TIMO 2017 Grade 6 Heat Set 5

There are 4 statements on a piece of paper as shown below, and only one of them is
correct. Write down the alphabet before the correct statement on the answer sheet.

Split 14, 24, 60, 65, 77, and 143 into 2 groups of 3 numbers such that the products of
numbers in both groups are equal. Find the sum of all numbers in the group that
contains 14.

A cricket has 6 legs and a spider has 8 legs. There are 12 animals and 90 legs. Find the
number of spiders.

John wrote a three-digit number. Peter guessed 123: one digit and its position are correct. He guessed 456: two digits are correct but both positions wrong. He guessed 354: all three digits are correct but all positions wrong. What number did John write?

There are 51 students in Grade 6. Among them, there are 22, 17 and 18 students who like
playing basketball, volleyball and football respectively. There are 5 students like playing
basketball and volleyball; 9 students like playing volleyball and football; 10 students like
playing basketball and football. 3 students like all three ball games. Find the number of
students who likes none of the ball games.

Find the value of 2+6+18+54+162+486+1458+4374+13122+39366+118098.

Calculate 1285²−285².

Find the value of 1+4+5+9+...+157+254+411. (Starting from the third number, each is the sum of the previous two.)

Find 1/2+1/6+1/12+1/20+1/30+1/42+1/56.

Convert 0.7868686... into the simplest fraction.

Find the sum of all positive factors of 2018.

When a natural number is divided by 20, the remainder is 4. When the number is
divided by 39, the remainder is also 4. When that number is divided by 65, what is the
remainder?

Split 9 into some integers (e.g.(8,1) / (2,7) / (4,2,3)...) so that the product among those
numbers is the largest. Find the value of that product.

When a 4-digit positive integer is divided by 3, 11, or 61 the remainders are all 2. Find
the least possible value of it.

A page has been torn from a book with even number of pages. The sum of page numbers of
the remaining pages is 761. Find the sum of two page numbers written on the torn page.

The perimeter of the shaded region in the attached figure is formed by arcs of 3 semi-
circles, the diameters of those semi-circles are 13m, 50m and 63m. Given that the 3
centers of the circles lie on the dotted line in the figure, find the perimeter of the shaded
22
region in meter. (Take π= ).
7

The figure below is formed by two squares ABFG and BCDE. If the length of CD and GA
are 16cm and 18cm respectively, find the area of ΔCEG in cm2.

The attached figure shows a triangle ΔABC , P and Q are points on AB and BC
respectively such that AP : PB = 1:1 and BQ : QC = 6:5 . If the area of ΔAPQ is 66cm2,
find the area of ΔABC in cm2.

The attached figure below shows a cubic container whose side length of the inner base is
20cm, and is filled with water with depth 16cm. A cuboid iron bar with a square base is
inserted into the container until the base of the cuboid touches the base of the container.
If the water level rose 9cm, find the side length of the square base of the iron bar in cm.

A solid cuboid is formed by merging 2000 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 maximum.