Olympiad problems

2001 Saint Petersburg Mathematical Olympiad, 9.4

Tags: St. Petersburg MO , GCD , number theory , Divisibility , greatest common divisor

Let $a,b,c\in\mathbb{Z^{+}}$ such that $$(a^2-1, b^2-1, c^2-1)=1$$ Prove that $$(ab+c, bc+a, ca+b)=(a,b,c)$$ (As usual, $(x,y,z)$ means the greatest common divisor of numbers $x,y,z$) [I]Proposed by A. Golovanov[/i]

2000 Saint Petersburg Mathematical Olympiad, 9.3

Tags: number theory , Polynomials , integer polynomials , Existence , St. Petersburg MO

Let $P(x)=x^{2000}-x^{1000}+1$. Do there exist distinct positive integers $a_1,\dots,a_{2001}$ such that $a_ia_j|P(a_i)P(a_j)$ for all $i\neq j$? [I]Proposed by A. Baranov[/i]

2001 Saint Petersburg Mathematical Olympiad, 9.2

Tags: quadratics , algebra , quadratic trinomial , roots , St. Petersburg MO

Define a quadratic trinomial to be "good", if it has two distinct real roots and all of its coefficients are distinct. Do there exist 10 positive integers such that there exist 500 good quadratic trinomials coefficients of which are among these numbers? [I]Proposed by F. Petrov[/i]

2001 Saint Petersburg Mathematical Olympiad, 9.3

Tags: geometry , St. Petersburg MO , pentagon , circumscribed

A convex pentagon $ABCDE$ is given with $AB=BC$, $CD=DE$ and $\angle A=\angle C=\angle E>90^{\circ}$. Prove that the pentagon is circumscribed [I]Proposed by F. Baharev[/i]

2023 Saint Petersburg Mathematical Olympiad, 6

Tags: combinatorics , St. Petersburg MO , graph theory

There are several gentlemen in the meeting of the Diogenes Club, some of which are friends with each other (friendship is mutual). Let's name a participant unsociable if he has exactly one friend among those present at the meeting. By the club rules, the only friend of any unsociable member can leave the meeting (gentlemen leave the meeting one at a time). The purpose of the meeting is to achieve a situation in which that there are no friends left among the participants. Prove that if the goal is achievable, then the number of participants remaining at the meeting does not depend on who left and in what order.

2000 Saint Petersburg Mathematical Olympiad, 9.2

Tags: geometry , Angle Chasing , St. Petersburg MO

Let $AA_1$ and $CC_1$ be altitudes of acute angled triangle $ABC$. A point $D$ is chosen on $AA_1$ such that $A_1D=C_1D$. Let $E$ be the midpoint of $AC$. Prove that points $A$, $C_1$, $D$, $E$ are concylic. [I]Proposed by S. Berlov[/i]