TIMO - 2020 - Grade 9 - Heat Round - Paper 2

There are 21 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 wrong answer and 0 marks will be given for a blank answer. Find the minimum number of candidate(s) to ensure that 5 candidates will have the same scores in the competition.

If $\overline{abcd}+\overline{bcda}+\overline{cdab}+\overline{dabc}=26664$, find $a+b+c+d$.

There are 53 pieces of white chopsticks, 21 pieces of yellow chopsticks and 41 pieces of brown chopsticks mixed together. Close your eyes. If you want to get 2 pairs of chopsticks that is not white colour and 2 pairs of chopsticks that is not yellow colour, at least how many piece(s) of chopstick(s) is / are needed to be taken?

15th December, 2019 is Sunday. Which date of the week will 15th November, 2028 be?

According to the pattern $1,6,15,28,45,66,?,\ldots$, find the missing number.

Find the greatest value of $-x^2+8x-51$.

Factorize $x^2-y^2+2y+6x+8$.

Simplify $\sqrt{11+4\sqrt7}$.

Find $c$ in $\begin{cases}6a+b+c+d+e=16\\a+6b+c+d+e=18\\a+b+6c+d+e=20\\a+b+c+6d+e=22\\a+b+c+d+6e=24\end{cases}$.

Rational $x>0$ satisfies $x=\sqrt{21+4\sqrt{21+4\sqrt{\cdots}}}$. Find $x$.

It is September now. Which month will it be after $11^{13}$ months?

Let $x>0$ be a 3-digit odd number with $x\equiv7\pmod9$ and $x\equiv7\pmod{12}$. Find the minimum value of $x$.

How many reduced positive fractions less than 1 have denominator 360?

Rational $x>0$ satisfies $x=3+\frac{28}{3+\frac{28}{3+\cdots}}$. Find $x$.

The 10-digit number $\overline{20A2B01062}$ is divisible by 33 and $A>B$. Find $A+B$.

How many diagonals does a convex 16-sided polygon have?

In triangle $ABC$, $AB=24$, $BC=36$, $AC=18$. Find $\frac{\sin A}{\sin B}$.

Find the area enclosed by the x-axis, y-axis and the line $5x-3y+45=0$.

The side lengths of a triangle are 11, 16, 19. Find the area of this triangle.

Find the shortest distance from $(6,7)$ to the line $y=\frac43x-5$.