TIMO - 2021 - Grade 8 - Heat Round - Paper 1

There are some chickens and rabbits in a cage. The number of rabbits is 3 times less 16 as the number of chickens. The total number of rabbit legs is 186 more than that of chicken. How many rabbit(s) is / are there?

There are 25 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, 1 mark will be deducted from a blank answer and 2 marks will be deducted from a wrong answer. Find the minimum number of candidate(s) to ensure that 3 candidates will have the same scores in the competition.

From the pattern $2,4,18,116,802,5604,?,\ldots$, find the number replacing the question mark.

There are six teams named A, B, C, D, E and F participating in a tournament. In 5 days, each team will play one game each day. They play against another team once in the tournament. There are 3 games every day, given that: 1) Team B is defeated by Team F on the first day. 2) Team A wins Team C on the second day. 3) Team E is defeated by Team B on the third day. 4) Team C wins Team F on the fourth day. Which team does Team D play with on the fifth day？

There are 5 suspects which are A, B, C, D, E in a case. They provide the testimony below respectively. It is known that only one of them did not lie and there is only one criminal. Who is the criminal? A: C is the criminal. B: A is lying and I am not the criminal. C: D and E are both not the criminal. D: C is lying and A is not the criminal. E: I am the criminal.

How many integer solutions $x$ satisfy $-4>\frac{7-2x}{4}>-7$?

Factorize $x^2-y^2-4x+6y-5$.

Given a non-zero real number $x$ satisfying $x^2-6x+1=0$, calculate $x^2+\frac{1}{x^2}$.

Real numbers $x,y$ satisfy $\lvert x-6\rvert+y^2-6y+9=0$. Find $3x-y$.

Given $f(x)=ax^2+bx+c$, $f(-4)=-10$, $f(4)=22$ and $f(-2)=1$, find $f(2)$.

Find the sum of positive factors of 1092.

Find the remainder when $119^{256}$ is divided by 12.

Find the last digit of $A=8+50+344+\cdots+(7^{44}+1)+(7^{45}+1)$.

A real number $x>0$ satisfies $x=4-\frac{6}{1+\frac{6}{4-\frac{6}{1+\frac{6}{4-\cdots}}}}$. Find $x$.

Given positive integers $x,y$ with $\frac{7}{15}<\frac{x}{y}<\frac{11}{18}$, find $x$ when $y$ is as small as possible.

Given points $A(2,-8)$, $B(6,9)$, $C(-8,-7)$ and $D(-2,4)$ on a coordinate plane, find the area of the convex quadrilateral whose vertices are these four points.

Find the area enclosed by the $x$-axis and the lines $9x+4y-108=0$ and $3x-4y+12=0$.

The radius of a sector is 9 and its perimeter is 52. Find the area of this sector.

Combine 612 squares with side length 1 unit to form a rectangle. Find the minimum perimeter of that rectangle.

Line $L$ passes through $A(9,8)$ and has slope $-4$. Find its $x$-intercept.