MathDB

2024 Dutch IMO TST

Part of Dutch IMO TST

Subcontests

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A functional equation on positive integers with two conditions

Find all functions f:Z>0→Z>0f:\mathbb{Z}_{>0} \to \mathbb{Z}_{>0} such that for all positive integers m,nm,n and aa we have a) f(f(m)f(n))=mnf(f(m)f(n))=mn and b) f(2024a+1)=2024a+1f(2024a+1)=2024a+1.

A three-variable functional inequality on non-negative reals

Find all functions f:R≥0→Rf:\mathbb{R}_{\ge 0} \to \mathbb{R} with 2x3zf(z)+yf(y)≥3yz2f(x)2x^3zf(z)+yf(y) \ge 3yz^2f(x) for all x,y,z∈R≥0x,y,z \in \mathbb{R}_{\ge 0}.

Compute side ratio in triangle if circle passes through incenter

Let ABCABC be a triangle. A point PP lies on the segment BCBC such that the circle with diameter BPBP passes through the incenter of ABCABC. Show that BPPC=cs−c\frac{BP}{PC}=\frac{c}{s-c} where cc is the length of segment ABAB and 2s2s is the perimeter of ABCABC.
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Arithmetic mean of divisors and of coprime numbers

For a positive integer nn, let α(n)\alpha(n) be the arithmetic mean of the divisors of nn, and let β(n)\beta(n) be the arithmetic mean of the numbers k≤nk \le n with gcd(k,n)=1\text{gcd}(k,n)=1. Determine all positive integers nn with α(n)=β(n)\alpha(n)=\beta(n).

A well-known geo configuration revisited

Let ABCABC be a triangle with orthocenter HH and circumcircle Γ\Gamma. Let DD be the reflection of AA in BB and let EE the reflection of AA in CC. Let MM be the midpoint of segment DEDE. Show that the tangent to Γ\Gamma in AA is perpendicular to HMHM.

Beetles moving on a 2023x2023 board, will their ways cross?

On a 2023×20232023 \times 2023 board, there are beetles on some of the cells, with at most one beetle per cell. After one minute, each beetle moves a cell to the right or to the left or to the top or to the bottom. After each further minute, the beetles continue to move to adjacent fields, but they always make a 90∘90^\circ turn, i.e. when a beetle just moved to the right or to the left, it now moves to the top or to the bottom, and vice versa. What is the minimal number of beetles on the board such that no matter where they start and how they move (according to the rules), at some point two beetles will end up in the cell?
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Replacing a number by multiples or digit permutations, can we reach 1?

Initially, a positive integer NN is written on a blackboard. We repeatedly replace the number according to the following rules: 1) replace the number by a positive multiple of itself 2) replace the number by a number with the same digits in a different order. (The new number is allowed to have leading digits, which are then deleted.) A possible sequence of moves is given by 5→20→140→041=415 \to 20 \to 140 \to 041=41. Determine for which values of NN it is possible to obtain 11 after a finite number of such moves.

Fire on n islands

Let nn be a positive integer. There are nn islands with n−1n-1 bridges connecting them such that one can travel from any island to another. One afternoon, a fire breaks out in one of the islands. Every morning, it spreads to all neighbouring islands. (Two islands are neighbours if they are connected by a bridge.) To control the spread, one bridge is destroyed every night until the fire has nowhere to spread the next day. Let XX be the minimum possible number of bridges one has to destroy before the fire stops spreading. Find the maximum possible value of XX over all possible configurations of bridges and island where the fire starts at.
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A game with binary numbers on board, can we force divisibility?

Player Zero and Player One play a game on a n×nn \times n board (n≥1n \ge 1). The columns of this n×nn \times n board are numbered 1,2,4,…,2n−11,2,4,\dots,2^{n-1}. Turn my turn, the players put their own number in one of the free cells (thus Player Zero puts a 00 and Player One puts a 11). Player Zero begins. When the board is filled, the game ends and each row yields a (reverse binary) number obtained by adding the values of the columns with a 11 in that row. For instance, when n=4n=4, a row with 01010101 yields the number 0⋅1+1⋅2+0⋅4+1⋅8=100 \cdot1+1 \cdot 2+0 \cdot 4+1 \cdot 8=10.
a) For which natural numbers nn can Player One always ensure that at least one of the row numbers is divisible by 44? b) For which natural numbers nn can Player One always ensure that at least one of the row numbers is divisible by 33?

A strangely asymmetric inequality with square roots

Let a,b,ca,b,c be real numbers such that 0≤a≤b≤c0 \le a \le b \le c and a+b+c=1a+b+c=1. Show that abb−a+bcc−b+acc−a<14.ab\sqrt{b-a}+bc\sqrt{c-b}+ac\sqrt{c-a}<\frac{1}{4}.

Polynomial

Given is a polynomial P(x)P(x) of degree n>1n>1 with real coefficients. The equation P(P(P(x)))=P(x)P(P(P(x)))=P(x) has n3n^3 distinct real roots. Prove that these roots could be split into two groups with equal arithmetic mean.