TIMO - 2020 - Grade 7 - Heat Round - Paper 4

There are 22 pieces of white chopsticks, 18 pieces of yellow chopsticks and 39 pieces of blue chopsticks mixed together. If you want to get 2 pairs of blue chopsticks and 2 pairs of chopsticks which are not blue in the dark, at least how many pieces of chopsticks are needed to be taken?

A, B, C, D and E are passing balls. Starting from A, 5th time pass will return to B. Within the pass, there is at least 1 time to A, how many passing methods are there?

If $\overline{abcd}+\overline{badc}=17743$, calculate $a+b+c+d$.

There are some chickens and rabbits in a cage. The number of chickens is 5 times as the number of rabbits. The total number of chicken legs is 72 more than that of rabbit. How many rabbit(s) is / are there?

According to the pattern $1,1,4,13,43,142,?,\ldots$, find the missing number.

Factorize $x^2+2x-63$.

How many negative integer solutions $x$ satisfy $8x+92>16$?

Given a negative real number $a
e-1$, if $a+\frac{10}{a+1}=-8$, find the minimum value of $a$.

Find $x$ if $(96+23x)-(11x-48)=0$.

Given real numbers $a,b$ with $a-b=6$ and $ab=6$, find $\frac{(a+b)^3}{a^2-b^2}$.

Given rational $x>0$ with $x=3+\frac{10}{3+\frac{10}{3+\cdots}}$, find $x$.

Find the sum of all positive factors of 786.

Given $x>0$ and $x+\frac1x=22$, find $x^2+\frac1{x^2}$.

Find the units digit of $A=13+13^2+13^3+13^4+\cdots+13^{24}+13^{25}$.

The difference between two square numbers is 253. Find the maximum value of the larger square.

$M(-18,-22)$ is the midpoint of $A(x,8)$ and $B(10,y)$. Find $x+y$.

An interior angle of an n-sided regular polygon is 17 times of an exterior angle. Find n.

In the figure, $ABCDEFGH$ is a cube. If the distance from $A$ to $F$ is $4\sqrt3$, find the volume of the cube.

A solid cuboid is formed by merging 770 cubes with side length 1cm. Find the minimum value of the total surface area of the cuboid.

The figure is a rectangle overlapped by a quarter circle and a right-angled triangle. Find the perimeter of the shaded region. Take $\pi=\frac{22}{7}$.