TIMO - 2018 - Grade 9 - Heat Round - Paper 4

Non-zero digits $A,B,C$ satisfy: $\overline{BAC}$ is divisible by 37, $\overline{BCA}$ is divisible by 5, and $\overline{ABC}$ has an odd number of factors. Find $\overline{CBA}$.

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

10000 children, numbered 1 to 10000, sit around a circle in order. Each child has an integer in hand. The child numbered 1 has the integer 1. Given that the sum of the integers of any 2017 consecutive children is equal to 2017. What is the integer held by the child numbered 1874?

There are 20 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 3 candidates will have the same scores in the competition.

Use numbers 1 to 5 to fill in the boxes in the table below such that all five numbers in every row, every column and 2 diagonals are different. What is the number that we should fill in the box X?

Let $\alpha,\beta\ne0$ be the roots of $x^2-2017x+1=0$. If $\frac1\alpha,\frac1\beta$ are the roots of $x^2-Sx+1=0$, find $S$.

Factorize $(x+y+z)^3-x^3-y^3-z^3$.

For real $x$, if $4^x+4^{x+1}=2^x+2^{x+2}$, find $x$.

Find the greatest integer $a$ such that $ax^2+x+2=0$ has real roots.

Given $0\le x\le2$ and $y=4x^2(2-x)+8x(2-x)^2+4(2-x)^3$, find the greatest value of $y$.

The 6-digit number $\overline{20A17B}$ is divisible by 72. Find $A+B$.

For real $x$, find the minimum value of $\sqrt{x(x+2)+82}$.

How many positive integers $x$ make $\sqrt{2017-x}$ an integer?

Real numbers $x,y$ satisfy $x^2+y^2=x+y$. Find the maximum value of $x+y$.

Real numbers $x,y$ satisfy $x+y=2$ and $x^2+y^2=5$. Find $x^3+y^3$.

Find the area enclosed by the x-axis, y-axis and the line $2x-y+8=0$.

Draw a square and a regular dodecagon (12-sided polygon) both inscribed in one circle. If the area of the dodecagon is 12 cm², find the area of the square in cm².

An exterior angle of a regular $n$-gon is $5^\circ$. Find $n$.

A triangle in the figure below has sides with lengths 3, 5 and 7. Find the radius of the inscribed circle of that triangle.

Square $OBCD$ has vertices $O(0,0)$, $B(0,6)$, $C(6,6)$, $D(6,0)$. Point $P$ lies inside it. Find the minimum value of $PO+PB+PC+PD$.