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

Michael has a box of $n$ balls labelled $1,2,\ldots,n$. His brother removes one ball. The sum of the labels on the remaining balls is 210. Find $n$.

1 November 2020 was Sunday. What day of the week is 1 November 2030?

Find the next term of the sequence $2,9,28,65,126,217,344,\ldots$.

A mathematics test has 10 questions. Each correct answer earns 2 marks, a blank answer earns 0, and a wrong answer loses 2 marks. What is the minimum number of candidates that guarantees two candidates with the same score?

There are 10 black, 20 yellow and 30 blue socks mixed together. Socks are drawn at random with eyes closed. How many socks must be drawn to guarantee two pairs that are not black? The socks in each pair must have the same colour.

For a real number $a$ satisfying $a^2-5a-1=0$, compute $a^4-25a^2-10a+5$.

Compute $1\times2\times3+2\times3\times4+3\times4\times5+\cdots+13\times14\times15$.

For real numbers $x,y$, find $x+y$ when $x^2-12x+3\lvert2x-3y\rvert$ attains its minimum.

Simplify $\sqrt{33-12\sqrt6}$.

How many integers $x$ satisfy $54\le x^2-3x\le180$?

Find the number of positive divisors of 2020.

Find the units digit of $1+3^1+3^2+3^3+\cdots+3^{2020}+3^{2021}$.

A three-digit number $n$ leaves remainder 5 on division by 17 and remainder 7 on division by 15. How many possible values of $n$ are there?

Let $A=\underbrace{11\cdots1}_{2020\text{ digits}}\times\underbrace{88\cdots8}_{2020\text{ digits}}$. How many digits does $A$ have?

How many primes $p$ make $p+81$ a perfect square?

2020 equilateral triangles of side length 1 cm are joined to form a quadrilateral. Find its largest possible perimeter, in cm.

One interior angle of a convex $n$-gon is selected. The sum of the other $n-1$ interior angles is $2020^\circ$. Find the selected angle.

Find the shortest distance from $A(0,1)$ to the line $y=-x$.

How many diagonals does a convex 20-gon have?

A circle is inscribed in square $ABCD$, as illustrated. $M,N,P,Q$ are the midpoints of $AB,BC,CD,DA$, respectively. Find the area of the white region if the shaded region has area $43\,\mathrm{cm}^2$. Use $\pi=3.14$.