Olympiad problems

2016 India Regional Mathematical Olympiad, 6

Tags: inequalities , algebra

Positive integers $a, b, c$ satisfy $\frac1a +\frac1b +\frac1c<1$. Prove that $\frac1a +\frac1b +\frac1c\le \frac{41}{42}$. Also prove that equality in fact holds in the second inequality.

2007 All-Russian Olympiad Regional Round, 11.2

Tags: polynomial , quadratic , algebra

Two quadratic polynomials $ f_{1},f_{2}$ satisfy $ f_{1}'(x)f_{2}'(x)\geq |f_{1}(x)|\plus{}|f_{2}(x)|\forall x\in\mathbb{R}$ . Prove that $ f_{1}\cdot f_{2}\equal{} g^{2}$ for some $ g\in\mathbb{R}[x]$.

2005 All-Russian Olympiad, 2

Tags: calculus , integration , quadratic , algebra

Find the number of subsets $A\subset M=\{2^0,\,2^1,\,2^2,\dots,2^{2005}\}$ such that equation $x^2-S(A)x+S(B)=0$ has integral roots, where $S(M)$ is the sum of all elements of $M$, and $B=M\setminus A$ ($A$ and $B$ are not empty).

2015 Mathematical Talent Reward Programme, MCQ: P 4

Tags: combinatorics , algebra

Let $n$ be an odd integer. Placing no more than one $X$ in each cell of a $n \times n$ grid, what is the greatest number of $X$ 's that can be put on the grid without getting $n$ $X$'s together vertically, horizontally or diagonally? [list=1] [*] $2{{n}\choose {2}}$ [*] ${{n}\choose {2}}$ [*] $2n $ [*] $2{{n}\choose {2}}-1$ [/list]

1987 Spain Mathematical Olympiad, 4

Tags: algebra , system

If $a$ and $b$ are distinct real numbers, solve the systems (a) $\begin{cases} x+y = 1 \\ (ax+by)^2 \le a^2x+b^2y \end{cases}$ and (b) $\begin{cases} x+y = 1 \\ (ax+by)^4 \le a^4x+b^4y \end{cases}$

1994 Romania TST for IMO, 1:

Tags: algebra , number theory

Let $p$ be a (positive) prime number. Suppose that real numbers $a_1, a_2, . . ., a_{p+1}$ have the property that, whenever one of the numbers is deleted, the remaining numbers can be partitioned into two classes with the same arithmetic mean. Show that these numbers must be equal.

1995 Baltic Way, 9

Tags: induction , logarithm , algebra

Prove that \[\frac{1995}{2}-\frac{1994}{3}+\frac{1993}{4}-\ldots -\frac{2}{1995}+\frac{1}{1996}=\frac{1}{999}+\frac{3}{1000}+\ldots +\frac{1995}{1996}\]

2022 AIME Problems, 13

Tags: algebra , polynomial

There is a polynomial $P(x)$ with integer coefficients such that $$P(x)=\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)}$$ holds for every $0<x<1.$ Find the coefficient of $x^{2022}$ in $P(x)$

2020 Middle European Mathematical Olympiad, 4#

Tags: algebra , equation , induction

Find all positive integers $n$ for which there exist positive integers $x_1, x_2, \dots, x_n$ such that $$ \frac{1}{x_1^2}+\frac{2}{x_2^2}+\frac{2^2}{x_3^2}+\cdots +\frac{2^{n-1}}{x_n^2}=1.$$

Russian TST 2015, P2

Tags: algebra , inequalities

Let $a,b,c,d$ be positive real numbers satisfying $a^2+b^2+c^2+d^2=1$. Prove that \[a^3+b^3+c^3+d^3+abcd\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)\leqslant 1.\]

2006 China Northern MO, 4

Tags: quadratic , function , algebra

Given a function $f(x)=x^{2}+ax+b$ with $a,b \in R$, if there exists a real number $m$ such that $\left| f(m) \right| \leq \frac{1}{4}$ and $\left| f(m+1) \right| \leq \frac{1}{4}$, then find the maximum and minimum of the value of $\Delta=a^{2}-4b$.

2019 India PRMO, 8

Tags: algebra

How many positive integers $n$ are there such that $3 \leq n \leq 100$ and $x^{2^{n}} + x + 1$ is divisible by $x^2 + x + 1$?

2015 Moldova Team Selection Test, 1

Tags: trigonometry , trigonometric inequality , algebra , inequalities , function

Let $c\in \Big(0,\dfrac{\pi}{2}\Big) , a = \Big(\dfrac{1}{sin(c)}\Big)^{\dfrac{1}{cos^2 (c)}}, b = \Big(\dfrac{1}{cos(c)}\Big)^{\dfrac{1}{sin^2 (c)}}$. \\Prove that at least one of $a,b$ is bigger than $\sqrt[11]{2015}$.

2012 Switzerland - Final Round, 2

Tags: algebra , functional , functional equation

Determine all functions $f : R \to R$ such that for all $x, y\in R$ holds $$f (f(x) + 2f(y)) = f(2x) + 8y + 6.$$

2008 Romania Team Selection Test, 5

Tags: greatest common divisor , modular arithmetic , algebra , polynomial , number theory

Find the greatest common divisor of the numbers \[ 2^{561}\minus{}2, 3^{561}\minus{}3, \ldots, 561^{561}\minus{}561.\]

1949 Kurschak Competition, 1

Tags: inequalities , algebra , trigonometry

Prove that $\sin x + \frac12 \sin 2x + \frac13 \sin 3x > 0$ for $0 < x < 180^o$.

1989 IMO Longlists, 4

Tags: calculus , rectangle , equation , area , algebra

Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. He knows that the sides of the carpet are integral numbers of feet and that his two storerooms have the same (unknown) length, but widths of 38 feet and 50 feet respectively. What are the carpet dimensions?

1966 IMO, 4

Tags: trigonometry , induction , algebra , trigonometric identities

Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]

2006 Bulgaria Team Selection Test, 3

Tags: algebra , polynomial , combinatorics

[b]Problem 6.[/b] Let $p>2$ be prime. Find the number of the subsets $B$ of the set $A=\{1,2,\ldots,p-1\}$ such that, the sum of the elements of $B$ is divisible by $p.$ [i] Ivan Landgev[/i]

1990 IMO Shortlist, 26

Tags: algebra , polynomial , number theory , cubic , iteration

Let $ p(x)$ be a cubic polynomial with rational coefficients. $ q_1$, $ q_2$, $ q_3$, ... is a sequence of rationals such that $ q_n \equal{} p(q_{n \plus{} 1})$ for all positive $ n$. Show that for some $ k$, we have $ q_{n \plus{} k} \equal{} q_n$ for all positive $ n$.

2024 Sharygin Geometry Olympiad, 15

Tags: algebra , geometry , ratio , circumcircle , inradius

The difference of two angles of a triangle is greater than $90^{\circ}$. Prove that the ratio of its circumradius and inradius is greater than $4$.

1986 Dutch Mathematical Olympiad, 1

Tags: algebra , functional

$f(x) = \frac{12x+9}{19x+86}, \,\, x \ne -\frac{86}{19}$ Prove that $\exists ! \,\,\, {x_o \in R} \,\,\, \forall h_1,h_2 \in R [f(x_0+h_1)f(x_0-h_1)=f(x_0+h_2)f(x_0-h_2)]$, and calculate $x_0$.

1998 USAMTS Problems, 2

Tags: quadratic , arithmetic series , algebra , quadratic formula

There are inﬁnitely many ordered pairs $(m,n)$ of positive integers for which the sum \[ m + ( m + 1) + ( m + 2) +... + ( n - 1 )+n\] is equal to the product $mn$. The four pairs with the smallest values of $m$ are $(1, 1), (3, 6), (15, 35),$ and $(85, 204)$. Find three more $(m, n)$ pairs.

2012 APMO, 1

Tags: geometry , polynomial , inequalities , algebra

Let $ P $ be a point in the interior of a triangle $ ABC $, and let $ D, E, F $ be the point of intersection of the line $ AP $ and the side $ BC $ of the triangle, of the line $ BP $ and the side $ CA $, and of the line $ CP $ and the side $ AB $, respectively. Prove that the area of the triangle $ ABC $ must be $ 6 $ if the area of each of the triangles $ PFA, PDB $ and $ PEC $ is $ 1 $.

2013 Dutch IMO TST, 5

Tags: algebra , inequalities

Let $a, b$, and $c$ be positive real numbers satisfying $abc = 1$. Show that $a + b + c \ge \sqrt{\frac13 (a + 2)(b + 2)(c + 2)}$