TIMO - 2019 - Grade 9 - Preliminary Round - Paper 3

According to the pattern $1,2,6,12,25,48,?,\ldots$, find the missing number.

There are n lines cutting a circle which separate the circle into 29 distinct parts, find the minimum value of n.

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

A competition has 20 questions. A correct answer earns 3 marks, a wrong answer loses 1 mark, and a blank answer earns 0 marks. Find the minimum number of candidates to ensure that 3 have the same score.

There are 15 red socks, 21 white socks and 17 blue socks. At least how many socks must be drawn at random to obtain 2 pairs that are not white and 2 pairs that are not red?

Find the largest value of $x$ given $x=3-\frac2{3-\frac2{3-\frac2{3-\cdots}}}$.

Let $x,y$ be real numbers with $x^2+y^2+5=4x+2y$. Find $x-y$.

If $\sqrt{18-12\sqrt2}=a\sqrt3-\sqrt b$ where $a,b$ are positive integers, find $a+b$.

If $x^4-8x^3+mx^2-nx+8$ is divisible by $x^2-6x+8$, find $m$.

Find the greatest value of $-x^2+12x-19$.

The 6-digit number $\overline{20A1B8}$ is divisible by 12 and $A<B$. Find the smallest value of $A+B$.

For real $x$, find the maximum value of $A=\frac{10}{\sqrt{x^2+14x+99}}$.

Given $7x\equiv107\pmod{11}$, find the minimum integral value of $\lvert x\rvert$.

It is May now. Which month will it be after $7^{20}$ months?

Let $x,y$ be real numbers with $x-y=6$ and $x^2+y^2=40$. Find $2x^3-2y^3$.

Find the area enclosed by the x-axis, y-axis and the line $x=9-\frac38y$.

Peter goes northeast for 8km, then goes northwest for 4km and goes southwest for 6km. How far is he now from the original position?

An interior angle of a regular $n$-gon is 12 degrees more than six times an exterior angle. Find $n$.

Find the area of the triangle with vertices $A(-3,6)$, $B(-1,3)$, $C(1,-5)$.

Find the shortest distance from $(2,5)$ to the line $4y=3x-6$.