Before you start the IVMO challenge, please read the following:
 There are 40 multiple choice questions in the challenge, each with 5 possible answers.
 Questions 1 – 25 each carry 2 marks, Questions 26 – 35 each carry 3 marks, and Questions 36 – 40 each carry 4 marks.
 The challenge is one hour long.
 Once you click on “Start” a clock will countdown and at the end of one hour your answers will be automatically saved and submitted.
 You can end the test before that by clicking on “Submit Test” on the final question.
 The grid of squares at the top will show you which questions you have answered.
 You can skip a question and come back to it later or revisit any question you have already answered by clicking on any question number in the grid.
 You will need paper for rough workings.
 You must not use a calculator or any other electronic device.
All the Best!!!
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Question 1 of 40
1. Question
2 point(s)Questions 1 – 25: Score 2 marks for each correct answer.
2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))))

Question 2 of 40
2. Question
2 point(s)Which of the following is not a multiple of 9?

Question 3 of 40
3. Question
2 point(s)A, B and C are three digits. B and C add up to 10.
A B x A C 4216 What is the value of A?

Question 4 of 40
4. Question
2 point(s)What is the whole number remainder when 242871 is divided by 897?

Question 5 of 40
5. Question
2 point(s)The working below shows the division, 259781 ÷ 11243 = 227 remainder 320, using the Vedic Transpose and Apply method. What are the digits for ABC?

Question 6 of 40
6. Question
2 point(s)2023 has a square factor whose square root ends in 7. What is that square root?

Question 7 of 40
7. Question
2 point(s)992²

Question 8 of 40
8. Question
2 point(s)What are the vinculum digits in the Nikhilam multiplication of 524 X 492 when using a working base of 500 and a real base of 1000?

Question 9 of 40
9. Question
2 point(s)What is the area of a rectangular floor measuring 15 ft 3 in. by 12 ft 9 in.? (1 foot = 12 inches)

Question 10 of 40
10. Question
2 point(s)Using Flag Division (Straight division) what is the third step in the calculation of 62995 ÷ 43?

Question 11 of 40
11. Question
2 point(s)79³

Question 12 of 40
12. Question
2 point(s)What are the last five digits in the recurring pattern of the decimal equivalent of 1 ? 49 
Question 13 of 40
13. Question
2 point(s)After the decimal point, how many nonrecurring digits are there in the decimal equivalent for 17 ? 384 
Question 14 of 40
14. Question
2 point(s)The digits 2, 3, 4, 5, 6 are placed in the grid to form two threedigit square numbers. Which number must be placed in the central square?

Question 15 of 40
15. Question
2 point(s)Images on a computer screen are made up of tiny dots called pixels. Each pixel is a square with edge length 1/96 of an inch. A certain screen has 2073600 pixels. What is its area in square inches?

Question 16 of 40
16. Question
2 point(s)Mr Elongate Musketeer owns 19 companies each with over ten thousand employees. Which of the following could be the total number of employees?

Question 17 of 40
17. Question
2 point(s)What fraction of the square is shaded?

Question 18 of 40
18. Question
2 point(s)What is the simplified form of, (2x^{2 }+3x +1)² −(x^{2 }−3x −1)² ?

Question 19 of 40
19. Question
2 point(s)Three identical rightangled triangles and a square are placed together with measurements as shown. What is the outside perimeter of the whole shape?

Question 20 of 40
20. Question
2 point(s)Which of the following is an equation of the straight line which passes through the point (3, 2) and also perpendicular to the line with equation, 5x −3y =19 ?

Question 21 of 40
21. Question
2 point(s)A set of five different integers have a mean value of 6 and their median is 5. The smallest number is 3 and when all five are multiplied the result is 4800. What is the largest of the five numbers?

Question 22 of 40
22. Question
2 point(s)What is the value of x for which, 2^{2x }×8^{x1} =16^{x+3} ?

Question 23 of 40
23. Question
2 point(s)A square has centre (3, 4) and one corner at (1, 5). Where is another corner?

Question 24 of 40
24. Question
2 point(s)Which of the following is a factor of 3x^{3} +11x^{2} +30x +72 ?

Question 25 of 40
25. Question
2 point(s)What is the radius of the circle with equation, x^{2} + y^{2} +2x − 4y +1= 0 ?

Question 26 of 40
26. Question
3 point(s)Questions 26 – 35: Score 3 marks for each correct answer.
Triangle ABC has vertices with coordinates, (3, 5), (6, 9) and (11, 7). In square units, what is the area of the triangle?

Question 27 of 40
27. Question
3 point(s)Two of the pyramids at Giza in Egypt are Khufu, with a height of 280 Royal Cubits, and Menkaure, with a height of 112 Royal Cubits. They can be treated as similar shapes. If the volume of Menkaure is 1.2 x 106 Cubic Royal Cubits, what is the volume of Khufu, in the same units?

Question 28 of 40
28. Question
3 point(s)Andria and Bertha both run 100 metres in times A and B seconds. The difference between the squares of their times is four times the sum of their times. The sum of their times is eight times the difference between their times. How long did it take for the slowest of the two runners?

Question 29 of 40
29. Question
3 point(s)Each square in this grid has side length one unit. Two circles, each with diameter one unit are placed in two opposite squares. What is the shortest distance between the two circles?

Question 30 of 40
30. Question
3 point(s)In this magic square, the products of numbers in each row, each column and each diagonal are all the same. The missing numbers are 2, 4, 5, 10, 25, 50 and 100. Which number must be placed in the square labelled M?

Question 31 of 40
31. Question
3 point(s)When a javelin is thrown it follows the path of a parabola and the maximum distance achieved is when the angle of throw is 45˚ to the horizontal. The graph of a record breaking throw is described by the equation,
y = x − x^{2} 98 What is the maximum height of this throw?
where, x and y are measured in metres. 
Question 32 of 40
32. Question
3 point(s)How many threedigit numbers can be written as the sum of five different powers of 3, including 3˚?

Question 33 of 40
33. Question
3 point(s)A cuboid, measuring 2 by 10 by 22 units, is placed inside a sphere with its eight vertices touching the surface. What is the edge length of the largest cube that can fit inside a sphere of the same size?

Question 34 of 40
34. Question
3 point(s)What is the value of p, given that, (4x^{2 } −3px +2)(x^{2 } + px +1) = 4x^{4 } + px^{3 } −7px^{2 } − px +2

Question 35 of 40
35. Question
3 point(s)What is the remainder when 1+3x +5x^{2}+7x^{3}+…+99x^{49} is divided by x −1 ?

Question 36 of 40
36. Question
4 point(s)Questions 36 – 40: Score 4 marks for each correct answer.
Three circles, each with a radius of one unit, are set out as shown with the circumference of the middle circle passing through the centres of the outer two circles. What is the area of the shaded region in terms of π?

Question 37 of 40
37. Question
4 point(s)What is the area of the shaded region given by,

Question 38 of 40
38. Question
4 point(s)Given that x< 1 ,which power series, to three terms,corresponds to the expansion of, 3x −1 2x² +3x−2 ? 2 
Question 39 of 40
39. Question
4 point(s)The funtion, y = e^{kx} satisfies the equation,
for

Question 40 of 40
40. Question
4 point(s)A pentagram is drawn within a regular pentagon. The area of the inner pentagon, PQRST, is one unit. Given that BC : BQ = BQ : PQ, what is the area of the large pentagon, ABCDE?