MathDB

2017 Team #1: Polynomials with equal floors

Source:

2/19/2017

Let

P(x)

,

Q(x)

be nonconstant polynomials with real number coefficients. Prove that if

\lfloor P(y) \rfloor = \lfloor Q(y) \rfloor

for all real numbers

y

, then

P(x) = Q(x)

for all real numbers

x

.

algebrapolynomial

2017 Algebra/NT #1: Polynomial with Integer Coefficients

Source:

2/19/2017

Let

Q(x)=a_0+a_1x+\dots+a_nx^n

be a polynomial with integer coefficients, and

0\le a_i<3

for all

0\le i\le n

.
Given that

Q(\sqrt{3})=20+17\sqrt{3}

, compute

Q(2)

.

algebrapolynomial

2017 Geometry #1: find AP/CP in a cyclic quad

Source:

2/19/2017

Let

A

,

B

,

C

,

D

be four points on a circle in that order. Also,

AB=3

,

BC=5

,

CD=6

, and

DA=4

. Let diagonals

AC

and

BD

intersect at

P

. Compute

\frac{AP}{CP}

.

geometry

2017 Combinatorics #1: Third roll is the sum of first two

Source:

2/19/2017

Kelvin the Frog is going to roll three fair ten-sided dice with faces labelled

0, 1, \dots, 9

. First he rolls two dice, and finds the sum of the two rolls. Then he rolls the third die. What is the probability that the sum of the first two rolls equals the third roll?

probability

2017 Theme #1

Source:

5/8/2018

Two ordered pairs

(a,b)

and

(c,d)

, where

a,b,c,d

are real numbers, form a basis of the coordinate plane if

ad \neq bc

. Determine the number of ordered quadruples

(a,b,c)

of integers between

1

and

3

inclusive for which

(a,b)

and

(c,d)

form a basis for the coordinate plane.

combinatoricsanalytic geometry

2017 General #1

Source:

5/8/2018

Find the sum of all positive integers whose largest proper divisor is

55

. (A proper divisor of

n

is a divisor that is strictly less than

n

.)

number theory

2017 Guts #1: Never lucky

Source:

2/21/2017

A random number generator will always output

7

. Sam uses this random number generator once. What is the expected value of the output?

1

Problems(7)