TIMO 2017 - Senior Secondary (Grades 10-12) - Heat Round - Paper 5

Selena's dessert was stolen. Hera says David stole it. David says Hera stole it. Ruby says she did not steal it. Steve says David is lying. Exactly one person tells the truth. Who stole the dessert?

Primes $x,y,z$ satisfy $4x^y+y^x=z$. Find the minimum possible $z$.

A rectangle is divided into $3\times5$ identical squares. Each is painted red or green. How many colourings have no rectangle of area at least four small squares whose four corner squares share one colour?

A box contains 100 pens: 40 blue, 30 red, 20 green and 10 yellow. Ivy draws pens without seeing their colours. How many pens must she take to guarantee at least three colours?

Five weights $A,B,C,D,E$ satisfy $A+B=7$, $B+C=9$, $C+D=11$, $D+E=8$, $A+C+E=10$, all in pounds. Find the weight of $E$ in pounds.

Factorise $x^2-2xy+y^2+x-y-6$.

Given $a^2-2a+3=0$, compute $a^4-a^2+6a+1$.

Given $C_n^r=\frac{n!}{r!(n-r)!}$, compute $C_7^1+2C_7^2+3C_7^3+\cdots+7C_7^7$.

For $x>0$ satisfying $x+1/x=12$, compute $x^3+1/x^3$.

Compute $2^2+5^2+8^2+\cdots+44^2+47^2+50^2$.

Find the greatest common divisor of $2^4\cdot3^2\cdot5^2$, $2^6\cdot3\cdot5^3\cdot7^4$ and $2^3\cdot3^2\cdot5^3\cdot11$.

Find the largest negative integer $x$ satisfying $4x\equiv8\pmod{20}$.

For real $x$, find the maximum of $\frac1{\sqrt{x^2+4x+8}}$.

Real numbers $a,b,c$ satisfy $a+b+c=6$. Find the minimum of $a^2+b^2+c^2$.

Let $f(n)=4n^3-6n^2+4n-1$. Find the units digit of $f(1)+f(2)+\cdots+f(2016)$.

$M,N,E$ are the midpoints of $BC,MC,AC$, respectively, as illustrated. If $BE=42$ cm, find $GH$ in cm.

A line through $A(6,1)$ intersects the circle $x^2+y^2+2x-8y-4=0$ at $B,C$. Find $AB\times AC$.

A triangle of side lengths 8, 15 and 17 is inscribed in a square. Find the minimum possible side length of the square.

Find the shortest distance from $(1,2)$ to the line $y=x$.

Quadrilateral $ABCD$ has $AB=15$ cm, $BC=7$ cm, $CD=20$ cm, $DA=25$ cm. Let $x$ be the angle between the diagonals with $0^\circ\le x\le90^\circ$. Find $x$.