Math Day mathematical competition.
Also available in French and German.

## Have fun!

The level of the exercises is marked with stars:
(★) = for all
(★★) = challenging

 (★) The twins Erik and Oskar have to share their favorite game for one year (January 1st to December 31st). The year is not a leap year. They decide that Erik will have the game on days which have an even number in the month (the 2nd of each month, the 4th of each month and so on) and Oskar on days which have an odd number in the month. However, they realize that there are more odd days than even days. In fact, how many more?

7

The months with 28 or 30 days have the same number of even and odd days. The seven months with 31 days have one odd day more. So the answer is 7 days.

 (★★) Two gentlemen, Mr Adams and Mr Beckam, arrive at a door at the same time, and have to agree upon who enters first. At regular intervals, they make an attempt to find an agreement. At every attempt, they can speak (saying “You go first”) or stay silent. If they both speak, nobody enters. If nobody speaks, nobody enters. If only one speaks, the silent gentleman enters. If they speak at the same time, Mr Adams will stay silent for at least one round, and Mr Beckam for at least two rounds. If nobody has spoken for at least 3 consecutive rounds, then Mr Beckam will speak in the next round. Neither gentleman knows the rules governing the other gentleman, and they do not agree on a strategy. Must there eventually come a round where one gentleman enters the door?

No

For example, they might both speak on every fourth turn and stay both silent in between (silent for three turns).

 (★★) There are four statues of an elf which only differ in their size. The sizes are XL (Extra Large), L (Large), M (Medium), S (Small) respectively. They have different weights decreasing in this order, i.e. the largest statue is the heaviest, the second-largest the second-heaviest etc. Your friend weighed the statues in pairs, and the weight of the various pairs, in grams, are 18, 24, 30, 30, 36, 42. What is the total weight in grams of the four statues?

60

Sum the weights and divide by 3, or sum the smallest weight (the two lightest statues) and the largest weight (the two larger statues). One could also have summed the two intermediate weights, which must be XL + S and L + M (and it is a coincidence that those two weights are the same, any of the two could be larger). Or one could have summed the second weight (XL + M) and the penultimate weight (L + S). In any case, the answer is 60.

 (★) A kid is playing with a car that can be remotely maneuvered. The car can only turn left or right at 90° angles. Moreover, it turns exactly once after each meter it has gone, and the kid can only decide whether it will be a left or a right turn. The kid is playing on a 3 meter times 4 meter rectangular carpet. Starting from one carpet corner parallel to one of the carpet sides, how many further carpet corners can be reached by the car?

3

The answer is 3, as the car can reach all further corners. By symmetry, it does not matter at which corner we start. With coordinates, let (0,0) be the starting corner, and let (4,3) be the opposite corner. Some paths going to the further 3 corners are:
(0, 0), (1, 0), (1, 1), (2, 1), (2, 2), (3, 2), (3, 3), (4, 3)
(0, 0), (1, 0), (1, 1), (2, 1), (2, 2), (3, 2), (3, 1), (4, 1), (4, 0)
(0, 0), (1, 0), (1, 1), (0, 1), (0, 2), (1, 2), (1, 3), (0, 3).

 (★★) There is a world similar to ours, but where people are either liars or truth-tellers. Liars always lie and truth-tellers always tell the truth. A child says: “Everyone in my family is a liar”. Is this assertion: True, False, or Forcibly neither true nor false?

Forcibly neither true nor false

The assertion cannot be true because, if true, the child would be a liar (as part of the family) telling the truth. The assertion can be false, for example if only half of the family members are liars.

 (★) You are preparing a yoga session where the participants will be seated on chairs arranged in a row. The chairs are next to one another with no space in between. The participants must however be able to stretch their arms sideways without touching one another, therefore between any two occupied seats, there must be two empty seats. After setting up the row of chairs by taking all the available space, you decide to remove two chairs from one side because doing this does not decrease the number of seats available to the participants. What is the remainder after division by 3 of the number of chairs in the row, after having removed the two chairs?

1

It is optimal to start the row with an occupied seat. Adding two chairs does not make a difference if and only if the last seat is also occupied, which means that the remainder after division by 3 equals 1.

 (★) You are allowed to draw N balls from an urn containing many balls of each of the following colors: blue, green, yellow, red. You win once you draw 1 blue ball, or 2 green balls, or 3 yellow balls, or 4 red balls. What is the least possible value of N for which you are guaranteed to win?

7

If you draw 1 green ball and 2 yellow balls and 3 red balls, then you lose (this is the most unfortunate situation). Once you draw any additional ball, you are guaranteed to win. Hence N=7.

 (★) In a shop, they sell triangular tiles, where each tile has the side lengths 8,12, and 18 centimeters. You like the shape, but you need a larger tile. The triangle you want is similar to the ones sold at the shop and the side lengths, in centimeters, are again integers. Moreover, two of its side lengths are 12, and 18 centimeters. What is the third side length, in centimeters?

27

Call X the missing side length. We must have 8/12=12/18=18/X hence X=27.

 (★★) Alice takes 4 hours to paint a fence while Bob takes 12 hours for the same task. How many hours will it take if they work together on the task?

3

If Alice and Bob work for X hours, the fraction of the complete task they get done is (1/4)X+(1/12)X. The task gets completed when this fraction equals 1, that is, if X=3.

 (★★) You are organizing a movie day at your school. You know that 50% of the pupils have watched "Aliens and Alligators" (movie A) and 40% of the pupils have watched "Basketball and Biscuits" (movie B). You know that 20% of the pupils who watched movie A also watched movie B. What percentage of the pupils who watched movie B also watched movie A?

25%

Say (without loss of generality) that we have 100 pupils in total. Then 10 pupils watched movies A and B, which corresponds to 25% of the pupils who watched movie B.

 (★) In your favorite restaurant, you can buy tofu nuggets to go. They are sold in packs of 3, 5, or 7. It is thus impossible to order, for instance, precisely 2 tofu nuggets. How many integers n≥1 exist for which you cannot buy precisely n nuggets?

3

If the amount of nuggets is a multiple of 3 we can buy several 3-bags. If it is a multiple of 3 plus 2 (at least 5), we can buy a bag of 5 and possibly several 3-bags, if it is a multiple of 3 plus 1 (at least 7), then we can buy a 7-bag and possibly several 3-bags. So the only amounts of nuggets that we cannot buy are 1,2, and 4.

 (★★★) Your class is preparing for an excursion and your teachers are filling lunch bags with sandwiches, one lunch bag for each pupil. The lunch bags and the sandwiches are all alike. Each sandwich can be cut into 2 or 3 equal parts. You know that with 14 sandwiches, one can fill 9 lunch bags but not 10 lunch bags. How many sandwiches were used to prepare lunch bags for the 24 pupils?

36

The amount of sandwich in each bag is in particular a rational number of the form X/6. From the information about 14 sandwiches, we know that 14/10≤ X/6 ≤ 14/9. We deduce that 8< X< 10, hence X=9. So each bag contains 9/6=3/2 sandwiches, so for 24 bags one needs 24· 3/2=36 sandwiches.

 (★) One of Alice, Bob, and Charles is a liar, the other two tell the truth. Alice said: I am not a liar. Bob said: Alice is not a liar. Charles said: Alice is a liar. Who is the liar?

Charles

Since Bob and Charles say contradictory things, one of them is a liar. So Alice cannot be the liar. Hence Charles is the liar.

 (★) Bob's grandparents like roses. On some days they want 3 white roses, 2 red roses, and 1 yellow rose. On the other days, they want 1 white rose, 2 red roses, and 3 yellow roses. Bob is at the flower shop and forgot his grandparents' preference for today. Bob needs to buy enough roses to be sure that the right ones are among them (the number of roses he buys may depend on the color, as one can buy single roses of any color). What is the least number of roses he needs to buy?

8

Bob needs 3 white roses, 2 red roses, and 3 yellow roses.

 (★) In a board game there are many direction cards, which are either North, East, West, or South. A player that has three direction cards of the same type wins. How many cards suffice for a player to win with absolute certainty? Select the smallest possible number.

9

The most unlucky case is having two cards of each type, namely 8 cards. With 9 cards one is sure to have three cards of the same type.

 (★) Forty children received an invitation to a birthday party, but not all of them came. At the party, the children played a game for teams of 8 players and no child was left without a team. They also played a game for teams of 5 players; two children were left without a team and they became referees for this game. How many children were at the party?

32

We look for a number from 0 to 40 which is a multiple of 8 and which leaves remainder 2 after division by 5. The only such number is 32.

 (★★) You have two identical apples, one banana, and one orange. You have to give them to four children (Alice, Bob, Charles, and David), so that each child receives exactly one piece of fruit. In how many different ways can you distribute the pieces of fruit to the children?

12

You only have to choose whom to give the banana to (4 choices) and whom the orange (3 choices left). There are 12 possibilities.

 (★) The frame for a painting should be in the form of a rectangle with a rectangular hole inside. To build such a frame you are given 4 pieces of wood. They all are 10 cm wide, but two of them are 40 cm long while two of them are 60 cm long. You can build different frames out of them by gluing the pieces of wood together. What is the largest hole that you can obtain inside the frame? Consider your result in square centimeters.

1600

You can build a 60cm×60cm frame with a 40cm× 40cm hole (1600 square centimeters). Or you can make a 50cm× 70cm frame with a 30cm× 50cm hole (1500 square centimeters). Or you can make a 40cm×80cm frame with a 20cm× 60cm hole (1200 square centimeters).

 (★★) You have to solve 200 mathematical problems, and fortunately the genie of the lamp will help you by fulfilling some wishes. Wish 1 will let you solve 60% of the unsolved problems, Wish 2 will let you solve 40% of the unsolved problems, Wish 3 will let you solve 50 unsolved problems (or all remaining problems, if you have less than 50 unsolved problems left). You can ask two distinct wishes in the order you prefer. Which wishes should you ask to solve most of the problems? For example, the answer (2,3) means that the first wish you ask is Wish 2 and the second wish you ask is Wish 3.

(1,3)

(1,2) and (2,1) will both leave 48 problems unsolved. (1,3) will leave 30 problems unsolved. (3,1) will leave 60 problems unsolved. (2,3) and (3,2) solve less problems than (1,3) and (3,1) respectively.

 (★★) You are driving a remote control toy car on a circuit in the shape of a regular polygon with 10 sides. At the end of each straight segment you turn left at an angle which is strictly between 0 and 180 degrees, and how much you turn is the measure of this angle. How much do you need to turn in total while doing one round on the full circuit, starting from the middle of a side? The answer is in degrees.

360

The sum of the exterior angles of any regular polygon is 360 degrees.

 (★★) In a foreign land there is a currency called AUR. There are golden coins with value 1AUR, 3AUR and 9AUR. What is the smallest number of coins you need to be able to pay any bill in the range from 1AUR to 44AUR? You can choose the coins freely, but you have to choose them before knowing the amount of the bill.

8

You can take 4 9AUR coins, 2 3AUR coins, 2 1AUR coin. This is optimal because with only 3 or fewer coins of 9AUR, you would need at least 10 coins to pay 44 AUR (3 × 9AUR + 5 × 3AUR + 2 × 1AUR), and with 4 coins of 9 AUR you need at least 8 coins to pay 44 AUR. The given coins allow you to pay any bill in the range; using the 9AUR coins one is left to pay an amount from 1AUR to 8AUR, which can be paid with the given 3AUR or 1AUR coin.

 (★) You have 8 sticks to make a closed polygon, and the sides of the polygon must be either horizontal or vertical. If each stick has length 1, what is the largest area that you can obtain by using all sticks? As unit measure for the area consider the square of the unit length.

4

Using all sticks, you can build a 2×2 square with area 4, a 3×1 rectangle with area 3, and an L-shaped figure with area 3.

 (★★) There are two fair dice with four faces. The first die is red and the second die is blue, and you throw them both at once. On the four faces of each die there are the numbers from 1 to 4. What is the most likely sum of the results of the two dice?

5

Consider the set of results of the first die and the second die, consisting of the 16 pairs made with the numbers from 1 to 4. The numbers 2 and 8 can be obtained each just with one pair. The numbers 3 and 7 can be obtained each with two pairs. The number 4=3+1=2+2 and 6=4+2=3+3 can be obtained each with three pairs. The number 5=4+1=3+2 can be obtained with four pairs.

 (★★) You are in a video call with some friends who are native speakers of the Combish language. You only remember four different words in this language: Xix, Yiy, Ziz, Wiw. Exactly one of them is extremely funny. You know that if you send some of these words to any of your friends, then that friend will start laughing immediately if and only if the funny word is among the words you chose. You can write exactly one message with some Combish words to each of your friends in the call. You can choose how many words and which words to write. You can send different individual messages, but all messages are sent at the same time. You are then able to check in the video call who is laughing. What is the minimum number of friends that you need in the call so that you are able to determine without doubt the funny word by using the above method?

2

One friend is not enough because from testing only one message you cannot determine the funny word in all possible cases. Two friends are enough: you send the first word only to the first friend, the second word only to the second, the third word to both, and the fourth word to none.

 (★★) In a group of koalas, the two lightest koalas together weigh 25% of the total weight of the group, while the three heaviest koalas together weigh 60% of the total weight of the group. How many koalas are in the group?

6

More than 5 because the five mentioned koalas do not make up the 100% of the weight. Each intermediate koala weights each at least 12,5% of the total weight and at most 20% of the total weight. Since only 15% of the total weight is missing, there is space for at most one intermediate koala. So there are 6 koalas.

 (★★) Amy and Ben play the Candy Game. At the beginning of the game, there are 10 candies. Amy and Ben take turns making moves. A move consists of removing either 2 or 3 candies. The first player that cannot make a move (because there are less than 2 candies left) loses. Amy makes the first move. If both Amy and Ben aim to win and play according to the best possible strategy, who wins the game?

Ben

Consider the number of remaining candies. When you see 0 or 1 candies on the table, you lose. When there are 2,3 or 4 candies, you win (for 4 candies take 3 candies, for 3 candies take 2 or 3 candies, for 2 candies take 2 candies). When there are 5 or 6 candies, you lose (taking either 2 or 3 candies puts the other player in a winning situation). When there are 7,8 or 9 candies you win (Taking either 2 or 3 candies puts the other player in a losing situation.) When there are 10 candies, you lose. (taking either 2 or 3 candies puts the other player in a winning situation). So the second player always wins, i.e. Ben.

 (★★) What is the maximum number of bishops you can place on a 4× 4 chessboard so that no two bishops are on the same diagonal line?

6

A1, A2, A3, A4, D2, D3 works, so the answer is at least 6. There are 7 diagonals going bottom left to top right, so the answer is at most 7. However the first and last diagonal are on a same diagonal line in the other direction, so you cannot place bishops on both of them. Thus the answer is at most 6.

 (★★★) In a bag there are balls, each of which is coloured red, yellow or green. We pick two balls in a row, without putting the first one back. The probability of picking two red balls is 1/17 and the one of picking two yellow balls is 1/15. What is the least possible number of balls in the bag to make this true?

15

Let n be the number we are looking for. Let r and y be the number of red and yellow balls, respectively. We are given that
(r(r-1))/(n(n-1))=1/17,
(y(y-1))(n(n-1))=1/15,
hence 7r(r-1)=n(n-1)=5y(y-1). This shows that n(n-1) is divisible by 5 and by 7, thus by 35. The least n≥ 1 for which this is the case is n=15. To see that n=15 is actually suitable, notice that r=6 and y=7 lead to the desired probabilities.

 (★★★) The new Luxembourgish car plates consist of two letters followed by a 4-digit number. As the letter O and the digit 0 look too similar on plates, one of the two symbols needs to be forbidden. (The considered alphabet has 26 letters.) Which symbol should we forbid to achieve the largest number of possible car plates: The letter O, The digit 0, or The choice does not matter?

The letter O

Forbidding the letter O leaves 252· 104 allowed car plates, while forbidding the digit 0 only leaves 262· 94. We conclude because 25 · 102>26 · 92.

 (★★★) How many integers n≥ 1 exist for which 16n2+25 is the square of a natural number?

1

We have (4n)2< 16n< 2+25< (4n+5)2, hence a perfect square of the form 16n2+25 must be among (4n+1)2, (4n+2)2, (4n+3)2 and (4n+4)2. Since 16n2+25 is odd, it is neither (4n+2)2 nor (4n+4)2. Putting 16n2+25=(4n+1)2 gives n=3. Putting 16n2+25=(4n+3)2 yields n=1/23, which is not an integer. Hence only the value n=3 is counted.
Alternative solution: Assume 16n2+25=x2, where x is a positive integer. Then 25=x2-16n2=(x+4n)(x-4n), and since 25=52 is the square of a prime, the only possibility is given by the equations x+4n=25 and x-4n=1, which give n=3 and x=13.

 (★★) There are five small statues of an elf, and you know their combined weight W. The statues differ in their sizes, which are XL (Extra Large), L (Large), M (Medium), S (Small), XS (Extra Small) and they have different weights decreasing in this order. Your friend weighed the statues in pairs, and wrote the list of values W1,W2, . . . , W10 from the largest to the smallest. Knowing only W2 + W10, which is the statue of which you can find out the weight?

L

Call the weight of a statue with its size, for example let XL be the weight of the largest and heaviest statue. Clearly W1=XL+L and, as the two weights we sum must be distinct, we have W2=XL+M. Similarly, we have W10=S+XS. Thus W2+W10=XL+M+S+XS. Considering that L=W-(W2+W10), we can compute the weight of L.

 (★) What is the maximum number of queens you can put on a 5 x 5 chessboard, so that no two queens are in the same diagonal, row or column?

5

For example, put them on a4, b2, c5, d3, e1. It cannot be more than 5, because there are 5 columns and you can put at most one queen per column.

 (★★) You have two identical apples, two identical oranges, and one banana. You have to give them to five children, so that each child receives exactly one piece of fruit. In how many different ways can you distribute the pieces of fruit to the children?

30

You only have to choose whom to give the banana to (5 choices) and whom the two oranges. Discarding the child who took the banana, there are four children left and you have to select a pair of children for the oranges (12 choices, as there are 4 choices for the first child, 3 choices for the second child). As the ordering does not matter, you divide by 2 and are left with 6 possibilities. Hence there are 30 possibilities in total.

 (★★) Alice and Zoe, when they run alone, always run at their usual constant speed. Alice runs 1 kilometer in 4:10 (4 minutes 10 seconds), while Zoe runs 1 kilometer in 5:00. They planned to run together on a 11 km long straight path along a river. But they just texted each other and found that, due to a misunderstanding, they are at opposite ends of the path. Now they start running towards each other. After how much time will they meet? Give your answers in minutes.

25

Alice runs 1km in 250 seconds, Zoe in 300 seconds, so Alice's speed is 6/5 of Zoe's speed. Hence they will meet after Alice made 6/11 of the total distance and Zoe 5/11 of the total distance. Thus Alice has run 6km, which will take 25 minutes.

 (★★) You arrive on an island that is inhabited by 7 dwarves. A dwarf can either be a truth-teller or a liar. The truth-tellers always speak the truth and the liars always lie. All dwarves queue in a straight line to greet you. They all look into your direction. The first dwarf in the line says: "All dwarves behind me are liars." All other dwarves say: "The dwarf right in front of me is a liar." How many dwarves are liars?

4

Dwarves just behind liars are truth-tellers, hence the first dwarf must be a liar. Dwarves just behind truth-tellers are liars, hence liars and truth-tellers alternate, giving a total of 4 liars.

 (★★) A very modern museum of very modern art has two floors, one on top of the other. Each floor consists of four corridors connected in the form of a square; it is possible to go from each corridor to the two neighbouring ones on the same floor; at the end of each corridor there is a staircase connecting vertically the two floors; the only entrance is also the only exit and it is located at one corner on the ground floor. You want to walk along each corridor exactly once (the direction doesn't matter to you). You can walk along different tours, according to the order in which you visit the corridors. How many different tours are there, assuming that you use the stairs exactly twice?

16

You must use the same staircase for going up and down. Then what you can choose is: the staircase (4 possibilities) and the direction of your visit on the lower floor (2 possibilities), and the direction of your visit on the upper floor (2 possibilities). This gives a total of 4×2×2 =16 tours.

 (★★★) You have a coin that when being thrown comes up heads more often than tails. You and a friend play the following game: You toss the coin twice. If the result is twice the same, you win. If the results of the tosses are different, your friend wins. Who is more likely to win? Answer 1 if you have a higher probability to win, answer 2 if your friend has a higher probability to win, answer 3 if you and your friend have the same probability to win.

1

Let h be the probability of coming up heads and 1-h the one of coming up tails. Getting the same result happens with probability h2+(1-h)2=1+2h2-2h while getting different results happens with probability 2h(1-h)=2h-2h2. The difference is (1+2h2-2h)-(2h-2h2)=1-4h+4h2=(1-2h)2. This is strictly positive, unless h=1/2, which we excluded. So getting the same result is more likely than getting two different results.

 (★★) You own a bag of letter-shaped pasta for children. There are 26 different letters. If you take 99 pieces of pasta from the bag, what is the largest integer number n such that you can be sure to have at least n pieces representing the same letter?

4

Pigeonhole principle. With 99 objects of 26 types, there are at least 4 (namely, the ceiling of 99/26) pieces of pasta of the same kind.

 (★★) A fairground booth boasts a game involving the wheel of fortune. The numbers 1 to 5 are written on the wheel, and when it is spun, every number occurs with equal probability. To play the game, you place a bet on any number between 2 and 10 (inclusive) of your choice. The wheel is then spun twice and if the sum of the two resulting numbers equals the number you bet on, you win a teddy-bear. Otherwise, you don't win anything. What number should you bet on to maximize your chance of winning the teddy-bear?

6

Call x the result of the first spin, and y the result from the second spin (both x ad y are integers from 1 to 5). When you count all the possible sums the pair (x,y) can add up to, 6 occurs five times, while all other numbers occur less often.