TIMO 2020 - Senior Secondary (Grades 10-12) - Heat Round - Paper 2

31 December 2020 was Thursday. What day of the week is 14 February 2026?

If the four-digit numbers $\overline{abcd}+\overline{bcda}+\overline{cdab}+\overline{dabc}=38885$, find $a+b+c+d$.

52 white, 66 yellow and 15 brown chopsticks are mixed together. How many must be drawn blindly to guarantee one nonbrown pair and three nonwhite pairs?

A competition has 50 questions. A correct answer earns 3 marks, a wrong answer loses 2, and a blank answer earns 0. How many candidates guarantee three with the same score?

Find the missing term in $2,6,12,20,30,42,\ ?,\ldots$.

Find $x$ if $\log_2(x^2)+\log_2\left(8^{\log_2 x}\right)=10$.

Factorise $x^3-7x+6$.

Simplify $\sqrt{21+6\sqrt{10}}$.

Find the coefficient of $x^3$ in $(3+x-2x^2)^4$.

Find the largest integer $p$ such that $(7-p)x^2+2px-p=0$ has no real root.

Today is Sunday. What day will it be after $201^{199}$ days?

Find the smallest positive integer $x$ satisfying $4x\equiv2\pmod9$ and $7x\equiv2\pmod{13}$.

The ten-digit number $20A191060B$ is divisible by 33. Find the maximum of $A+B$.

The positive rational number $x$ satisfies $x=4+\cfrac{16}{6+\cfrac{16}{6+\cfrac{16}{6+\cdots}}}$. Find $x$.

Find the units digit of $A=7+2\cdot7^2+3\cdot7^3+\cdots+99\cdot7^{99}+100\cdot7^{100}$.

Find the area of quadrilateral $ABCD$ with $A(-7,-9)$, $B(-6,-5)$, $C(-3,-5)$, $D(10,-6)$.

Find the maximum of $\frac{1+\sin2x}{\sin x+\cos x}$.

Find the area bounded by $2x+12=3\lvert y\rvert$ and the line $y=2x-4$.

$O$ is the centroid of the large triangle, whose area is 174. Find the shaded area.

A square has side 12. The smaller circle's radius is half the larger circle's radius, as illustrated. Find the smaller circle's area in exact form using radicals and $\pi$.