TIMO - 2019 - Grade 7 - Preliminary Round - Paper 3

According to the pattern $2,12,30,56,90,132,?,\ldots$, find the number replacing the question mark.

Peter goes east for 20 km, then goes south for 24 km, then goes west for 13 km. How far is he now from the original position?

If $\overline{abcd}+\overline{badc}=3487$, calculate $a+b+c+d$.

If $x,y,z$ are all primes and $2x^y+y^x=2z$, find the minimum value of $z$.

There are 10 pieces of red socks, 6 pieces of white socks and 22 pieces of blue socks in the box. If you want to get 2 pairs of socks with different colour, at least how many pieces of socks are needed to be drawn?

Find $x$ if $(37+21x)-(8x-67)=0$.

Given positive integers $x,y$ with $7x-2y=151$, find the least value of $\lvert x-y\rvert$.

How many integer solutions $x$ satisfy $-56\le5x+14<89$?

Positive real numbers $a,b$ satisfy $a^2-b^2=51$ and $a=3+b$. Find $a+b$.

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

Find the sum of all positive factors of 513.

From 1 to 1000, how many integers are divisible by 3 and 5, but not divisible by 9?

Find the units digit of $5^{2019}+1^{2019}+3^{2019}$.

Given $x>0$ and $x+\frac1x=5$, find $x^4+\frac1{x^4}$.

The difference between two square numbers is 611. Find the possible maximum value of the larger square.

A cube with side length 18 is cut into cubes with side length 6. Find the increase in total surface area of the cubes.

An iron wire is bent into two circles, each of radius 21. It is then bent into three identical rectangles with integer side lengths. What is the maximum area of one rectangle? Take $\pi=\frac{22}{7}$.

Small cubes with side length 1 are combined according to the pattern shown below. Find the total volume of the 15th layer.

The longest length of a right triangle is 12. Find the maximum area of the triangle.

An interior angle of a n-sided regular polygon is 8 times of an exterior angle. Find n.