This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 6

1997 Moscow Mathematical Olympiad, 5
Tags: Grade 9 , 1997
Let $1+x+x^2+...+x^{n-1}=F(x)G(x)$, where $n>1$ and where $F$ and $G$ are polynomials whose coefficients are zeroes and units. Prove that one of the polynomials $F$ and $G$ can be represented in the form $(1+x+x^2+...x^{k-1})T(x),$ where $k>1$ and $T$ is a polynomial whose coefficients are zeroes and units.

1997 Moscow Mathematical Olympiad, 4
Tags: Grade 9 , 1997
Along a circular railroad, $n$ trains circulate in the same direction at equal distances between them. Stations $A, B$ and $C$ on this railroad (denoted as the trains pass them) form an equilateral triangle. Ira enters station $A$ at the same time as Alex enters station $B$ in order to take the nearest train. It is knows that if they enter the stations at the same time as the driver Roma passes a forest, then Ira takes her train earlier than Alex; otherwise Alex takes the train earlier than or simultaneously with Ira. What part of the railroad goes through the forest (between which stations)?

2016 Saint Petersburg Mathematical Olympiad, 6
Tags: geometry , incircle , incenter , Petersburg , Grade 9
Incircle of $\triangle ABC$ touch $AC$ at $D$. $BD$ intersect incircle at $E$. Points $F,G$ on incircle are such points, that $FE \parallel BC,GE \parallel AB$. $I_1,I_2$ are incenters of $DEF,DEG$. Prove that angle bisector of $\angle GDF$ passes though the midpoint of $I_1I_2 $.

1997 Moscow Mathematical Olympiad, 1
Tags: Grade 9 , 1997
In a triangle one side is $3$ times shorter than the sum of the other two. Prove that the angle opposite said side is the smallest of the triangle’s angles.

1997 Moscow Mathematical Olympiad, 3
Tags: Grade 9 , 1997
Convex octagon $AC_1BA_1CB_1$ satisfies: $AB_1=AC_1$, $BC_1=BA_1$, $CA_1=CB_1$ and $\angle{A}+\angle{B}+\angle{C}=\angle{A_1}+\angle{B_1}+\angle{C_1}$. Prove that the area of $\triangle{ABC}$ is equal to half the area of the octagon.

1997 Moscow Mathematical Olympiad, 2
Tags: Grade 9 , 1997
$9$ different pieces of cheese are placed on a plate. Is it always possible to cut one of them into two parts so that the $10$ pieces obtained were divisible into two portions of equal mass of $5$ pieces each?