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

There are 6 pieces of white chopsticks, 8 pieces of yellow chopsticks and 10 pieces of blue chopsticks mixed together. If you want to get 4 pairs of chopsticks of the same color in the dark, at least how many pieces of chopsticks are needed to be taken out?

Robert goes east for 28km, goes south for 28km, then goes west for 7km. How many km is he now from the original position?

Given $A:B=3:5$, $B:C=7:11$, $C:D=2:3$ and $A+B+C+D=1161$, find $B$.

If $\overline{abcd}+\overline{bcda}+\overline{cdab}+\overline{dabc}=37774$, calculate $a+b+c+d$.

The five statements below contain exactly one correct statement. Write the letter of the correct statement.

Find $x$ in the system $\begin{cases}3x+4y=-27,\\4x-3y=-11.\end{cases}$

Find $x$ if $\lvert x+5\rvert-\lvert x-9\rvert=2$.

Find $\frac{1}{14\times34}+\frac{1}{34\times20}+\frac{1}{46\times26}+\frac{1}{20\times46}+\frac{1}{7\times20}+\frac{1}{20\times13}$. Give your answer as a fraction in simplest form.

Given $\lvert x+5\rvert+(y-4)^2=0$, find $A=(y+x)+(y+x)^2+(y+x)^3+\cdots+(y+x)^{2017}+(y+x)^{2018}$.

For positive integers $x,y$ with $5x+3y=37$, find the minimum value of $x+y$.

When a 4-digit number is divided by 4, 6 or 8, the remainder is 3. What is the largest possible value of that 4-digit number?

If $x^2-y^2+8y-16\equiv(x+ay+b)(x+cy+d)$, find $a\times c+b\times d$.

Find the last digit of $A=1+4+9+16+25+\cdots+841+900+961$.

Given $x>0$ and $x+\frac{1}{x}=7$, find $x^2+\frac{1}{x^2}$.

A rational number $x>0$ satisfies $x=3+\frac{4}{3+\frac{4}{3+\cdots}}$. Find $x$.

If the side lengths of a triangle are 24, 32 and 40, find the area of the triangle.

Points $A(2,4)$, $B(5,4)$, $C(5,-9)$ and $D$ are the vertices of a rectangle on a coordinate plane. Find its area.

Find the largest area of a quadrilateral whose perimeter is 24.

In the figure, $\triangle ADB\sim\triangle ABC\sim\triangle CDB$. Find $\angle BAD$.

The figure below shows that a circle is inscribed in an equilateral triangle. The equilateral triangle is also inscribed in another circle. Find the ratio of the radius of the smaller circle is divided by that of the larger circle.