TIMO - 2018 - Grade 7 - Heat Round - Paper 2

Given nonzero digits $A,C$, the two-digit numbers formed by them satisfy:
1. $\overline{CA}$ is divisible by 8.
2. $\overline{AC}$ is a cube.
Find the two-digit number $\overline{CA}$.

Split 2, 3, 5, 7, 14, 33, 65 and 143 into 2 groups of 4 numbers such that the products of numbers in both groups are equal. Find the sum of the numbers of the group that contains 7.

30 children, numbered 1 to 30, sit around a circle in order. Each child has an integer in hand. The child numbered 1 has the integer 1. The child numbered 2 has the integer 2. Given that the sum of the integers of any 9 consecutive children is equal to 30. What is the integer held by the child numbered 30?

There are 20 problems in a mathematics competition. The scores of each problem are allocated in the following ways: 2 marks will be given for a correct answer, 0 marks will be given for a blank answer or a wrong answer. Find the minimum number of candidate(s) to ensure that 2 candidates will have the same scores in the competition.

There are 4 pairs of white chopsticks, 5 pairs of yellow chopsticks and 6 pairs of brown chopsticks mixed together. Close your eyes. If you want to get each 3 pairs of chopsticks with different colours, at least how many piece(s) of chopstick(s) is / are needed to be taken?

Find $x$ if $(2x+3)+(4x-9)=0$.

Given $\lvert x-4\rvert+y^2-2y+1=0$, calculate $(7y-2x)+(7y-2x)^2+\cdots+(7y-2x)^{2017}$.

How many positive integer solutions $x$ satisfy $20x-17<1225$?

Given real numbers $a,b$ satisfying $a+b=7$ and $ab=12$, find $a^2+b^2$.

$a$ is a negative real number and $a
e-2$. If $a+\frac1{a+2}=-4$, find $a$.

Find the number of positive factor(s) of the product of 2017 and 1221.

Find $2017^2-2016^2+2015^2-2014^2+2013^2-2012^2+\cdots+3^2-2^2+1^2-0^2$.

Find the units digit of $A=1+2+2^2+2^3+\cdots+2^{2016}+2^{2017}$.

Given $x>0$ and $x+\frac1x=1221$, find $x^2+\frac1{x^2}$.

Given the sum of 10 distinct integers is 5959, find the maximum possible value of their Greatest Common Divisor.

A solid cuboid is formed by merging 1221 cubes with side length 1cm. Find the minimum value of the total surface area of the cuboid.

A triangle has three sides with lengths 20cm, 48cm and 52cm. Find the area of the triangle in cm².

Small cubes with side length 1cm are stacked according to the pattern shown below. If there are 10 layers, find the surface area.

Triangle $ABC$ is right-angled with $\angle ABC=90^\circ$. If the distance from $B$ to the midpoint of $AC$ is 6 cm, find the maximum area of triangle $ABC$.

The figure is a square of side length 14 overlapped by two quarter circles. Find the shaded area. Take $\pi=\frac{22}{7}$.