Olympiad problems

Russian TST 2018, P3

Tags: algebra , polynomial

Let $a < b$ be positive integers. Prove that there is a positive integer $n{}$ and a polynomial of the form \[\pm1\pm x\pm x^2\pm\cdots\pm x^n,\]divisible by the polynomial $1+x^a+x^b$.

JOM 2015 Shortlist, A2

Tags: inequality , inequalities

Let $ a, b, c $ be positive real numbers greater or equal to $ 3 $. Prove that $$ 3(abc+b+2c)\ge 2(ab+2ac+3bc) $$ and determine all equality cases.

1997 Austrian-Polish Competition, 7

Tags: inequalities , max , algebra

(a) Prove that $p^2 + q^2 + 1 > p(q + 1)$ for any real numbers $p, q$, . (b) Determine the largest real constant $b$ such that the inequality $p^2 + q^2 + 1 \ge bp(q + 1)$ holds for all real numbers $p, q$ (c) Determine the largest real constant c such that the inequality $p^2 + q^2 + 1 \ge cp(q + 1)$ holds for all integers $p, q$.

2010 China Team Selection Test, 2

Tags: algebra , polynomial , floor function , triangle inequality , inequalities

Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$, and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.

1899 Eotvos Mathematical Competition, 1

Tags: algebra , geometry

The points $A_0, A_1, A_2, A_3, A_4$ divide a unit circle (circle of radius $1$) into five equal parts. Prove that the chords $A_0, A_1, A_0, A_2$ satisfy $$(A_0A_1 \cdot A_0A_2)^2= 5$$

2012 Princeton University Math Competition, A8

Tags: floor function , algebra

If $n$ is an integer such that $n \ge 2^k$ and $n < 2^{k+1}$, where $k = 1000$, compute the following: $$n - \left( \lfloor \frac{n -2^0}{2^1} \rfloor + \lfloor \frac{n -2^1}{2^2} \rfloor + ...+ \lfloor \frac{n -2^{k-1}}{2^k} \rfloor \right)$$

2021 Iranian Geometry Olympiad, 1

Tags: geometry , symmetry , reflection

With putting the four shapes drawn in the following figure together make a shape with at least two reflection symmetries. [img]https://cdn.artofproblemsolving.com/attachments/6/0/8ace983d3d9b5c7f93b03c505430e1d2d189fd.png[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

1975 Kurschak Competition, 2

Tags: geometry , rhombus , inscribed , geometric inequalities

Prove or disprove: given any quadrilateral inscribed in a convex polygon, we can find a rhombus inscribed in the polygon with side not less than the shortest side of the quadrilateral.

2001 District Olympiad, 2

Tags: number theory

Consider the number $n=123456789101112\ldots 99100101$. a)Find the first three digits of the number $\sqrt{n}$. b)Compute the sum of the digits of $n$. c)Prove that $\sqrt{n}$ isn't rational. [i]Valer Pop[/i]

2006 Silk Road, 4

Tags: induction , geometry

A family $L$ of 2006 lines on the plane is given in such a way that it doesn't contain parallel lines and it doesn't contain three lines with a common point.We say that the line $l_1\in L$ is [i]bounding[/i] the line $l_2\in L$,if all intersection points of the line $l_2$ with other lines from $L$ lie on the one side of the line $l_1$. Prove that in the family $L$ there are two lines $l$ and $l'$ such that the following 2 conditions are satisfied simultaneously: [b]1)[/b] The line $l$ is bounding the line $l'$; [b]2)[/b] the line $l'$ is not bounding the line $l$.