MathDB

2022 Algebra/NT #5

Source:

3/11/2022

Given a positive integer

k

, let

||k||

denote the absolute difference between

k

and the nearest perfect square. For example,

||13||=3

since the nearest perfect square to

13

is

16

. Compute the smallest positive integer

n

such that

\frac{||1|| + ||2|| + ...+ ||n||}{n}=100

.

algebra

2022 Geometry 5

Source:

3/14/2022

Let triangle

ABC

be such that

AB = AC = 22

and

BC = 11

. Point

D

is chosen in the interior of the triangle such that

AD = 19

and

\angle ABD + \angle ACD = 90^o

. The value of

BD^2 + CD^2

can be expressed as

\frac{a}{b}

, where

a

and

b

are relatively prime positive integers. Compute

100a + b

.

geometry

2022 Team 5

Source:

3/14/2022

Let

ABC

be a triangle with centroid

G

, and let

E

and

F

be points on side

BC

such that

BE = EF = F C

. Points

X

and

Y

lie on lines

AB

and

AC

, respectively, so that

X

,

Y

, and

G

are not collinear. If the line through

E

parallel to

XG

and the line through

F

parallel to

Y G

intersect at

P\ne G

, prove that

GP

passes through the midpoint of

XY

.

geometrybarycentermidpointparallel

2022 Combinatorics 5

Source:

3/18/2022

Five cards labeled

1, 3, 5, 7, 9

are laid in a row in that order, forming the five-digit number

13579

when read from left to right. A swap consists of picking two distinct cards, and then swapping them. After three swaps, the cards form a new five-digit number n when read from left to right. Compute the expected value of

n

.

combinatorics

5

Problems(4)