TIMO 2021 - Senior Secondary (Grades 10-12) - Heat Round - Paper 1

Charlie travels 26 km east, 42 km south, 6 km north and 11 km west. How far is he from his starting point, in km?

A box contains 2020 balls coloured red, green, blue, yellow, white or black. Drawing any 1920 balls guarantees every colour. What is the minimum number to draw to guarantee four different colours?

17 February 2020 was Monday. What day of the week is 19 January 2028?

There are 38 candies. A and B alternate taking 1 to 4 candies, with A starting. Whoever takes the last candy wins. A first takes two candies. How many should B take on his first turn to guarantee a win?

Primes $x,y$ satisfy $x^2-3y^3=1$. Find $x+y$.

Solve $\log_6(x^2+11)+\log_6 18-2\log_2 x=3$.

Let $k=2^{2020}$. Find the remainder on dividing $x^{2020}-10x+18$ by $x^2-4$. Give the answer in terms of $k$.

Find the coefficient of $x^3$ in $(1+3x-x^2-x^3)^3$.

Find $x$ satisfying $x=\sqrt{x}+30$.

Factorise $2x^2+5xy-3y^2+14y-8$.

The nine-digit number $168A248BC$ is divisible by 66. Find the largest possible $A\times B\times C$.

Find the remainder when $222^{2020}$ is divided by 16.

The product of three distinct primes is 17 times their sum. Find the largest of the primes.

Find the greatest integer strictly less than $(2+\sqrt3)^4$.

Express $\left(\cos\frac\pi{1010}+i\sin\frac\pi{1010}\right)^{2020}$ in the form $a+bi$.

Find the area bounded by the $y$-axis and the lines $2x=y+8$, $3y=9-5x$.

Given $\tan x=\frac{\sqrt5}{2}$, find $\cos2x+\sin2x$.

The figure consists of a regular octagon and eight identical sectors. The octagon's side length is 2 and each sector's radius is 1. Find the shaded area in exact form using $\pi$ and radicals.

Find the sum of the slopes of the two angle bisectors of $5x+y-2=0$ and $x-2y-1=0$.

In the figure, $ABC$ is right-angled and $BED$ is isosceles with $BD=BE$. Also $BD=DC$ and $\angle BCD=a$. Find the ratio of the area of $BED$ to that of $ABC$, in terms of $a$.