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

Given that the mean, median, range and the only mode of 200 integers are also 100. If A is the smallest integer among those 200 integers, find the minimum value of A.

There are 14 questions in a mathematics competition. The scores of each question are allocated in the following ways: 3 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 4 candidates will have the same scores in the competition.

A box contains 150 coloured balls: 40 blue, 40 red, 50 green and 20 yellow. Sherry takes some balls from the box without looking at the colours of the balls. What is the least number of balls that she must take to ensure that she has 30 balls with the same colour?

15th January 2021 is Friday. Which day of the week will 29th February 2028 be?

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 in each day. They play against another team once in the tournament. So, there are 3 games every day. Given that: 1) Team A wins Team E on the first day. 2) Team A is defeated by Team D on the second day. 3) Team E wins Team F on the third day. 4) Team D is defeated by Team C on the fourth day. Which team does Team C play against on the fifth day?

Find the minimum value of $3x^2-24x+59$.

Factorize $4x^2-y^2+24x-12y$.

If $\alpha,\beta$ are the roots of $x^2-4x-1=0$, find $\alpha^2+\beta^2$.

If $4x^4+nx^3+mx^2+26x+21$ is divisible by $x^2-2x-3$, find $mn$.

Positive integers $x,y$ satisfy $\begin{cases}x+3y^2=57\\3y+xy=48\\x+3y=21\end{cases}$. Find $x-2y$.

How many reduced positive fractions less than 1 have denominator 1980?

Find the remainder when $2020^{2020}$ is divided by 13.

The 9-digit number $\overline{221A9C2B2}$ is divisible by 132. Find the maximum value of $A$.

If $\sqrt{17+\sqrt{288}}=a+b\sqrt c$ with positive integers $a,b,c$, find the maximum value of $\frac{a+b}c$.

Given $x>0$, find the minimum value of $15x^2-14x+\frac8{x^2}+22$.

Line $L$ passes through $A(8,-5)$ and has slope $-\frac57$. Find its x-intercept.

Find the area enclosed by the x-axis and the lines $6x-y+66=0$ and $3x-5y+6=0$.

The side lengths of a triangle are 14, 18, 28. Find the area of this triangle. (Answer in terms of surd form)

In the triangle shown, $\angle A=60^\circ$, $\angle B=75^\circ$, and $BC=\sqrt6$. If the area of $ABD$ is 2, find the shortest distance from $D$ to $AB$.

In the figure, $ABCD$ is a square with side $a$. Points $E,F$ lie on diagonal $AC$, and the areas of $ABED$, $BEDFC$, $DFC$ are in the ratio $1:3:2$. Find $EF$ in surd form in terms of $a$.