IKMC 2027 - Grades 7-8 - Self-study SET 5 - Thematic Batch 3

The four examinations and their candidates are:
– Mathematics: An, Bình and Cường.
– Literature: Bình, Yến and Phương.
– English: Yến, Phương and Giang.
– Physics: An, Cường and Giang.
What is the smallest number of examination sessions so that no candidate takes two examinations at the same time? A session may contain one examination or several simultaneous examinations.

John cuts a 3 × 3 block from the May 1918 calendar, with each of its nine cells containing a date from 1 to 31. The three dates on the main diagonal are all prime. What is the date in the middle cell?

Editorial note: Use the usual Monday-first calendar convention. The source also prints an unrelated triangular-grid figure; it is retained separately for source fidelity and is not needed for this calendar problem.

Georg builds the solid shown from small cubes. How many more small cubes must he add to complete a large cube whose edge is three times a small cube’s edge?

The shortest road from Atown to Cetown passes through Betown. Two signposts stand along this road. What distance was written on the broken sign?

Using the figure, find $\angle BCD$.

Which of the following numbers is greatest?

A glass jar contains 1000 g of a 10% poison solution. The solution is poured into another container, leaving 0.1% of the solution in the jar. The jar is then rinsed repeatedly with distilled water. At each rinse the poison mixes completely with the water, but after pouring, 0.1% of the mixture remains. How many rinses are needed to leave at most $1/10^{12}$ g of poison, if each rinse uses 1000 g of distilled water?

How many solids can be built from six equal-sized cubes in three different colours if their top view is as shown and cubes of the same colour are stacked together?

Boo’s turtle family initially has 12 members and prepares food for a 20-day hike up Fansipan. Some of Boo’s friends join the trip, and the food now lasts only 15 days. How many friends join, assuming everyone eats the same amount per day?

Two smaller squares A and B lie inside a large square as shown. The side of A is half the side of the large square. The side of B is one third of the large square’s diagonal. If B has area 16, what is the area of A?

Given $2x+2y=14$ and $x^2-y^2=21$, find $x-y$.

Points P,Q,R,S,T,U,V,W trisect the four sides of the quadrilateral as shown. K is an interior point. Given $S_{APKW}=8$, $S_{PKSB}=10$ and $S_{DWKT}=18$, find $S_{KSCT}$.

Kangaroos in four countries are counted. Country I has 10.4% of the total, country II 39.25%, country III 30.1%, and country IV 20.25%. What is the smallest possible total number of kangaroos?

A cube has one red, one green and one yellow face; its other three faces are white. A3 ×3 grid has six cells numbered 1 to 6 as shown. First roll the cube so that the green face touches cell 1. Next roll it so that the red face touches cell 2. Continue rolling through the numbered cells until the cube is on cell 6. What colour is then on top?

ABCD is a square and CED is an equilateral triangle, as shown. Find $\angle AEB$.

Two hikers set off from the same wooden cabin along the same road at the same time. Their average speeds are 3.5 km/h and 4.5 km/h. How many kilometres farther has the faster hiker travelled after 2 hours?

Original printed choices: A) 1.3; B) 2.6; C) 8.3; D) 9.6; E) 16.6

Editorial task: Solve the literal problem, identify the source discrepancy, and explain any intended correction.

ABCD is a square of side 1 cm. Find the area of rectangle ACFE shown.

A game drops a ball through staggered rows of pins as shown. At each pin the ball bounces either left or right. In how many ways can it land in bin B?

Kangarina discovers that someone has eaten her jar of honey. She suspects four neighbours: Anty the antelope, Beary the bear, Foxy the fox and Leo the lion. Anty says Beary always tells the truth. Beary says Leo ate the honey. Foxy and Leo each deny eating it. Who ate the honey if exactly one of the four neighbours says something true?

Dala and Hillary walk along the Kakada trail and return along it. Exactly three fifths of their walking time is spent going uphill, and the rest going downhill. Their average downhill speed is 6 km/h. What is their average uphill speed?