TIMO - 2017 - Grade 8 - Heat Round - Paper 5

Selena’s dessert is stolen, she asked her 4 guests and received replies as the followings: 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 the truth, who stole Selena’s dessert?

Split 85, 104, 119, 121, 143, 161, 176 and 230 into 2 groups of 4 numbers such that the products of numbers in both groups are equal. Find the sum of the numbers of the group that contains 104.

John writes a 4-digit number and asks Peter to guess it. Peter asks, “Is it 4321?” John replies, “Three digits are correct and their positions are also correct.” Peter then asks, “Is it 2420?” John replies, “Two digits are correct, but both are in the wrong positions.” What number did John write?

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 to ensure that she has at least 25 pens of the same colour?

Exactly one of the following five statements is correct. Write the letter of the correct statement.

Given $a^2+a-3=0$, find $a^4+7a$.

Calculate $1\times2\times3+2\times3\times4+3\times4\times5+\cdots+20\times21\times22$.

If $(x+2)^4=bx^4+cx^3+dx^2+ex+f$, find $b-c+d-e+f$.

For positive integers $a,b$ satisfying $\sqrt a-\sqrt b=\sqrt{36-\sqrt{1292}}$, find $a-b$.

Calculate $1+\frac13+\frac19+\frac1{27}+\cdots$.

Find the number of positive factors of 2016.

Find the last two digits of $7^{2016}$.

Split 25 into some integers (e.g.(12,13)/ (20,5)/ (18,4,3)...) so that the product among those numbers is the largest. Find the value of that product.

Find the number of digits of $A=\underbrace{999\cdots999}_{2016\text{ digits}}\times\underbrace{666\cdots666}_{2016\text{ digits}}$.

Find the prime number $p$ for which $p+64$ is a perfect square.

In the figure, rectangle ABCD is formed by 7 smaller identical rectangles. The perimeter of the rectangle ABCD is 340 cm. Find the area of the rectangle ABCD in cm².

In a convex $n$-sided polygon, one interior angle is selected. If the sum of the remaining $n-1$ interior angles is $2016^\circ$, find $n$.

In the figure, $ABCD$ is a trapezium with $AB\parallel DC$, $\angle ADC=\angle AEB=90^\circ$ and $CD=2AB$. Find $\frac{AC}{BD}$.

A triangle has sides with lengths 13cm, 14cm and 15cm. Find the area of the triangle in unit cm².

Find $A+B+C+D+E+F+G$ in the figure below.