TIMO 2018 - Senior Secondary (Grades 10-12) - Heat Round - Paper 4

Nonzero digits $A,B,C$ satisfy: $BAC$ is divisible by 3, $BCA$ is divisible by 5, and $ABC$ has an odd number of positive divisors. Find $CBA$.

The mean, median, range and unique mode of 200 integers all equal 200. If $A$ is the largest integer, find the greatest possible $A$.

30000 children numbered 1–30000 sit in order around a circle. Each holds an integer; child 1 holds 2. Every 2017 consecutive children hold integers summing to 4034. What integer does child 24601 hold?

A competition has 25 questions. A correct answer earns 2 marks, a wrong answer loses 1, and a blank answer earns 0. How many candidates are needed to guarantee two with the same score?

Fill the grid using 1–5 so that every row, every column and each of the two diagonals contains five different numbers. Find the number in box $X$.

Nonzero $\alpha,\beta$ are roots of $x^2-2017x+1=0$. If $1/\alpha^2$ and $1/\beta^2$ are roots of $x^2-Sx+1=0$, find $S$.

Find the constant term of $\left(2x-\frac1x\right)^3\left(x+\frac1x\right)^4$.

For real $x$, solve $5^x+5^{x+1}=2^x+2^{x+1}+2^{x+2}+2^{x+3}$.

Find the largest integer $a$ such that $ax^2+(a+1)x+(a+2)=0$ has a real root.

For $0\le x\le2$, find the maximum of $y=4x^4+12x^3(2-x)+12x^2(2-x)^2+4x(2-x)^3$.

The six-digit number $20A17B$ is divisible by 56. Find $A+B$.

For real $x$, find the minimum of $\sqrt{x(x+2)(x+4)(x+6)+160}$.

How many positive integers $x$ make $\sqrt{2017-x}$ an integer?

Form four-digit numbers using each digit 1, 2, 3 and 4 exactly once. Find the sum of all these numbers.

Find the largest integer $n$ satisfying $n^{2000}\le7^{3000}$.

Find the area bounded by the $x$-axis and the lines $2x-y=0$, $2x+3y-8=0$.

A regular hexagon and a regular dodecagon are inscribed in the same circle. The dodecagon has area $12\,\mathrm{cm}^2$. Find the hexagon's area, in $\mathrm{cm}^2$, in radical form.

Let $O$ be the origin, $A_1,A_2,\ldots$ distinct points on $y=\sqrt{x}$ and $B_1,B_2,\ldots$ points on the positive $x$-axis. Triangles $OB_1A_1$, $B_1B_2A_2$, $B_2B_3A_3$, etc. are equilateral with side lengths $l_1,l_2,l_3,\ldots$. Find $l_1+l_2+\cdots+l_{300}$.

Acute angles $x,y$ satisfy $\sin y=3\cos(x+y)\sin x$. Find the maximum of $\tan y$.

Find the shortest distance from $(-2,26)$ to $5x-12y+10=0$.