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

How many positive integers less than 500 end in digit 7 and have an even number of positive divisors?

There are 34 marbles. A and B alternate taking 1, 2 or 3 marbles, with A starting. Whoever takes the last marble wins. A first takes three. How many should B take on his first turn to guarantee a win?

36 children numbered 1–36 sit clockwise around a circle. Starting at child 1, count clockwise and remove every third child, continuing the count at the next remaining child after each removal. What is the number of the last child?

Each competitor plays 30 chess games, earning 2 points for a win, 1 for a draw and 0 for a loss. How many different final scores are possible?

Mr Yellow, Mr Blue, Mr Black and Mr White wear yellow, blue, black and white shirts, one colour each. Nobody wears his surname's colour. Mr White is older than Mr Blue; the blue-shirted man is older than Mr White; the black-shirted man is youngest. What colour does Mr White wear?

$\alpha,\beta$ are nonzero roots of $x^2-20x-30=0$. The numbers $\frac{\alpha+\beta}{\alpha^2\beta}$ and $\frac{\alpha+\beta}{\alpha\beta^2}$ are roots of $x^2-Sx+T=0$. Find $S+T$.

Find the coefficient of $x^4$ in $(1-2x+3x^2)^3$.

Convert the repeating base-three number $(12.020202\ldots)_3$ to a reduced fraction in base ten.

Solve $\log_5(2x+7)-\log_5 x=2$.

For real $x$, find the minimum of $\frac{x^2-3x+4}{x^2+3x+4}$.

Find the greatest integer strictly less than $(3+\sqrt8)^3$.

Find the remainder when $2015^{2018}$ is divided by 3.

Three distinct primes have product equal to 11 times their sum. How many unordered triples of primes satisfy this condition?

How many perfect squares are multiples of 44 and at most 201800?

An integer $k$ satisfies $5<k<20$. The equation $x^2-2(2k-3)x+4k^2-14k+8=0$ has two integer roots. Find $k$.

$A,B,C,D$ lie on a circle. $X,B,A$ are collinear, as are $X,D,C$. Given $XA=18$, $XB=10$, $XC=24$, find $XD$.

Right triangle $ABC$ has $AB=7$, $BC=24$, $AC=25$. $AD$ bisects $\angle BAC$. Find the area of $ADC$.

For real $x,y$, find the minimum of $\sqrt{x^2+y^2}+\sqrt{(x-3)^2+(y-4)^2}$.

For acute $x$, find the maximum of $9\sin^2x+6\cos x+7$.

Find the sum of the slopes of the two angle bisectors of $2x+y=0$ and $2x-4y-3=0$.