Archive for the ‘Math Competitions’ Category.

Puzzle. A group of 20 students formed a line to take an oral exam with Professor Chill. They were afraid to enter the classroom, so they decided to do a drawing. They wrote the numbers from 1 to 20 on pieces of paper, placed them in a hat, and each student picked one. The student who drew the number 1 went into the classroom first. Then, the remaining 19 students repeated the process, writing down numbers from 1 to 19, and the one who drew the number 1 took the exam next. They continued this process until all 20 students had taken their exams. Remarkably, each student drew a different number each time. Olga drew number 14 in the first round. How many students took the exam before she entered the classroom?

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## Mafia in a Math Battle

The Ural Math Battle in 2016 had several mafia-themed problems of various difficulty with the same initial setup.

Puzzle Setup. Among 100 residents of Saint-San, m are mafiosi, and the rest are civilians. A commissioner arrived to the town after getting this information. In an attempt to expose the mafia, this commissioner asked each of the residents to name s mafia suspects from among the other 99 residents. The commissioner knows that none of the mafiosi would name other mafiosi, but each civilian would name at least k mafia members. What is the maximum number of mafia members the commissioner can definitively identify after his survey?

1. The most difficult case was m = s = 3 and k = 2.
2. In the next case, where m = 3 and s = k = 2, the puzzle had a different task: prove that the commissioner can find at least one mafioso.
3. In the third case, where m = s = 10 and k = 6, the question was whether the commissioner can find at least three mafiosi.
4. In the fourth case, where m = s = 10 and k = 7, the question was whether the commissioner can find all the mafiosi.
5. The last case was for younger students with m = 6, s = 10, and k = 6. The question was whether the commissioner can find all the mafiosi.

When I asked ChatGPT to translate the first and the most difficult case of this puzzle from Russian, ChatGPT decided to solve it too. At the end of its ridiculous solution, it concluded that the commissioner could identify all 21 mafiosi out of the given 3. So, if you comment on this blog that the answer to the first case is 21, I will know that you are a bot.

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## Candy Game

I recently saw another puzzle on Facebook, a generalization of a problem from the 2002 Belarus Olympiad. In the problem, there are red and white boxes. Given how Russia and Belarus are filled with propaganda, my first question was whether the Belarusian flag was red and white. But in fact, the official flag is red and green; however, the opposition uses a red and white one. It could either be a coincidence or a sneaky way of protesting. Anyway, here is the problem.

Puzzle. There are two boxes filled with candy. The red box has R candies, and the white box has W candies. Alice and Bob are playing a game where Alice starts, and both players have the same options each turn: Either move one candy from the red box to the white box or take two candies from any box and eat them. The player who can’t move loses. For which values of R and W is each of the following true?

• Alice, following her optimal strategy, wins but might lose if she makes a mistake.
• Alice wins no matter what.
• Bob, following his optimal strategy, wins but might lose if he makes a mistake.
• Bob wins no matter what.

The list of options is weird, but I decided to keep it to emphasize …. Oops, I do not want to spoil it. You can decide for yourself what I wanted to emphasize.

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## Pay Lives to Touch Glasses

Here is one of my all-time favorite problems.

Puzzle. Four glasses are placed on the four corners of a square rotating table; each glass is either right-side up or upside down. You need to turn them all in the same direction, either all facing up or all down. You may do so by grasping any two glasses and turning either none, one of them, or both over.
There are two catches: 1) You are blindfolded, and 2) the table is spun after each round. Assuming a bell rings when you have them all facing the same way, how do you do it?

When I first heard this problem, the person who tortured me with it forgot to mention the bell. The problem was impossible: there was no feedback. Then, the bell was mentioned. It was an aha moment: you get information, but only a confirmation that the problem is solved. I tried to think about the puzzle backwards. What could be my last step? Assume that I know that the glasses alternate around the table: up, down, up, down. Then, as my last step, I can reach for the diagonal glasses and turn them over.

Here is a wonderful variation, proposed by Michael Hotiner from Ukraine, that appeared ten years ago at a puzzle competition. This variation is not cheap; the players must pay with their lives, albeit virtual ones.

Puzzle setup. Consider a computer game where at level M, there is an M-sided rotating table with glasses at each corner. The setup is similar to the four-glasses puzzle above. The player is blindfolded, and the table rotates between rounds. As soon as level M is over, if all the glasses face one direction, the bell rings, and the player moves to the next level, M+1.
At each round, the player can decide on the number N of glasses to touch. Upon deciding, they have to pay N! lives for that round. Then, the player can touch the glasses one by one, choosing the next glass depending on how the previous glasses are oriented. After all the glasses are touched, the player can decide which ones to turn upside down.

Let’s see what happens at the very beginning of this game. The game starts at level 1, with only one glass. The round is solved before it starts. The cost is 0. The bell rings, and the game immediately goes to level 2.

Consider level 2. If the bell doesn’t ring immediately, the glasses face different directions. The cheapest way to proceed is to choose N = 1 and turn any glass. The cost is 1.

Consider level 3. A player can solve it in one round by touching all three glasses and turning them the same way. The cost is 3! = 6. There is a cheaper method that costs 4 lives in two rounds. In the first round, the player can touch any two glasses and turn both of them up. If the bell doesn’t ring, then the third glass is upside down. In the next round, the player can touch two glasses. If one is upside down, that glass should be turned up. If both glasses are right side up, then the player should turn both of them over to finish the round.

1. Find a way to pass level 3 using only three lives.
2. Find the cheapest way to pass level 4.
3. Find the cheapest way for level 6.
4. Find a way to pass level 5 using only 30 lives.
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Here are some problems that I liked from the YuMSh (Youths Math School in St. Petersburg) Olympiad.

Problem for 6th grade. Twenty people from an island of knights and knaves have a party. Knights always tell the truth, and knaves always lie. Each party-goer got a card with a different number from 1 to 20. When they were asked about their numbers, each answered with a number from 1 to 20. The sum of all the answers is 156. What is the minimum possible number of liars that have to be at the party?

Problem for 7th grade. Alice and Bob bought a deck of playing cards (52 cards total) and took turns gluing the cards on the wall one at a time. Alice was first. The game is lost if, after a move, the wall has 4 cards of the same suit or 4 cards of consecutive values (for example, 8-9-10-jack). Can Alice or Bob guarantee themselves a win, regardless of their opponent’s moves?

Problem for 7th grade. Buddhist monks gather in an infinite cave where a finite number of prime numbers are written on the wall. The numbers might not be distinct. Every second, one of the monks performs one of the following operations.

1. Adds to one of the numbers one of its digits.
2. Shuffles the digits of one of the numbers.

Every time they do it, they erase the old number and write the new one. The rule is that the new number has to be greater than the old one. If a composite number gets written on the wall of this cave, then the world collapses into nothingness. Can the monks save the world for eternity?

Problem for 8th grade. The incenter of a triangle is equidistant from the midpoints of the sides of the triangle. Prove that the triangle is equilateral.

Problem for 9th grade. Bob was given 30 distinct natural numbers. He wrote down all the 435 pairwise sums. It appears that among those sums, 230 are divisible by 3. How many of the original 30 numbers are divisible by 3?

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## A Number Theory Problem from the 43rd Tournament of Towns

Problem. Find the largest number n such that for any prime number p greater than 2 and less than n, the difference np is also a prime number.

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## More Gnomes

I recently posted a cute Shapovalov’s puzzle about gnomes. Here is another easier gnome puzzle, also by Alexander Shapovalov.

Puzzle. All gnomes are knights or knaves: knights always tell the truth, and knaves always lie. There is a gnome on every cell of a 4 by 4 chessboard. It is known that both knights and knaves are present in this group. Every gnome states, “Among my neighbors, the number of knaves is the same as the number of knights”. How many knaves are there, if by neighbors they mean orthogonally adjacent gnomes?

The next gnome puzzle has a different author, Alexander Gribalko. Gnomes in this puzzle are not knights or knaves but rather friendly and polite beings.

Puzzle. Nine gnomes repeated the following procedure three times. They arranged themselves on a 3 by 3 chessboard with one gnome per cell and greeted all their orthogonal neighbors. Prove that not all pairs of gnomes greeted each other.

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## The 1978 Ukrainian Math Olympiad

Ukraine is on my mind. Here is a problem for 9-graders from the 1978 Ukrainian Math Olympiad:

Problem. Prove that for every natural number n, the following number is not an integer.

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Here is a problem from the 1978 Kyiv Olympiad for 7 graders.

Is it possible to place seven points on a plane so that among any three of them, two will be at distance 1 from each other?

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## Archimedes Helps Again

Below, you can find a lovely problem from the 2016 All-Russian Olympiad suggested by Bogdanov and Knop. I took some liberties translating it.

Problem. King Hiero II of Syracuse has 11 identical-looking metallic ingots. The king knows that the weights of the ingots are equal to 1, 2, …, 11 libras, in some order. He also has a bag, which would be ripped apart if someone were to put more than 11 libras worth of material into it. The king loves the bag and would kill if it was destroyed. Archimedes knows the weights of all the ingots. What is the smallest number of times he needs to use the bag to prove the weight of one of the ingots to the king?

And a bonus question from me.

Bonus. Add one more weighing to prove the weight of three more ingots.

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