MathDB

3

Part of 2024 OMpD

Problems(3)

number of triples such that the roots of ax^2+bx+c=0 are both integers

Source: 2024 5th OMpD L3 P3 - Brazil - Olimpíada Matemáticos por Diversão

10/16/2024
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}} .
algebra
confused cockroach in a cube ABCDEFGH and visits each vertice

Source: 2024 5th OMpD L2 P3 - Brazil - Olimp&amp;iacute;ada Matem&amp;aacute;ticos por Divers&amp;atilde;o

10/16/2024
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&
combinatorics3D geometrygeometry
f a differentiable function such that f(0)= 0 and 0&lt;f&#039;(t)\leq1 for all t\in[0,1]

Source: 2024 5th OMpD LU P3 - Brazil - Olimp&iacute;ada Matem&aacute;ticos por Divers&atilde;o

10/16/2024
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.
functioncalculus