/sci/ - Science & Math

>>10447649
y-you say that like it isn't

Previous Thread >>10445459

[math]
\text{Let } S \text{ be the set of ordered triples } (a, b, c) \text{ of distinct elements of a finite set } A \text{.}
\\
\text{Suppose that}
\\ \text{ } \\
\quad 1.~(a,b,c) \in S \text{ if and only if } (b,c,a) \in S;
\\ \text{ } \\
\quad 2.~(a,b,c) \in S \text{ if and only if } (c,b,a) \notin S;
\\ \text{ } \\
\quad 3.~(a,b,c) \text{ and } (c,d,a) \text{ are both in } S \text{ if and only if } (b,c,d) \text{ and } (d,a,b) \text{ are}
\\
\quad \text{both in } S \text{.}
\\ \text{ } \\
\text{Prove that there exists a one-to-one function } g \text{ from } A \text{ to } R \text{ such that } g(a) <
\\
g(b) < g(c) \text{ implies } (a,b,c) \in S \text{. Note: } R \text{ is the set of real numbers.}
[/math]

>>10446075
it's an entire book

Previous Thread >>10442502

[math]
\text{Suppose that each of 20 students has made a choice of anywhere from 0 to}
\\
\text{6 courses from a total of 6 courses offered. Prove or disprove: there are 5}
\\
\text{students and 2 courses such that all 5 have chosen both courses or all 5 have}
\\
\text{chosen neither course.}
[/math]

Previous Thread >>10439604

[math]
\text{Let } C_1 \text{ and } C_2 \text{ be circles whose centers are } 10 \text{ units apart, and whose radii}
\\
\text{are } 1 \text{ and } 3 \text{. Find, with proof, the locus of all points } M \text{for which there exists}
\\
\text{points } X \text{ on } C_1 \text{ and } Y \text{ on } C_2 \text{ such that } M \text{ is the midpoint of the line segment}
\\
XY \text{.}
[/math]

[math]
\text{Find the least number } A \text{ such that for any two squares of combined area 1, }
\\
\text{a rectangle of area } A \text{ exists such that the two squares can be packed in the }
\\
\text{rectangle (without interior overlap). You may assume that the sides of the }
\\
\text{squares are parallel to the sides of the rectangle.}
[/math]

[math]
\text{For a positive real number } \alpha \text{, define}
\\
\qquad \qquad \qquad \qquad \qquad S (\alpha) = \lbrace \lfloor n \alpha \rfloor \rbrace \colon n = 1,2,3, … \rbrace .
\\
\text{Prove that } \lbrace 1,2,3 … \rbrace \text{ cannot be expressed as the disjoint union of three sets}
\\
S(\alpha),S(\beta) and S(\gamma). [\text{As usual, }\lbrace x \rbrace \text{ is the greatest integer} \le x.]
[/math]

Suppose <span class="math">q_0,q_1,q_2,\ldots[/spoiler] is an infinite sequence of integers satisfying the following two conditions:

(i) <span class="math">m-n[/spoiler] divides <span class="math">q_m - q_n[/spoiler] for <span class="math">m > n \geq 0[/spoiler],

(ii) there is a polynomial <span class="math">P[/spoiler] such that <span class="math">|q_n| < P(n)[/spoiler] for all <span class="math">n[/spoiler].

Prove that there is a polynomial <span class="math">Q[/spoiler] such that <span class="math">q_n = Q(n)[/spoiler] for all <span class="math">n[/spoiler].

A calculator is broken so that the only keys that still work are the <span class="math">\sin[/spoiler], <span class="math">\cos[/spoiler], <span class="math">\tan[/spoiler], <span class="math">\sin^{-1}[/spoiler], <span class="math">\cos^{-1}[/spoiler], and <span class="math">\tan^{-1}[/spoiler] buttons. The display initially shows <span class="math">0[/spoiler]. Given any positive rational number <span class="math">q[/spoiler], show that pressing some finite sequence of buttons will yield <span class="math">q[/spoiler]. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.

Given a nonisosceles, nonright triangle <span class="math">ABC[/spoiler], let <span class="math">O[/spoiler] denote the center of its circumscribed circle, and let <span class="math">A_1[/spoiler], <span class="math">B_1[/spoiler], and <span class="math">C_1[/spoiler] be the midpoints of sides <span class="math">BC[/spoiler], <span class="math">CA[/spoiler], and <span class="math">AB[/spoiler], respectively. Point <span class="math">A_2[/spoiler] is located on the ray <span class="math">OA_1[/spoiler] so that <span class="math">\Delta OAA_1[/spoiler] is similar to <span class="math">\Delta OA_2A[/spoiler]. Points <span class="math">B_2[/spoiler] and <span class="math">C_2[/spoiler] on rays <span class="math">OB_1[/spoiler] and <span class="math">OC_1[/spoiler], respectively, are defined similarly. Prove that lines <span class="math">AA_2[/spoiler], <span class="math">BB_2[/spoiler], and <span class="math">CC_2[/spoiler] are concurrent, i.e. these three lines intersect at a point.

A calculator is broken so that the only keys that still work are the <span class="math">\sin[/spoiler], <span class="math">\cos[/spoiler], <span class="math">\tan[/spoiler], <span class="math">\sin^{-1}[/spoiler], <span class="math">\cos^{-1}[/spoiler], and <span class="math">\tan^{-1}[/spoiler] buttons. The display initially shows <span class="math">0[/spoiler]. Given any positive rational number <span class="math">q[/spoiler], show that pressing some finite sequence of buttons will yield <span class="math">q[/spoiler]. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.

Given a nonisosceles, nonright triangle <span class="math">ABC[/spoiler], let <span class="math">O[/spoiler] denote the center of its circumscribed circle, and let <span class="math">A_1[/spoiler], <span class="math">B_1[/spoiler], and <span class="math">C_1[/spoiler] be the midpoints of sides <span class="math">BC[/spoiler], <span class="math">CA[/spoiler], and <span class="math">AB[/spoiler], respectively. Point <span class="math">A_2[/spoiler] is located on the ray <span class="math">OA_1[/spoiler] so that <span class="math">\Delta OAA_1[/spoiler] is similar to <span class="math">\Delta OA_2A[/spoiler]. Points <span class="math">B_2[/spoiler] and <span class="math">C_2[/spoiler] on rays <span class="math">OB_1[/spoiler] and <span class="math">OC_1[/spoiler], respectively, are defined similarly. Prove that lines <span class="math">AA_2[/spoiler], <span class="math">BB_2[/spoiler], and <span class="math">CC_2[/spoiler] are concurrent, i.e. these three lines intersect at a point.

Let <span class="math">|U|[/spoiler], <span class="math">\sigma(U)[/spoiler], and <span class="math">\pi(U)[/spoiler] denote the number of elements, the sum, and the product, respectively, of a finite set <span class="math">U[/spoiler] of positive integers. (If <span class="math">U[/spoiler] is the empty set, <span class="math">|U| = 0[/spoiler], <span class="math">\sigma(U) = 0[/spoiler], <span class="math">\pi(U) = 1[/spoiler].) Let <span class="math">S[/spoiler] be a finite set of positive integers. As usual, let <span class="math">{n \choose k}[/spoiler] denote <span class="math">\frac{n!}{k!(n-k)!}[/spoiler]. Prove that <div class="math">\sum_{U \subseteq S} (-1)^{|U|} {m - \sigma(U) \choose |S|} = \pi(S)</div> for all integers <span class="math">m \geq \sigma(S)[/spoiler].

Let <span class="math">a_1,a_2,a_3,\ldots[/spoiler] be a sequence of positive real numbers satisfying <span class="math">\sum^n_{j=1} a_j \geq \sqrt{n}[/spoiler] for all <span class="math">n \geq 1[/spoiler]. Prove that, for all <span class="math">n \geq 1[/spoiler], <div class="math">\sum^n_{j=1} a^2_j > \frac{1}{4}\left(1+\frac{1}{2}+\cdots + \frac{1}{n}\right).</div>

A convex hexagon <span class="math">ABCDEF[/spoiler] is inscribed in a circle such that <span class="math">AB = CD = EF[/spoiler] and diagonals <span class="math">AD[/spoiler], <span class="math">BE[/spoiler], and <span class="math">CF[/spoiler] are concurrent. Let <span class="math">P[/spoiler] be the intersection of <span class="math">AD[/spoiler] and <span class="math">CE[/spoiler]. Prove that <span class="math">CP/PE = (AC/CE)^2[/spoiler].

The sides of a 99-gon are initially colored so that consecutive sides are red, blue, red, blue, ... red, blue, yellow. We make a sequence of modifications in the coloring, changing the color of one side at a time to one of the three given colors (red, blue, yellow), under the constraint that no two adjacent sides may be the same color. By making a sequence of such modifications, is it possible to arrive at the coloring in which consecutive sides are red, blue, red, blue, red, blue, ... red, yellow, blue?

Apparently there are two problems today.

Show that, for any fixed integer <span class="math">\,n \geq 1,\,[/spoiler] the sequence
<div class="math">
2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots \pmod n
</div>is eventually constant.

[The tower of exponents is defined by <span class="math">a_1 = 2, \; a_{i+1} =
2^{a_i}[/spoiler].
Also <span class="math">a_i\; (\bmod\; n)[/spoiler] means the remainder which results from dividing <span class="math">a_i[/spoiler] by <span class="math">n[/spoiler].]

For any nonempty set <span class="math">\,S\,[/spoiler] of numbers, let <span class="math">\,\sigma(S)\,[/spoiler] and <span class="math">\,\pi(S)\,[/spoiler] denote the sum and product, respectively, of the elements of <span class="math">\,S\,[/spoiler]. Prove that
<div class="math">
\sum \frac{\sigma(S)}{\pi(S)} = (n^2 + 2n) - \left(
1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right)
(n+1),
</div>where ``<span class="math">\Sigma[/spoiler]'' denotes a sum involving all nonempty subsets <span class="math">S[/spoiler] of <span class="math">\{1,2,3, \ldots,n\}[/spoiler].

In triangle ABC, angle A is twice angle B, angle C is obtuse, and the three side lengths a, b, c are integers. Determine, with proof, the minimum possible perimeter.

An acute-angled triangle ABC is given in the plane. The circle with diameter AB intersects altitude CC' and its extension at points M and N and the circle with diameter AC intersects altitude BB' and its extensions at P and Q. Prove that the points M, N, P, Q lie on a common circle.

Find, with proof, the number of positive integers whose base-<span class="math">n[/spoiler] representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by <span class="math">\pm 1[/spoiler] from some digit further to the left. (Your answer should be an explicit function of <span class="math">n[/spoiler] in simplest form.)

Suppose that necklace A has 14 beads and necklace B has 19. Prove that for any odd integer n >= 1, there is a way to number each of the 33 beads with an integer from the sequence

n, n+1, n+2, ..., n+32

so that each integer is used once, and adjacent beads correspond to relatively prime integers. (Here a ``necklace'' is viewed as a circle in which each bead is adjacent to two other beads.)