MathDB

1

2d geometry, points in disks

n

be a positive integer and let

x_1,\ldots,x_n

be

n

nonzero points in

\mathbb R^2

. Suppose

\langle x_i,x_j\rangle

(scalar or dot product) is a rational number for all

i,j

(

1\le i,j\le n

). Let

S

denote all points of

\mathbb R^2

of the form

\sum_{i=1}^na_ix_i

where the

a_i

are integers. A closed disk of radius

R

and center

P

is the set of points at distance at most

R

from

P

(includes the points distance

R

from

P

). Prove that there exists a positive number

R

and closed disks

D_1,D_2,\ldots

of radius

R

such that
(a) Each disk contains exactly two points of

S

; (b) Every point of

S

lies in at least one disk; (c) Two distinct disks intersect in at most one point.

Problem 6

1

FE R^2->R, bounding absolute value

Let

(a_1,b_1),\ldots,(a_n,b_n)

be

n

points in

\mathbb R^2

(where

\mathbb R

denotes the real numbers), and let

\epsilon>0

be a positive number. Can we find a real-valued function

f(x,y)

that satisfies the following three conditions?
1.

f(0,0)=1

; 2.

f(x,y)\ne0

for only finitely many

(x,y)\in\mathbb R^2

; 3.

\sum_{r=1}^n\left|f(x+a_r,y+b_r)-f(x,y)\right|<\epsilon

for every

(x,y)\in\mathbb R^2

.
Justify your answer.

Problem 5

1

improper integral, arctan

Evaluate

\int^\infty_0\frac{\operatorname{arctan}(\pi x)-\operatorname{arctan}(x)}xdx

(where

0\le\operatorname{arctan}(x)<\frac\pi2

for

0\le x<\infty

).

Problem 4

1

surjectivity of sum of subseries of harmonic series

Consider the harmonic series

\sum_{n\ge1}\frac1n=1+\frac12+\frac13+\ldots

. Prove that every positive rational number can be obtained as an unordered partial sum of this series. (An unordered partial sum may skip some of the terms

\frac1k

.)

Problem 3

1

2015! divides determinant of integer powers

Let

(a_i)_{1\le i\le2015}

be a sequence consisting of

2015

integers, and let

(k_i)_{1\le i\le2015}

be a sequence of

2015

positive integers (positive integer excludes

0

). Let

A=\begin{pmatrix}a_1^{k_1}&a_1^{k_2}&\cdots&a_1^{k_{2015}}\\a_2^{k_1}&a_2^{k_2}&\cdots&a_2^{k_{2015}}\\\vdots&\vdots&\ddots&\vdots\\a_{2015}^{k_1}&a_{2015}^{k_2}&\cdots&a_{2015}^{k_{2015}}\end{pmatrix}.

Prove that

2015!

divides

\det A

.

Problem 2

1

origami, find volume of folded shape (VTRMC 2015 P2)

The planar diagram below, with equilateral triangles and regular hexagons, sides length

2

cm, is folded along the dashed edges of the polygons, to create a closed surface in three-dimensional Euclidean spaces. Edges on the periphery of the planar diagram are identified (or glued) with precisely one other edge on the periphery in a natural way. Thus, for example,

BA

will be joined to

QP

and

AC

will be joined to

DC

. Find the volume of the three-dimensional region enclosed by the resulting surface. https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMy9jL2ZiZjc1ZjY5Nzk5YzRiMjhjODNlZDBiZjU1MzljYzZkNTVhOGQ3LnBuZw==&rn=VlRSTUMgMjAxNS5wbmc=

Problem 1

1

Find all n such that n^4+6n^3+11n^2+3n+31 is perfect square.

Find all n such that

n^{4}+6n^{3}+11n^{2}+3n+31

is a perfect square.

2015 VTRMC

Subcontests

2d geometry, points in disks

FE R^2->R, bounding absolute value

improper integral, arctan

surjectivity of sum of subseries of harmonic series

2015! divides determinant of integer powers

origami, find volume of folded shape (VTRMC 2015 P2)

Find all n such that n^4+6n^3+11n^2+3n+31 is perfect square.

2015 VTRMC

Subcontests

2d geometry, points in disks

FE R^2-&gt;R, bounding absolute value

improper integral, arctan

surjectivity of sum of subseries of harmonic series

2015! divides determinant of integer powers

origami, find volume of folded shape (VTRMC 2015 P2)

Find all n such that n^4+6n^3+11n^2+3n+31 is perfect square.

FE R^2->R, bounding absolute value