IMAS 2017 — Upper Primary Grades 5–6 — Round 2

Arranged 80 triangles in a row and color them black and white in a pattern as shown below. How many more black triangles than white triangles are there?

The results of a math quiz of a certain class are as follows: 4 students got 100 points; the scores of 6 students are from 90 to 99; the scores of 18 students are from 80 to 89; while of 12 remaining students are from 70 to 79 and 10 students got below 69. The average of the class is 81.4. What is the total score of the class?

Let AB and CD be two perpendicular diameters of circle O. Draw two lines through any point P on the circle perpendicular to AB and CD, with intersections at F and E, respectively. If the diameter of circle O is 8 cm, what is the length, in cm, of EF?

The figure below is composed of 36 small equilateral triangles, with each having an area of 1 cm² . What is the area, in cm² , of triangle ABC?

After removing the decimal part of a certain positive number, 5 times the sum of the integral part and the original positive number is 22.1. What is the value of this positive number?

A box contains identical balls where 7 are black, 5 are white and 8 are red balls. What is the least number of balls that must be taken out from the box to get balls of each color?

In the figure, ABCD is a parallelogram, where E and F are midpoints of BC and CD respectively. Now connect AE, AF, DE, BF, BD. The area of ABCD is 4 cm² . With three of A, B, C, D, E, F as vertices and present line segments as sides, how many triangles of area 1cm² can you find in the figure?

A round table has 20 seats. Some seats are occupied such that a new person will always sit adjacent to someone wherever he is already seated. What is the least number of seats already occupied?

Rotate an equilateral triangle inscribed in a circle 40 degrees clockwise and counter-clockwise, as shown in the figure below. How many triangles are there in the figure?

A rectangle is divided into 12 unit squares such that 10 are white and 2 are black, as shown in the figure below. To form a centrally symmetric picture by adding some white squares but no black squares, what is the least number of white squares needed?

In the figure below, ABCD is a right trapezoid where $\angle ABC=\angle BCD=90^\circ$ , AB = 3 cm, CD = 9 cm. Points E and F are on CD and BC respectively. If BF = 2 cm and AE with EF divides the trapezoid into three parts with equal area, what is the area, in cm² , of ABCD?

A factory produces an order of parts. If the output per hour is 4 parts more than the original speed, the time spent is $\frac1{10}$ less than the originally estimated time. If the speed is 6 parts less than the original, the time spent is $\frac15$ more than the original estimate. How many parts does the factory originally produce per hour?

A three-digit number is said to be "lucky" if it is divisible by 6 and by swapping its last two digits will give a number divisible by 6. How many "lucky" numbers are there?

There is a sequence of five positive integers. Each number right after the first term is at least twice the number before it. If the sum of the five numbers is 2018, what is the least possible value of the last number?

Some chess pieces are put on a $8\times8$ chess board, with at most 1 piece in each square. After taking all pieces on any chosen 4 rows and 4 columns, there is at least 1 piece left on the board. Find the least number of pieces originally on the board.