Olympiad problems

2021/2022 Tournament of Towns, P3

Grasshopper Gerald and his 2020 friends play leapfrog on a plane as follows. At each turn Gerald jumps over a friend so that his original point and his resulting point are symmetric with respect to this friend. Gerald wants to perform a series of jumps such that he jumps over each friend exactly once. Let us say that a point is achievable if Gerald can finish the 2020th jump in it. What is the maximum number $N{}$ such that for some initial placement of the grasshoppers there are just $N{}$ achievable points? [i]Mikhail Svyatlovskiy[/i]

2020/2021 Tournament of Towns, P2

Tags: combinatorics , Tournament of Towns

Maria has a balance scale that can indicate which of its pans is heavier or whether they have equal weight. She also has 4 weights that look the same but have masses of 1001, 1002, 1004 and 1005g. Can Maria determine the mass of each weight in 4 weightings? The weights for a new weighing may be picked when the result of the previous ones is known. [i]The Jury[/i] (For the senior paper) The same question when the left pan of the scale is lighter by 1g than the right one, so the scale indicates equality when the mass on the left pan is heavier by 1g than the mass on the right pan. [i]Alexey Tolpygo[/i]

2019 Tournament Of Towns, 1

Tags: game , game strategy , combinatorics , Tournament of Towns , ToT

The King gives the following task to his two wizards. The First Wizard should choose $7$ distinct positive integers with total sum $100$ and secretly submit them to the King. To the Second Wizard he should tell only the fourth largest number. The Second Wizard must figure out all the chosen numbers. Can the wizards succeed for sure? The wizards cannot discuss their strategy beforehand. (Mikhail Evdokimov)

2020/2021 Tournament of Towns, P4

Tags: combinatorics , game , Tournament of Towns

There is a row of $100N$ sandwiches with ham. A boy and his cat play a game. In one action the boy eats the first sandwich from any end of the row. In one action the cat either eats the ham from one sandwich or does nothing. The boy performs 100 actions in each of his turns, and the cat makes only 1 action each turn; the boy starts first. The boy wins if the last sandwich he eats contains ham. Is it true that he can win for any positive integer $N{}$ no matter how the cat plays? [i]Ivan Mitrofanov[/i]

2019 Tournament Of Towns, 3

Tags: number theory , Tournament of Towns , ToT , Divisibility

The product of two positive integers $m$ and $n$ is divisible by their sum. Prove that $m + n \le n^2$. (Boris Frenkin)

2020/2021 Tournament of Towns, P3

Tags: geometry , Tournament of Towns

Two circles $\alpha{}$ and $\beta{}$ with centers $A{}$ and $B{}$ respectively intersect at points $C{}$ and $D{}$. The segment $AB{}$ intersects $\alpha{}$ and $\beta{}$ at points $K{}$ and $L{}$ respectively. The ray $DK$ intersects the circle $\beta{}$ for the second time at the point $N{}$, and the ray $DL$ intersects the circle $\alpha{}$ for the second time at the point $M{}$. Prove that the intersection point of the diagonals of the quadrangle $KLMN$ coincides with the incenter of the triangle $ABC$. [i]Konstantin Knop[/i]

2020/2021 Tournament of Towns, P2

Tags: geometry , Tournament of Towns

There were ten points $X_1, \ldots , X_{10}$ on a line in this particular order. Pete constructed an isosceles triangle on each segment $X_1X_2, X_2X_3,\ldots, X_9X_{10}$ as a base with the angle $\alpha{}$ at its apex. It so happened that all the apexes of those triangles lie on a common semicircle with diameter $X_1X_{10}$. Find $\alpha{}$. [i]Egor Bakaev[/i]

2022/2023 Tournament of Towns, P7

Tags: combinatorics , geometry , Tournament of Towns

There are $N{}$ friends and a round pizza. It is allowed to make no more than $100{}$ straight cuts without shifting the slices until all cuts are done; then the resulting slices are distributed among all the friends so that each of them gets a share off pizza having the same total area. Is there a cutting which gives the above result if a) $N=201$ and b) $N=400$?

2021/2022 Tournament of Towns, P1

Tags: combinatorics , cards , Tournament of Towns

The wizards $A, B, C, D$ know that the integers $1, 2, \ldots, 12$ are written on 12 cards, one integer on each card, and that each wizard will get three cards and will see only his own cards. Having received the cards, the wizards made several statements in the following order. [list=A] [*]“One of my cards contains the number 8”. [*]“All my numbers are prime”. [*]“All my numbers are composite and they all have a common prime divisor”. [*]“Now I know all the cards of each wizard”. [/list] What were the cards of $A{}$ if everyone was right? [i]Mikhail Evdokimov[/i]

2022/2023 Tournament of Towns, P5

Tags: geometry , nonagon , Tournament of Towns

On the sides of a regular nonagon $ABCDEFGHI$, triangles $XAB, YBC, ZCD$ and $TDE$ are constructed outside the nonagon. The angles at $X, Y, Z, T$ in these triangles are each $20^\circ$. The angles $XAB, YBC, ZCD$ and $TDE$ are such that (except for the first angle) each angle is $20^\circ$ greater than the one listed before it. Prove that the points $X, Y , Z, T$ lie on the same circle.

2021/2022 Tournament of Towns, P1

Tags: number theory , prime numbers , Tournament of Towns

Let us call a positive integer $k{}$ interesting if the product of the first $k{}$ primes is divisible by $k{}$. For example the product of the first two primes is $2\cdot3 = 6$, it is divisible by 2, hence 2 is an interesting integer. What is the maximal possible number of consecutive interesting integers? [i]Boris Frenkin[/i]

Kvant 2021, M2652

Tags: combinatorics , Process , Tournament of Towns

A hundred tourists arrive to a hotel at night. They know that in the hotel there are single rooms numbered as $1, 2, \ldots , n$, and among them $k{}$ (the tourists do not know which) are under repair, the other rooms are free. The tourists, one after another, check the rooms in any order (maybe different for different tourists), and the first room not under repair is taken by the tourist. The tourists don’t know whether a room is occupied until they check it. However it is forbidden to check an occupied room, and the tourists may coordinate their strategy beforehand to avoid this situation. For each $k{}$ find the smallest $n{}$ for which the tourists may select their rooms for sure. [i]Fyodor Ivlev[/i]

Kvant 2021, M2651

Tags: combinatorics , invariant , Tournament of Towns

In a room there are several children and a pile of 1000 sweets. The children come to the pile one after another in some order. Upon reaching the pile each of them divides the current number of sweets in the pile by the number of children in the room, rounds the result if it is not integer, takes the resulting number of sweets from the pile and leaves the room. All the boys round upwards and all the girls round downwards. The process continues until everyone leaves the room. Prove that the total number of sweets received by the boys does not depend on the order in which the children reach the pile. [i]Maxim Didin[/i]

2021/2022 Tournament of Towns, P4

Tags: algebra , Tournament of Towns

What is the minimum $k{}$ for which among any three nonzero real numbers there are two numbers $a{}$ and $b{}$ such that either $|a-b|\leqslant k$ or $|1/a-1/b|\leqslant k$? [i]Maxim Didin[/i]