Đề thi toán IMC 2024 Key stage II

Individual Contest

Time limit: 90 minutes

English Version

Elementary Mathematics International Contest

Information:  You are allowed 90 minutes for this paper, consisting of 15 questions

to which only numerical answers are required.  Each question is worth 10 points. No partial credits are given. There

are no penalties for incorrect answers, but you must not give more than the number of answers being asked for. For questions asking for several answers, full credit will only be given if all correct answers are found.  Diagrams shown may not be drawn to scale.

Instructions:  Write down your name, your contestant ID and your team’s name on

the answer sheet.  Enter your answers in the space provided on the answer sheet.  You are allowed to use HB, B, and 2B pencils and ball-point pens with

black or blue ink.  You cannot use instruments such as protractors, calculators and

electronic devices.  At the end of the contest, you must hand in the envelope containing the

question sheets, the answer sheet and all scrap paper.

India International Mathematics

Competition 2024

Lucknow, 26th to 31st July 2024

1. There are some closed boxes, each box containing either a silver coin or a gold

coin. It is known that 2

5 of all the coins inside the boxes are silver and the rest

are gold. After opening 1

4 of the closed boxes, we observed that 1

3 of them

contain silver coins. What fraction of the remaining closed boxes contain silver coins? 2. A number is called primalistic if it can be expressed in the form of 1 pqr , where p, q and r are three different prime numbers. What is the primalistic number that is closest to 360? 3. A box contains a sufficient amount of red, blue and yellow balls, each of which

has a value of 3, 5 and 6 points, respectively. Larry draws some balls from the box such that the number of yellow balls he draws is less than both the number of red balls drawn and the number of blue balls drawn. If Larry draws the smallest possible number of balls from the box, such that at least one ball of each colour is drawn and the total value of the drawn balls is 66 points, what is the number of red balls drawn by Larry? 4. When a positive integer is divided by a positive integer divisor, the remainder is

26. When the same positive integer is doubled and then is divided by the same divisor, the remainder is 7. What is the value of the divisor? 5. What is the value of x in the following equation:

25 101 ... 4 9 9 14 14 19 2019 2024 5 10 15 ... 200 x x x x               ?

6. Fill in each of the fifteen circles with the numbers from 1 to 15, where each

number can be used exactly once, so that the sums of all the numbers on each side of the pentagon are all equal. Let S be the greatest possible sum of each side and T be the least possible sum of each side. What is the value of S T  ? 7. Let a, b and c be three different digits such that

1 1 1 11 3321 ab bc bc ca ca ab       .

What is the numerical value of ab bc ca   ?