VMO Competition #1: November 21 st, 2014 Math Relays Problems

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1 VMO Competition #1: November 21 st, 2014 Math Relays Problems 1. I have 5 different colored felt pens, and I want to write each letter in VMO using a different color. How many different color schemes of the letters are there? 2. Special Instructions: From this problem on, including this problem, every solution to a problem with a prime problem number p must come with the memorization of p + 2 digits of π. For example, for this problem, one of your teammates must recant four digits of π when he/she submits the solution to this problem. Call an integer n good if the product of the positive integers that divide n is n 2. How many good integers are there in the interval from 1 to 100 inclusive? Please note that 0 is good, whereas 2 and 4 are not. 3. Determine the repeating segment of the decimal form of Special Instructions: From this problem on, including this problem, every solution to a problem with an even problem number must be accompanied by the name of a mathematician. You may not repeat names. A 900 litre supply of gasoline is available at one edge of an endless desert, but there is no source in the desert itself. A truck can carry enough gasoline to go 450km, and it can build its own refueling stations at any spot along the way. Assuming that the truck travels 1km per 1 litre of gasoline used, how far into the desert (in km) could the truck possibly go? 5. The five members of your team will line up in a single file and hold a red or black card on your head. You can see the cards of the people in front of you in the line but not your own hat, nor those of anyone behind you. The judge starts at the back of the line and asks the last person the color of his or her card. He or she must answer red or black. Everyone can hear the person s answer, but the judge does not reveal whether that answer was correct. The judge then move on to the second last person in the line and repeat the same process. As a team, at least 4 of the 5 people must answer correctly. You are allowed to devise a strategy ahead of time to help you. Ask a judge/event organizer when you are ready to play the game. Your team gets only one try.

2 6. Let O be the center of a circle of radius 5. Pick three points, A, B, and C on the circle such that ABC is an acute, scalene triangle, and let M, N, and P be the midpoints of BC, CA, and AB, respectively. Let the perpendicular bisectors of AB and AC intersect ray AM in points D and E respectively, and let lines BD and CE intersect in point F, inside of triangle ABC. Let H be the foot of the altitude from A to BC. AH intersects BD and CE at two other points R and S. Construct the incircle of triangle F RS, and denote its incentre as I. Connect AI, BI, and CI, and let AI intersect the circle with center O at W, BI intersect the circle with center O at X, and CI intersect the circle with center O at Y. Construct the excircles of triangle W XY, and let the centers of the excircle tangent to W X, XY, and W Y be denoted O 1, O 2, and O 3 respectively. The lengths O 1 M, O 2 N, and O 3 P have lengths 6, 7, and 8 respectively. Connect AO 1, BO 2, and CO 3. The area of quadrilaterals AO 1 O 2 B, BO 2 O 3 C, and CO 3 O 1 A are 36, 40, and 44 respectively. Also, all three of the above quadrilaterals (AO 1 O 2 B, BO 2 O 3 C, and CO 3 O 1 A) are cyclic. Determine the area of the circle with center O. 7. Special Instructions: Solve the given Rubik s cube in 4 or less moves to receive the problem slip for problem 7. Your team will be penalized for taking more than 4 moves to solve the cube. You may turn the cube for experimentation. An unlimited supply of gasoline is available at one edge of an 800 kilometer long desert, but there is no source in the desert itself. A truck can carry enough gasoline to go 500 kilometers, and it can build its own refueling stations at any spot along the way. How can the truck driver get across the desert as quickly as possible (in other words, how can the truck driver minimize the total distance travelled)? 8. Find the problem in this location: SLBCP Y RYZJC GL PCDCPCLAC QCARGML Problem 8: I have cups A and B. A has a liter of water in it whereas B is empty. First, I pour 1 2 of cup A s water into cup B, then I pour 1 3 of cup B s water into cup A, then I pour 1 4 of cup s A s water into cup B, then I pour 1 5 of cup B s water into cup A... and so on. After the 2014 th pouring, how much water is in cup A, assuming that the pouring of liquids is perfectly precise?

3 9. Fold the following 4 2 grid into a cube with edge lengths of a 1 1 square: 10. In this problem, I will prove that 0=2. Consider the equation cos 2 x = 1 sin 2 x which holds as a consequence of the Pythagorean theorem. Then, by taking the square root, cos x = (1 sin 2 x) 1/2 so that 1 + cos x = 1 + (1 sin 2 x) 1/2 But evaluating this when x = π implies 1 1 = 1 + (1 0) 1/2 or Q.E.D. 0 = 2 What went wrong in the proof? 11. At what exact time between 3:14:16pm and 3:44:16pm are the minute and hour hands of the clock perpendicular to each other?

4 12. Go play the following game with one of the event organizers: one of the event organizers will write any 16 consecutive positive integers on a piece of paper. You and the event organizer will take turns crossing off one of the numbers at a time. If the last two numbers are not relatively prime, then your team scores a point. Otherwise, please move on to problem 13. Your team can send any number of team members to play the game; however, your team has only one try at the game. 13. What is the maximum number of bishops that can be placed on a chessboard such that no two bishops share a diagonal? 14. Play the following game with one of the event organizers in a light blue T-shirt. The event organizer goes first. On his first turn, he writes down either an A or a B; on each of his subsequent turns, he adds either an A or B at the (right) end of the already written string. Your team, on your turns, has two options: Do nothing Interchange any two already written letters The game ends after each player has had n turns, where n is an arbitrary positive integer. Your goal is to end up with a string that is a palindrome (a palindrome is a sequence of letters or numbers that read the same forwards and backwards). The event organizer s goal is to thwart you. Your team is allowed to choose the value for n ahead of time, where n > 8. You can send as many people in your team as you wish to play the game. However, your team gets only one try at the game.

5 15. Special Instructions: Paper airplane incoming. Problem 15: A number can be factorized into the product of three primes. The sum of the squares of these three primes is Find the number.

6 16. A teacher wrote a large number on the board and asked the students to tell about the divisors of the number one by one. The 1 st student said, The number is divisible by 2. The 2 nd student said, The number is divisible by 3. The 3 rd student said, The number is divisible by 4. (and so on) The 30 th student said, The number is divisible by 31. The teacher then commented that exactly two students, who spoke consecutively, spoke wrongly. Which two students spoke wrongly? 17. Find the remainder when MMXIV MMXIII is divided by XIII. Answer in the numerical form as given in the problem. 18. In your group, you must create an unsolvable 5-person human knot (everyone stands in a circle and puts both hands in and grabs two more hands making sure not to grab the hands of somebody standing next to them.). That is, no matter what your team does to try to get untangled, the knot will not resolve in a circle (or unknot). Show the human knot to the judges once you ve figured it out. 19. The surface of a soccer ball consists of tessellated regular hexagons and pentagons, where every side of a regular pentagon is in line with a side of another hexagon. If there are 12 pentagons on the soccer ball, how many hexagons are there? 20. Waley (one of the event organizers in a light blue T-shirt) has a rule for determining which triples of positive integers are good and which are bad. Your team may give Waley a triple of positive integers, and he ll say whether it s a good triple or a bad triple. Your team may ask for as many different triples as you wish. Figure out Waley s rule.

7 21. Determine three functions on the Cartesian plane with bounds on their x-inputs which graph the letters V, M, and O such that: There exist two horizontal lines on the coordinate plane such that one of the lines touches the bottommost point(s) of the three letters, and the other line touches the topmost point(s) of the three letters. Each letter has one vertical line of symmetry. The length of the curve for each letter is at least 20 units. No two curves touch each other. Note: The O does not need to be a function. 22. Two people play a number game. One person starts with the number 1, triples it, adds one to the result, and hands it to the second person. The second person does the same thing: triples it, add one to the result, and hands it back to the first person. Then the process repeats over and over again. Given that 3 19 = , find the resulting number after the second person s 9 th turn. 23. You have a scale and some denomination of weights. To weigh something, you are allowed to put any combination of weights on either side of the scale. For example, if you have a 1g and 3g weight, in addition to being able to measure 1g and 3g, you can also measure a 2g weight (by placing 3g on one side and 1g on the side which contain the substance to be weighed), and you can also measure 4g by putting both weights on the same side of the scale. You want to be able to weigh all integer weights between 1g and 1000g inclusive with the minimum number of weights. How many weights and of what denominations do you need to measure all the weights from 1g to 1000g? 24. How many ways are there to walk up a set of 10 stairs by taking some combination of 1, 2, or 3 steps? 25. How many of the Math Relays problems today had integers as answers? Playing a game against a staff member does not constitute as a problem with an integer solution.