IMAS 2017 — Middle Primary Grades 3–4 — Round 2

Arranged 30 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?

It is known that $ A \times B \times C = 30 $, $ B \times C \times D = 90 $ and $ C \times D \times E = 120 $, what is the value of $ A \times C \times E $?

Replace each $\Delta$ by "+" or "-" in the expression $1\Delta2\Delta3\Delta8\Delta15$ and compute the value of the expression. How many different positive integers can we have for the expression?

A rectangle is divided into 6 identical squares as shown in the figure below. Color four squares black such that each row has at least one black square. Two color methods are considered the same if they are identical after rotation. How many different color methods are there?

Two cars start moving from opposite ends of a road towards each other at a constant speed. The faster car travels at 40 km per hour. After two hours, the faster car is 20 km past the midpoint and 6 km away before meeting the slower car. What is the speed of the slower car in km per hour?

Select three different numbers from 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 such that their average is 5. How many different possible combinations are there?

The pages of a book begin from No. 1, 2, 3, … such that two consecutive numbers appear on both sides of a single page. When one page is torn, the sum of the remaining page numbers is 1133. What is the sum of those two numbers on the torn page?

The number of boys in a class is twice as the number of girls. In a math test, the average score of the class is 86, but average score of the girls is 90. What is the average score of the boys?

The road map of a certain city consists of $3\times3$ squares with each of the smaller squares having a side length of 20 meters as shown in the figure. A street sweeper starts cleaning from point $A$ and every road thereafter until finally returning to $A$. What is the shortest distance, in meters, in which he can do the task?

The perimeter of rectangle $ ABCD $ is 34 cm. Draw four squares extending outward from its four sides, as shown in the figure below. If the sum of the areas of the four new squares is $ 338 \, cm^2 $, what is the area, in $ cm^2 $, of rectangle $ ABCD $?

Millie has 60 red beads, 50 black beads and a magic machine. Each time 4 red beads are inserted into the machine, it gives out 1 black bead and for every 5 black beads is inserted, the machine gives out 2 red beads. After having operated the machine 30 times, Millie has no more red beads. Find the number of black beads she has now.

A 3-digit number $ \overline{abc} $ is said to be a lucky number if $ a \times b \times c $ is also a three-digit number. What is the least possible lucky number $ \overline{abc} $?

Rotate an equilateral triangle inscribed in a circle 40 degrees clockwise and counter-clockwise, as 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? Please draw the centrally symmetric picture.

There are 12 different positive integers which satisfy the condition that the product of every 5 numbers is even and that the sum of all 12 numbers is odd. Find the least possible sum of these 12 positive integers.