VTMO 2026 - Grade 5 - Final - SET 8 - Self-study

Last week, Bien Dong Hotel was closed for repairs on Monday and Tuesday. Rooms could be booked on the other five days. On Saturday, 18 rooms were booked, strictly more than on any other day. On Wednesday, 15 rooms were booked, strictly fewer than on any other day. Each room was booked on exactly four of the five days. How many rooms does the hotel have?

Pi thinks of a two-digit number with all three properties:
(a) Subtracting 4 gives a number divisible by 4.
(b) Subtracting 5 gives a number divisible by 5.
(c) Subtracting 6 gives a number divisible by 6.
What is Pi’s number?

A box contains seven cards numbered 1 to 7. The first person shakes out three cards, and the second person then shakes out two cards from those remaining. The first person says, “I know for certain that the sum of your two cards is even.” What is the sum of the first person’s three cards?

A square is divided into six congruent rectangles in the arrangement shown. The sum of the perimeters of all six rectangles is 100 cm. What is the area of the square, in square centimetres?

An auditorium has 12 rows of 10 seats each, and 108 students are seated. What is the greatest number of rows you can select such that the selected rows all have different numbers of students?

The shaded figure has a boundary made of horizontal and vertical segments, with all angles right angles. The given lengths are shown. If its area is $A$ and its perimeter is $P$, find $A+P$ (using the numerical values in the given units).

Each symbol in the grid represents a different number. The sum of each row is shown on the right. Find the values of the heart, triangle, square and diamond.

Rectangle $ABCD$ has area $600\text{ cm}^2$. Point $F$ lies on $BC$, $D$ is the midpoint of $EG$, and $AEGF$ is a trapezoid as shown, with $AE$ parallel to $FG$. Find the area of $AEGF$ in square centimetres.

Each minute, machine A makes 24 more items than machine B. The number A makes in 25 minutes is twice the number B makes in 20 minutes. How many items does machine A make each minute?

The large square is divided into smaller rectangles as shown. The sum of the perimeters of the shaded rectangles is 48 cm. Find the area of the large square in square centimetres.

Each $2\times2$ square pattern represents a digit, and each row of three patterns represents a three-digit number. The four numbers are 913, 895, 628 and 238, in some order. Find the value of $B$.

Two boats travel between points A and B on the same river.
• Boat 1 takes 90 hours downstream from A to B and 210 hours upstream from B to A.
• Boat 2 takes 35 hours downstream from A to B and 45 hours upstream from B to A.
What is the ratio of boat 1’s speed in still water to boat 2’s speed in still water?

Practice convention: the river current is constant, and each boat keeps a constant speed relative to the water on both journeys.

The three pictures show different views of the same die. Which letter is on the face opposite O?

Enter the answer code:
• 0001 if the answer is V.
• 0002 if the answer is T.
• 0003 if the answer is M.
• 0004 if the answer is O.
• 0005 if the answer is N.
• 0006 if the answer is another character.