TIMO - 2018 - Grade 8 - Heat Round - Paper 4

Find the 3-digit number $\overline{ACA}$. The non-zero digits $A,C$ satisfy: (1) $\overline{CAC}$ is divisible by 9; (2) $\overline{CCA}$ is divisible by 7; (3) $\overline{ACC}$ has an odd number of factors.

Split 7, 34, 51, 64, 69, 70, 92 and 720 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 21 consecutive children is equal to 7. 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 wrong answer. Find the minimum number of candidate(s) to ensure that 3 candidates will have the same scores in the competition.

There are 3 pieces of white chopsticks, 4 pieces of yellow chopsticks and 5 pieces of brown chopsticks mixed together. Close your eyes. If you want to get a pair of yellow chopsticks, at least how many pieces of chopsticks are needed to be taken?

Find $x$ if $\lvert x+7\rvert-\lvert x-4\rvert=9$.

Factorize $x^2-y^2+2y-1$.

How many integer solutions $x$ satisfy $-24\leq7x+4\leq18$?

Given $x\neq4,5,6,7$ and $\frac{x-11}{(x-5)(x-6)}=\frac{x-11}{(x-4)(x-7)}$, find $x$.

A negative real number $a\neq-1$ satisfies $a+\frac{1}{a+1}=-3$. Find $a$.

Find the remainder when $2017^{2017}$ is divided by 16.

For a real number $x$, find the minimum value of $x^2+6x+25$.

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

Given $x>0$ and $x+\frac{1}{x}=2017$, find $x^2+\frac{1}{x^2}$.

If $(x-1)$ is a factor of $2017x^2-1221x-c$, find $c$.

In the figure, $AB=AC$. Point $D$ lies on $AB$, $E$ lies on the extension of $AC$, and $DE$ intersects $BC$ at $F$. Given $\frac{FC}{BF}=\frac18$ and $\frac{AB}{AE}=\frac34$, find $\frac{DF}{FE}$.

Given points $A(1,3)$, $B(5,2)$ and $C(7,4)$ on a coordinate plane, find the area of triangle $ABC$.

The lengths of 3 sides of a right-angled triangle are all integers. One of the right-angled sides is 2017. Find the length of the other right-angled side.

Hexagon $ABCDEF$ is equiangular, with $AB=2015$, $BC=2016$, $CD=2017$ and $DE=2018$. Find its perimeter.

How many diagonal(s) does a convex nonagon (9-sided polygon) have?