TIMO - 2017 - Grade 7 - Heat Round - Paper 1

There are four statements on a piece of paper, and only one is correct. Write the letter before the correct statement on the answer sheet.

Split 21, 24, 36, 38, 65, 77, 95 and 143 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 36.

There are some 1 dollars, 2 dollars and 5 dollars coins. The number of 1 dollars coins is 12 times of that of 2 dollars coins. The total value of coins is 100 dollars. Find the number of 5 dollars coins.

John wrote a 4-digit number on a paper and asked Peter to guess it. Peter asked: “Is the number 8921?” John replied: “Three digits are correct. The positions of those digits are also correct.” Peter asked again: “Is the number 4782?” John replied: “Two digits are correct, but the positions of those digits are both wrong.” Peter asked again: “Is the number 3501?” John said: “Two digits are correct. The positions of those digits are also correct.” What is the number written by John?

There are 42 students attending the reading lesson. They have borrowed 260 books in total from the school library. Everyone can borrow at most 7 books. At least how many students have borrowed 7 books?

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

Find $1\times2+2\times3+3\times4+\cdots+20\times21$.

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

If $a,b$ are positive integers and $\sqrt a+\sqrt b=\sqrt{36+\sqrt{1292}}$, find $a+b$.

Find $\frac1{512}+\frac1{256}+\frac1{128}+\cdots+256+512$.

Find the number of positive factors of 2020.

The ratio of the sum, the difference and the product of two positive integers is $10:4:105$. Find the greater number.

Split 23 into a sum of integers, for example $(12,11)$, $(20,3)$ or $(18,2,3)$, so that their product is largest. Find that product.

Find the number of digits of $A$ if $A=\underbrace{999\ldots999}_{2016}\times\underbrace{666\ldots666}_{2016}$.

Find the prime numbers $p$ for which $p+36$ is a perfect square.

The area of triangle ABC is 72cm² and is twice of the area of parallelogram DEFC. Find the area of the shaded region in cm².

A triangle has sides with lengths 15cm, 20cm and 25cm. Find the area of the triangle in cm².

The figure shows triangle $ABC$, with $P,Q$ on $AB,BC$ respectively, such that $AP:PB=1:1$ and $BQ:QC=2:5$. If the area of triangle $APQ$ is 66 cm², find the area of triangle $ABC$ in cm².

If the sum of all interior angles of an $n$-sided polygon is $1800^\circ$, find $n$.

A solid cuboid is formed by merging 1331 cubes with side length 1cm. Find the total surface area of the cuboid when the cuboid is arranged in a way that the total surface area is at its minimum.