TIMO 2019 - Senior Secondary (Grades 10-12) - Preliminary Round - Paper 3

Find the missing term in $2,6,15,28,55,78,\ ?,\ldots$.

$n$ straight lines cut a circle into 232 regions. Find the minimum possible $n$.

Primes $x,y,z$ satisfy $6x^y+y^x=3z$. Find the minimum possible $z$.

A competition has 31 questions. A correct answer earns 2 marks; a wrong or blank answer loses 2. How many candidates are needed to guarantee three with the same score?

A box contains 17 red, 13 white and 16 blue socks. How many socks must be drawn at random to guarantee two pairs that are not red and three pairs that are not blue?

$2x^4-19x^3+mx^2-nx+36$ is divisible by $x^2-4x+3$. Find $m+n$.

If $\sqrt[2019]{x}+\frac1{\sqrt[2019]{x}}=2$, compute $x^{2019}+\frac1{x^{2019}}$.

Find $x$ if $3x=2+\cfrac5{4+\cfrac5{4+\cfrac5{4+\cdots}}}$.

Find the smallest integer $k$ for which $(k^2-5)x^2+kx-2=0$ has no real root.

Factorise $2x^4+5x^3-11x^2-20x+12$.

Find the smallest positive integer $x$ satisfying $4x\equiv3\pmod{11}$ and $6x\equiv2\pmod8$.

It is September now. What month will it be after $9^{2019}$ months?

For real $x$, find the maximum of $\frac1{\sqrt{2x^2+32x+199}}$.

Find the largest integer $n$ for which $\frac{4000!}{15^n}$ is a positive integer.

The ten-digit number $20190316AB$ is divisible by 66. Find the largest possible $A\times B$.

Find the area bounded by the $x$-axis and the lines $x=2y-3$, $2x+y=9$.

Find the area of quadrilateral $ABCD$ with $A(-3,5)$, $B(5,7)$, $C(6,-2)$, $D(-2,-1)$.

A wire is cut and bent into three circles, each of radius 21. If the same wire is instead made into six identical triangles, what is the maximum area of one triangle? Use $\pi=22/7$.

Peter travels 12 km east, then $12\sqrt2$ km southwest, then 16 km east. How far is he from his starting point?

A cuboid container is 15 cm long, 10 cm wide and 18 cm high. Its water is 3 cm deep. Eight cubes of side 5 cm are placed inside. What is the largest possible rise in water level, in cm?