20|+|4-y|=0$.

A three-digit number leaves remainder 3 when divided by 6, remainder 6 when divided by 11, and remainder 7 when divided by 16. Find it.

If $x\equiv 8\pmod{11}$, find the largest three-digit value of $x$.

Find the last digit of $3^{999}+5^{111}+9^{1111}$.

How many positive fractions less than 1, in lowest terms, have denominator 2332?

Find the sum of all positive factors of 240.

$M(13,5)$ is the midpoint of $A(x,2)$ and $B(-3,y)$. Find $x+y$.

A right triangle has hypotenuse 24. Find its maximum area.

A solid cuboid is made by joining 648 unit cubes. Find the minimum total surface area.

Parallelogram $ABCD$ has area 128. Points $E,F,G,H$ are the midpoints of $AB,BC,CD,DA$, respectively. Find $[EFGH]$.

Unit cubes are stacked according to the pattern shown. The illustrated solid has three levels and nine cubes. Find the total volume for 15 levels.