TIMO - 2017 - Grade 9 - Heat Round - Paper 5

Selena’s dessert is stolen, she asked her 4 guests and received replies: Hera: “David’s stolen it.” David: “Steve’s stolen it.” Ruby: “I have not stolen it.” Steve: “David is lying.” Given that only one of them had told lies, who stole Selena’s dessert?

If $x,y,z$ are prime numbers and $x^y+y^x=z$, find $z$.

Divide a rectangle into $3\times6$ identical small squares and colour each red or green. How many colourings have no rectangle of area at least four whose four corners are painted the same colour?

A box contains 100 coloured pens: 40 blue, 30 red, 20 green and 10 yellow. Ivy takes some pens from the box without looking at the colours of the pens. What is the least number of pens she must take if she wants to have pens with all 4 colours?

There are five standard weights A, B, C, D and E. Given the weight A and the weight B weigh 7 pounds in total, the weight B and weight C weigh 9 pounds in total, the weight C and weight D weigh 11 pounds in total, the weight D and weight E weigh 8 pounds in total, and the weight A, C and E weigh 10 pounds in total. How many pounds do all five weights weigh?

Find $1+\frac1{1+2}+\frac1{1+2+3}+\cdots+\frac1{1+2+3+\cdots+8+9+10}$.

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

Given $C_r^n=\frac{n!}{r!(n-r)!}$, find $C_1^6+2C_2^6+3C_3^6+\cdots+6C_6^6$.

Given $x>0$ and $x+\frac1x=12$, find $x^2+\frac1{x^2}$.

Find $1-3+9-27+\cdots+729-2187+6561$.

Find the Greatest Common Divisor of 34417 and 34959.

Find the smallest positive integer solution of $5x\equiv8\pmod{13}$.

For real $x$, find the maximum value of $\frac1{x^2+4x+5}$.

Given $y>0$, find the minimum value of $y+\frac1y$.

Positive integers $a,b$ satisfy $a<b$ and $a+b+ab=104$. Find the minimum value of $a$.

The triangle has sides with lengths 3, 5 and 7. Find the value of the largest interior angle of the triangle.

In the figure, $BD:DC=CE:EA=2:1$ and $AD$ intersects $BE$ at $P$. The area of $ABC$ is 70 cm². Find the area of $CEPD$ in cm².

If a right-angled triangle with side lengths 5, 12 and 13 can be inscribed in a square, find the minimum value of the side length of the square (express your answer in surd form).

A triangle has sides with lengths 16cm, 20cm and 24cm. Find the area of the triangle in cm².

In the figure, $HD$ is tangent to the circle at $C$, and $AE$ is a diameter. Lines $AE$ and $DC$ meet at $H$. Find $\angle HAC$ if $\angle ACD=75^\circ$.