MathDB

3

Part of 2022 Rioplatense Mathematical Olympiad

Problems(4)

Rioplatense 2022 - Level 3 - Problem 3

Source:

12/6/2022
Let nn be a positive integer. Given a sequence of nonnegative real numbers x1,…,xnx_1,\ldots ,x_n we define the transformed sequence y1,…,yny_1,\ldots ,y_n as follows: the number yiy_i is the greatest possible value of the average of consecutive terms of the sequence that contain xix_i. For example, the transformed sequence of 2,4,1,4,12,4,1,4,1 is 3,4,3,4,5/23,4,3,4,5/2. Prove that a) For every positive real number tt, the number of yiy_i such that yi>ty_i>t is less than or equal to 2t(x1+⋯+xn)\frac{2}{t}(x_1+\cdots +x_n). b) The inequality y1+⋯+yn32n≤x12+⋯+xn232n\frac{y_1+\cdots +y_n}{32n}\leq \sqrt{\frac{x_1^2+\cdots +x_n^2}{32n}} holds.
algebra
Rioplatense 2022 - Level 2 - Problem 3

Source:

12/6/2022
Let ABCABC be a triangle with AB<ACAB<AC. There are two points XX and YY on the angle bisector of BA^CB\widehat AC such that XX is between AA and YY and BXBX is parallel to CYCY. Let ZZ be the reflection of XX with respect to BCBC. Line YZYZ cuts line BCBC at point PP. If line BYBY cuts line CXCX at point KK, prove that KA=KPKA=KP.
geometry
Rioplatense 2022 - Level A - Problem 3

Source:

12/6/2022
On the table there are several cards. Each card has an integer number written on it. Beto performs the following operation several times: he chooses two cards from the table, calculates the difference between the numbers written on them, writes the result on his notebook and removes those two cards from the table. He can perform this operation as many times as he wants, as long as there are at least two cards on the table. After this, Beto multiplies all the numbers that he wrote on his notebook. Beto's goal is that the result of this multiplication is a multiple of 71007^{100}.
a) Prove that if there are 207207 cards initially on the table then Beto can always achieve his goal, no matter what the numbers on the cards are. b) If there are 128128 cards initially on the table, is it true that Beto can always achieve his goal?
number theory
Rioplatense 2022 - Level 1 - Problem 3

Source:

12/6/2022
On the table there are NN cards. Each card has an integer number written on it. Beto performs the following operation several times: he chooses two cards from the table, calculates the difference between the numbers written on them, writes the result on his notebook and removes those two cards from the table. He can perform this operation as many times as he wants, as long as there are at least two cards on the table. After this, Beto multiplies all the numbers that he wrote on his notebook. Beto's goal is that the result of this multiplication is a multiple of 71007^{100}. Find the minimum value of NN such that Beto can always achieve his goal, no matter what the numbers on the cards are.
number theory