MathDB

2024 OMpD

Part of Mathematicians for Fun Olympiad

Subcontests

(4)
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"Given A and B, how many roads are part of the route connecting A to B?"

Lavidópolis is a city with 2024 neighborhoods. Lavi Dopes was elected mayor, and since he saw that there were no roads in the city, he asked Gil Bento, the monster engineer, to design the city's roads according to the following rules: 1. Any two neighborhoods are connected by at most one two-way road; 2. For any two neighborhoods, there is exactly one route from one neighborhood to another, which may pass through some intermediate neighborhoods, but never passes through the same neighborhood more than once.
Mayor Lavi Dopes wants to try for re-election, but since he knows nothing about the city and only shows up during campaign times (he spent all this time stealing... I mean, thinking about math problems), he wants to find a pair of neighborhoods such that the number of roads that are part of the route connecting them is maximized among all pairs of neighborhoods. To do this, he starts asking Gil Bento various questions, all in the following manner: he chooses two of the 2024 neighborhoods, say A and B, and asks: "Given neighborhoods A and B, how many roads are part of the route connecting A to B?"
Knowing that Gil Bento always answers correctly to each question, determine the minimum number of questions that Lavi Dopes needs to ask to achieve his goal, regardless of how Gil Bento has designed the roads of Lavidópolis.

a_n+a_{n-1} is a perfect square for all n

Let a0,a1,a2,…a_0, a_1, a_2, \dots be an infinite sequence of positive integers with the following properties: - a0a_0 is a given positive integer; - For each integer n≥1n \geq 1, ana_n is the smallest integer greater than an−1a_{n-1} such that an+an−1a_n + a_{n-1} is a perfect square. For example, if a0=3a_0 = 3, then a1=6a_1 = 6, a2=10a_2 = 10, a3=15a_3 = 15, and so on.
(a) Let TT be the set of numbers of the form ak−ala_k - a_l, with k≥l≥0k \geq l \geq 0 integers. Prove that, regardless of the value of a0a_0, the number of positive integers not in TT is finite.
(b) Calculate, as a function of a0a_0, the number of positive integers that are not in TT.

euclidean norm in functions

Let n n be a positive integer. Determine the largest possible value of k k with the following property: there exists a bijective function ϕ:[0,1]→[0,1]k \phi: [0, 1] \to [0, 1]^k and a constant C>0 C > 0 such that, for all x,y∈[0,1] x, y \in [0, 1] ,
∥ϕ(x)−ϕ(y)∥≤C∥x−y∥k. \| \phi(x) - \phi(y) \| \leq C \| x - y \|^k.
Note: ∥⋅∥ \| \cdot \| denotes the Euclidean norm, that is, ∥(a1,…,an)∥=a12+⋯+an2 \| (a_1, \ldots, a_n) \| = \sqrt{a_1^2 + \cdots + a_n^2} .
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number of triples such that the roots of ax^2+bx+c=0 are both integers

For each positive integer n n , let f(n) f(n) be the number of ordered triples (a,b,c) (a, b, c) such that a,b,c∈{1,2,…,n} a, b, c \in \{1, 2, \ldots, n\} and that the two roots (possibly equal) of the quadratic equation ax2+bx+c=0 ax^2 + bx + c = 0 are both integers.
(a) Prove that for every positive real number C C , there exists a positive integer nC n_C such that for all integers n≥nC n \geq n_C , we have f(n)>C⋅n f(n) > C \cdot n .
(b) Prove that for every positive real number C C , there exists a positive integer nC n_C such that for all integers n≥nC n \geq n_C , we have f(n)<C⋅n20252024 f(n) < C \cdot n^{\frac{2025}{2024}} .

confused cockroach in a cube ABCDEFGH and visits each vertice

A confused cockroach is initially at vertex AA of a cube ABCDEFGHABCDEFGH with edges measuring 11 meter, as shown in the figure. Every second, the cockroach moves 11 meter, always choosing to go to one of the three adjacent vertices to its current position. For example, after 11 second, the cockroach could stop at vertex BB, DD, or EE.
(a) In how many ways can the cockroach stop at vertex GG after 33 seconds? (b) Is it possible for the cockroach to stop at vertex A after exactly 20232023 seconds? (c) In how many ways can the cockroach stop at A after exactly 20242024 seconds?
Note: One way for the cockroach to stop at a vertex after a certain number of seconds differs from another way if, at some point, the cockroach is at different vertices in the trajectory. For example, there are 22 ways for the cockroach to stop at CC after 22 seconds: one of them passes through AA, BB, CC, and the other through AA, DD, CC.
https://cdn.discordapp.com/attachments/954427908359876608/1299721377124847616/Screenshot_2024-10-16_173123.png?ex=671e3b5b&is=671ce9db&hm=76962ee2949d8324c2f7022ef63f8b7d3c6fe3aabf4ecf526f44249439f204ac&

f a differentiable function such that f(0)= 0 and 0&lt;f&#039;(t)\leq1 for all t\in[0,1]

Let f:R→R f: \mathbb{R} \to \mathbb{R} be a differentiable function such that f(0)=0 f(0) = 0 and 0<f′(t)≤1 0 < f'(t) \leq 1 for all t∈[0,1] t \in [0, 1] . Show that:
(∫01f(t) dt)2≥∫01f(t)3 dt. \left( \int_0^1 f(t) \, dt \right)^2 \geq \int_0^1 f(t)^3 \, dt.
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calculate the perimeter of triangle MNP

Let ABCDABCD be a convex quadrilateral, and MM, NN, and PP be the midpoints of diagonals ACAC and BDBD, and side ADAD, respectively. Also, suppose that ∠ABC+∠DCB=90\angle{ABC} + \angle{DCB} = 90 and that AB=6AB = 6, CD=8CD = 8. Calculate the perimeter of triangle MNPMNP.

Let ABCDE be a convex pentagon whose vertices lie on a circle

Let ABCDE ABCDE be a convex pentagon whose vertices lie on a circle Γ \Gamma . The tangents to Γ \Gamma at C C and E E intersect at X X , and the segments CE CE and AD AD intersect at Y Y . Given that CE CE is perpendicular to BD BD , that XY XY is parallel to BD BD , that AY=BD AY = BD , and that ∠BAD=30∘ \angle BAD = 30^\circ , what is the measure of the angle ∠BDA \angle BDA ?
Proposed by João Gilberti Alves Tavares

matrices A, B such that ABBA - BAAB = A - B.

Let n n be a positive integer, and let A A and B B be n×n n \times n matrices with real coefficients such that
ABBA−BAAB=A−B. ABBA - BAAB = A - B.
(a) Prove that Tr(A)=Tr(B) \text{Tr}(A) = \text{Tr}(B) and that Tr(A2)=Tr(B2) \text{Tr}(A^2) = \text{Tr}(B^2) .
(b) Prove that det⁡A=det⁡B \det A = \det B .
Note: Tr(X) \text{Tr}(X) denotes the trace of X X , which is the sum of the elements on its main diagonal, and det⁡X \det X denotes the determinant of X X .
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kawaii subsets

We say that a subset T T of {1,2,…,2024}\{1, 2, \dots, 2024\} is kawaii if T T has the following properties: 1. T T has at least two distinct elements; 2. For any two distinct elements x x and y y of T T , x−y x - y does not divide x+y x + y .
For example, the subset T={31,71,2024} T = \{31, 71, 2024\} is kawaii, but T={5,15,75} T = \{5, 15, 75\} is not kawaii because 15−5=10 15 - 5 = 10 divides 15+5=20 15 + 5 = 20 .
What is the largest possible number of elements that a kawaii subset can have?

find all the possible values for ompd

Let O,M,PO, M, P and DD be distinct digits from each other, and different from zero, such that O<M<P<DO < M < P < D, and the following equation is true: OMPD‾×(OM‾−D‾)=MDDMP‾−OM‾ \overline{\text{OMPD}} \times \left( \overline{\text{OM}} - \overline{\text{D}} \right) = \overline{\text{MDDMP}} - \overline{\text{OM}}
(a) Using estimates, explain why it is impossible for the value of OO to be greater than or equal to 33. (b) Explain why OO cannot be equal to 11. (c) Is it possible for MM to be greater than or equal to 55? Justify. (d) Determine the values of MM, PP, and DD.