AMO 2027 Grades 4-5 SET 19 - Self-study

Pi and John spent the same amount of money. Pi had 4136 dollars left, while John had 148 dollars left. If they had 7380 dollars altogether at first,

(a) How much did Pi spend?

(b) How much did John have at first?

Answer entry: Enter the integer dollar amounts for (a);(b), in that order, separated by one semicolon, without units, commas or spaces.

The diagram below shows square $ABCD$ such that

$$AE=EF=FB=BG=GH=HC$$
$$=CI=IJ=JD=DK=KL=LA=20\text{ cm}.$$

Find the area of square $MNPQ$ in cm².

Calculate the value of

$$T=1+2+3+\cdots+40.$$

a) The speed of a boat in still water is 15 km/h and the rate of current is 3 km/h. Find the distance travelled downstream in 30 minutes.

b) A man rows 18 km upstream in 5 hours and also 28 km downstream in 5 hours. Find the velocity of the stream.

c) If a man can swim downstream at 6 km/h and upstream at 2 km/h, find his speed in still water.

Motion convention: in each part the current and still-water speeds are constant, with the same still-water speed in both directions.

Answer entry: Enter the integer answers for (a);(b);(c), in that order, separated by semicolons, without units or spaces. Use km for (a) and km/h for (b) and (c).

The sum of the digits in a 2 - digit number is 12. If the tens digit is 3 times the ones digit, find the number.

Last year Mr. Jon Q. Public received an inheritance. He paid 20% in federal taxes on the inheritance, and paid 10% of what he had left in state taxes. He paid a total of 10,500 dollars for both taxes. How many dollars was his inheritance?

A pump can fill a tank with water in 2 hours. Because of a leak, it took $2\frac23$ hours to fill the tank. How many hours the leak can drain all the water of the tank?

Tank convention: filling starts with an empty tank; pump and leak rates are constant. For the requested draining time, start with a full tank and turn the pump off.

Mary and Jane played a game of chess everyday over a period of time. Whoever lost a game had to give the winner a piece of chocolate. In the end, though Jane won 14 games, she had to give 11 pieces of chocolates to Mary. How many games did Mary and Jane play in total?

Practice convention: no games were drawn. The 11 chocolates are Jane’s net payment to Mary after cancelling chocolates won against chocolates lost.

Danzel’s age is $2/9$ of his mother’s age. Three years ago, his mother was 42 years old. In how many years’ time will Danzel’s age be $2/7$ of his mother’s age?

In the following cryptarithm, all different letters stand for different digits. Find the product of A and B.

$$\overline{AB}\times B=\overline{AA9}.$$

Each letter is one base-ten digit; different letters have different digits, and leading digits A and B are nonzero.

The pie chart below shows the choice of preference of breakfast of 400 students in a school. How many students choose either bread or others for breakfast?

In one side of the road in front of my school, there are 505 trees. The trees are 10m away from each other. How far is the last tree away from the first one?

Measure distance along the ordered row of trees; the 10 m spacing is between consecutive trees.

After Theodore spent 25% of his salary and Sophie spent 30% of her salary, both of them had the same amount of money left. If Theodore’s salary was 2100, dollars what was Sophie's salary?

There are 14 rows of seats in a cinema. The first row has 15 seats, the second row has 17 seats, the third row has 19 seats, and so on. How many seats are there in the cinema altogether?

Continue the rule that each successive row has exactly 2 more seats than the previous row.

What will be the day on 26th January four years later if today is 26th January, 2021?

Use the Gregorian calendar and the original reference date 26 January 2021, not the collection year 2027. Give the weekday in English or Vietnamese.