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Difference between revisions of "2018 UNCO Math Contest II Problems"

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==Problem 9==
 
==Problem 9==
  
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Call a set of integers Grassilian if each of its elements is at least as large as the number of
 +
elements in the set. For example, the three-element set <math>\{2, 48, 100\}</math> is not Grassilian, but the
 +
six-element set <math>\{6, 10, 11, 20, 33, 39\}</math> is Grassilian. Let <math>G(n)</math> be the number of Grassilian subsets of <math>\{1, 2, 3, ..., n\}</math>. (By definition, the empty set is a subset of every set and is Grassilian.)
 +
(a) Find <math>G(3)</math>, <math>G(4)</math>, and <math>G(5)</math>.
 +
(b) Find a recursion formula for <math>G(n + 1)</math>. That is, find a formula that expresses <math>G(n + 1)</math> in
 +
terms of <math>G(n), G(n  1),\;dots</math>
 +
(c) Give an explanation that shows that the formula you give is correct.
  
 
[[2018 UNCO Math Contest II Problems/Problem 9|Solution]]
 
[[2018 UNCO Math Contest II Problems/Problem 9|Solution]]

Revision as of 03:27, 13 January 2019

Twenty-sixth Annual UNC Math Contest Final Round January 20, 2018

Rules: Three hours; no electronic devices. The positive integers are 1, 2, 3, 4, . . . A prime is an integer strictly greater than one that is evenly divisible by no integers other than itself and 1. The primes are 2, 3, 5, 7, 11, 13, 17, . . .

Problem 1

A printer used 1890 digits to number all the pages in the Seripian Puzzle Book. How many pages are in the book? (For example, to number the pages in a book with twelve pages, the printer would use fifteen digits.)

Solution

Problem 2

Segment AB is perpendicular to segment BC and segment AC is perpendicular to segment BD. If segment AB has length 15 and segment DC has length 16, then what is the area of triangle ABC?


Solution

Problem 3

Find all values of B that have the property that if (x, y) lies on the hyperbola 2y^2-x^2 = 1, then so does the point (3x + 4y, 2x + By).

Solution

Problem 4

How many positive integer factors of $36,000,000$ are not perfect squares?

Solution

Problem 5

Find the length of segment BC formed in the middle circle by a line that goes through point A and is tangent to the leftmost circle. The three circles in the figure all have radius one and their centers lie on the horizontal line. The leftmost and rightmost circles are tangent to the circle in the middle. Point A is at the rightmost intersection of the rightmost circle and the horizontal line.

Solution

Problem 6

Circling the square. Exactly one of these polynomials is a perfect square; that is, can be written as $(p(x))^2$ where $p(x)$ is also a polynomial. Circle the choice that is a perfect square, and for that choice, find the square root, the polynomial $p(x)$.

(A) $36-49x^2 + 14x^4$

(B) $36-48x^2 + 14x^4-x^6$

(C) $9-12x + 4x^2 + 12x^3-8x^4 + 4x^6$

(D) $36-49x^2 + 15x^4-x^6$

Solution

Problem 7

Let $x = 2A + 10B$ where $A$ and $B$ are randomly chosen with replacement from among the

positive integers less than or equal to twelve. What is the probability that $x$ is a multiple of $12$?

Solution

Problem 8

Let $p(x) = x^{2018} + x^{1776}-3x^4-3$. Find the remainder when you divide $p(x)$ by $x^3-x$

Solution

Problem 9

Call a set of integers Grassilian if each of its elements is at least as large as the number of elements in the set. For example, the three-element set $\{2, 48, 100\}$ is not Grassilian, but the six-element set $\{6, 10, 11, 20, 33, 39\}$ is Grassilian. Let $G(n)$ be the number of Grassilian subsets of $\{1, 2, 3, ..., n\}$. (By definition, the empty set is a subset of every set and is Grassilian.) (a) Find $G(3)$, $G(4)$, and $G(5)$. (b) Find a recursion formula for $G(n + 1)$. That is, find a formula that expresses $G(n + 1)$ in terms of $G(n), G(n 1),\;dots$ (c) Give an explanation that shows that the formula you give is correct.

Solution

Problem 10

Solution

Problem 11

Solution

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