Difference between revisions of "2018 UNCO Math Contest II Problems"
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==Problem 8== | ==Problem 8== | ||
| − | + | Let <math>p(x) = x^{2018} + x^{1776}-3x^4-3</math>. Find the remainder when you divide <math>p(x)</math> by <math>x^3-x</math> | |
[[2018 UNCO Math Contest II Problems/Problem 8|Solution]] | [[2018 UNCO Math Contest II Problems/Problem 8|Solution]] | ||
Revision as of 03:26, 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, . . .
Contents
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.)
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?
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).
Problem 4
How many positive integer factors of 
are not perfect squares?
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.
Problem 6
Circling the square. Exactly one of these polynomials is a perfect square; that is, can be
written as 
where 
is also a polynomial. Circle the choice that is a perfect square,
and for that choice, find the square root, the polynomial .
(A) 
(B) 
(C) 
(D) 
Problem 7
Letwhere
and
are randomly chosen with replacement from among the
where 
and 
are randomly chosen with replacement from among the
positive integers less than or equal to twelve. What is the probability that 
is a multiple of
?
Problem 8
Let 
. Find the remainder when you divide by 
