TIMO - 2021 - Grade 9 - Preliminary Round - Paper 1

Two primes are called a pair of twin primes if they have just one natural number between them, e.g. (3, 5) or (5, 7). How many pairs of twin primes are there from 1 to 80?

4 years ago, Candace was 12 years older than Miley. 6 years later, the sum of their ages is 56 years. Find Miley's age right now.

There is an eight-car train waiting for passengers at a station. The ticket seller wants to make sure that there are 23 people sitting in the same car. At least how many tickets should he sell?

On Halloween, Hannah’s candies were stolen. She asked her 4 guests and received replies as follows: Cupcake: “Chocolate’s stolen it.” Chocolate: “Jelly’s stolen it.” Gummy: “I have not stolen it.” Jelly: “Chocolate is lying.” Given that only one of them had told lies, who stole Hannah’s candies?

If $x,y,z$ are all prime numbers and $x^y+y^x=z$, find $z$.

For $a\ne b$, define $a\otimes b=\frac{ab}{a-b}$. Compute $20\otimes18-18\otimes20$.

Given $x>0$ and $x-\frac1x=2$, find $\left(x^2-\frac1{x^2}\right)^2$.

Let $x,y$ be real numbers such that $(2x-y)^2+(x-1)^2=0$. Find $A=(x-2)^{2021}+(y-2)^{2020}$.

Given $\frac13+\frac1{15}+\frac1{35}+\cdots+\frac1{(2n+1)(2n+3)}=\frac{1010}{2021}$, find the natural number $n$.

If $A=\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}$, find $A$.

Find the number of positive factors of 108.

Find the last digit of $3^{2019}+4^{2020}+5^{2021}$.

Given that $\frac{40!}{10^n}$ is an integer, find the largest possible value of $n$.

Given a natural number n. When n is divided by 6, the remainder is 2. When n is divided by 7, the remainder is 4. What is the smallest positive integer p such that p + n is a multiple of 42?

How many fractions in lowest terms with denominator 30 are there such that it is positive and smaller than 1?

In right-angled triangle $ABC$, right at $A$, the altitude $AH=6$ cm and the area of $ABH$ is 40% of the area of $ABC$. Find $BC$.

A solid cuboid is formed by merging 210 cubes with side length 1cm. Find the smallest possible value of the total surface area of the cuboid.

The difference between an interior angle and an exterior angle of a regular polygon is $156^\circ$. How many sides does it have?

A fish tank measuring 60cm by 30cm contains water to a height of 24cm. Then, 60% of a rock with volume of 600cm³ is submerged in water. Find the new water level.

A square with side length 10 cm is overlapped by two quarter-circles as shown. Find the area of the shaded region. Take $\pi=3.14$.