MathDB

1

numbers in a grid

n \geq 2

be an integer. Lucia chooses

n

real numbers

x_1,x_2,\ldots,x_n

such that

\left| x_i-x_j \right|\geq 1

for all

i\neq j

. Then, in each cell of an

n \times n

grid, she writes one of these numbers, in such a way that no number is repeated in the same row or column. Finally, for each cell, she calculates the absolute value of the difference between the number in the cell and the number in the first cell of its same row. Determine the smallest value that the sum of the

n^2

numbers that Lucia calculated can take.

4

1

prove midpoints

In an acute-angled triangle

ABC

, let

D

be a point on the segment

BC

. Let

R

and

S

be the feet of the perpendiculars from

D

to

AC

and

AB

, respectively. The line

DR

intersects the circumcircle of

BDS

at

X

, with

X \neq D

. Similarly, the line

DS

intersects the circumcircle of

CDR

at

Y

, with

Y \neq D

. Prove that if

XY

is parallel to

RS

, then

D

is the midpoint of

BC

.

5

1

find primes satisfying a condition

Find all pairs of primes

(p,q)

such that

6pq

divides

p^3+q^2+38

2

1

Permutation of \(1,\ldots,2n\) in the sum of rows and columns

In each cell of an

n \times n

grid, one of the numbers

0

,

1,

or

2

must be written. Determine all positive integers

n

for which there exists a way to fill the

n \times n

grid such that, when calculating the sum of the numbers in each row and each column, the numbers

1, 2, \ldots, 2n

are obtained in some order.

3

1

prove lines are parallel

Let

ABC

an acute triangle and

D,E

and

F

be the feet of altitudes from

A,B

and

C

, respectively. The line

EF

and the circumcircle of

ABC

intersect at

P

, such that

F

it´s between

E

and

P

. Lines

BP

and

DF

intersect at

Q

. Prove that if

ED=EP

, then

CQ

and

DP

are parallel.

1

Sum of numbers with largest prime divisor odd

An integer

n \geq 2

is said to be tuanis if, when you add the smallest prime divisor of

n

and the largest prime divisor of

n

(these divisors can be the same), you obtain an odd result. Calculate the sum of all tuanis numbers that are less or equal to

2023

.

2023 Pan-American Girls’ Mathematical Olympiad

Subcontests

numbers in a grid

prove midpoints

find primes satisfying a condition

Permutation of \(1,\ldots,2n\) in the sum of rows and columns

prove lines are parallel

Sum of numbers with largest prime divisor odd

2023 Pan-American Girls&rsquo; Mathematical Olympiad

Subcontests

numbers in a grid

prove midpoints

find primes satisfying a condition

Permutation of \(1,\ldots,2n\) in the sum of rows and columns

prove lines are parallel

Sum of numbers with largest prime divisor odd

2023 Pan-American Girls’ Mathematical Olympiad