Olympiad problems

2024 Chile TST Ibero., 4

Tags: TST , inequalities , algebra , TitusLemma , Chile

Prove that if $ a $, $ b $, and $ c $ are positive real numbers, then the following inequality holds: \[ \frac{a + 3c}{a + b} + \frac{c + 3a}{b + c} + \frac{4b}{c + a} \geq 6. \]

Let $\triangle ABC$ be an acute-angled triangle. Let $P$ be the midpoint of $BC$, and $K$ the foot of the altitude from $A$ to side $BC$. Let $D$ be a point on segment $AP$ such that $\angle BDC = 90^\circ$. Let $E$ be the second point of intersection of line $BC$ with the circumcircle of $\triangle ADK$. Let $F$ be the second point of intersection of line $AE$ with the circumcircle of $\triangle ABC$. Prove that $\angle AFD = 90^\circ$.

2023 Chile TST IMO, 2

Tags: number theory , Chile , TST

Determine the number of pairs of positive integers $ (p, k) $ such that $ p $ is a prime number and $ p^2 + 2^k $ is a perfect square less than 2023. A number is called a perfect square if it is the square of an integer.

2023 Chile Classification NMO Juniors, 4

Tags: geometry , Chile

In the convex quadrilateral $ABCD$, $M$ is the midpoint of side $AD$, $AD = BD$, lines $CM$ and $AB$ are parallel, and $3\angle LBAC = \angle LACD$. Find the measure of angle $\angle ACB$.

2023 Chile Classification NMO Seniors, 2

Tags: combinatorics , Chile

There are 7 numbers on a board. The product of any four of them is divisible by 2023. Prove that at least one of the numbers on the board is divisible by 119.

2024 Chile Classification NMO Seniors, 3

Tags: combinatorics , Chile , prime numbers

Is it possible to place 100 consecutive numbers around a circle in some order such that the product of each pair of adjacent numbers is always a perfect square? (Recall that a number is a perfect square if it is the square of an integer.)

2024 Chile Classification NMO Juniors, 3

Tags: number theory , Chile

Bus tickets from a transportation company are numbered with six digits, ranging from 000000 to 999999. A ticket is considered "lucky" if the sum of the first three digits equals the sum of the last three digits. For example, ticket 721055 is lucky, whereas 003101 is not. Determine how many consecutive tickets a person must buy to guarantee obtaining at least one lucky ticket, regardless of the starting ticket number.