TIMO - 2019 - Grade 8 - Preliminary Round - Paper 3

According to the pattern $3,12,27,48,75,108,?,\ldots$, what number replaces the question mark?

Peter goes southeast for 15km, then goes southwest for 16km and goes northeast for 8km. How far is he now from the original position?

Given $\overline{abcd}+\overline{badc}=16676$, calculate $a+b+c+d$.

There are n lines cutting a circle which separate the circle into 11 parts, find the minimum value of n.

There are 20 pieces of red socks, 7 pieces of white socks and 30 pieces of blue socks in the box. If you want to get 2 pairs of socks with red colour and 2 pairs of socks that are not red colour, at least how many pieces of socks are needed to be drawn at random?

Find $y$ if $\lvert3x-2y+7\rvert+(x+y-16)^2=0$.

Factorize $x^2-y^2-5x+3y+4$.

How many non-positive integer solutions $x$ satisfy $-108\leq15x+9\leq65$?

Given $x+\frac{1}{x}=3$, find $x^3+\frac{1}{x^3}$.

Given $f(x)=ax^2+bx+c$, $f(-3)=6$, $f(3)=18$ and $f(-1)=2$, find $f(1)$.

Find the remainder when $2019^{111}$ is divided by 5.

A positive integer $x$ satisfies $x^2\equiv2\pmod7$ and $x>20$. Find its minimum value.

Find the last digit of $C=1^{2019}+2^{2019}+3^{2019}+4^{2019}+5^{2019}$.

How many positive integers smaller than 51 that are co-prime with 51?

Find the sum of positive factors of 60.

Given right-angled triangles $\triangle ABC\sim\triangle DEF$ with legs $AB=18$ and $AC=24$, find $\frac{DE}{EF}$.

Given points $A(-5,2)$, $B(1,6)$ and $C(4,2)$ on a coordinate plane, find the area of triangle $ABC$.

Anna stacks 10 cubes to get the figure on the right. Find its total surface area given that the side length of each cube is 2cm.

An interior angle of a regular $n$-sided polygon is $(6n+42)^\circ$. Find the maximum value of $n$.

Given that the area of a parallelogram is 2020. Find the smallest value of the perimeter of the parallelogram if all side lengths are integers.