Olympiad problems

1997 Romania National Olympiad, 2

Tags: algebra , function

Find the range of the function $f: \mathbb{R} \to \mathbb{R},$ $$f(x)=\frac{3+2\sin x}{\sqrt{1+\cos x}+\sqrt{1-\cos x}}.$$

2010 Moldova Team Selection Test, 1

Tags: algebra , polynomial

Let $ p\in\mathbb{R}_\plus{}$ and $ k\in\mathbb{R}_\plus{}$. The polynomial $ F(x)\equal{}x^4\plus{}a_3x^3\plus{}a_2x^2\plus{}a_1x\plus{}k^4$ with real coefficients has $ 4$ negative roots. Prove that $ F(p)\geq(p\plus{}k)^4$

2018 Junior Balkan Team Selection Tests - Romania, 2

Tags: algebra , inequalities

Let $k > 2$ be a real number. a) Prove that for all positive real numbers $x,y$ and $z$ the following inequality holds: $$\sqrt{x + y }+\sqrt{y + z }+\sqrt{z + x} > 2\sqrt{\frac{(x + y)(y + z)(z + x)}{xy + yz + zx}}$$ b) Prove that there exist positive real numbers $x, y$ and $z$ such that $$\sqrt{x + y }+\sqrt{y + z}+\sqrt{z + x} <k\sqrt{\frac{(x + y)(y + z)(z + x)}{xy + yz + zx}}$$ Leonard Giugiuc

2001 National Olympiad First Round, 4

Tags: polynomial , algebra

How many real solution does the equation $\dfrac{x^{2000}}{2001} + 2\sqrt 3 x^2 - 2\sqrt 5 x + \sqrt 3$ have? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ \text{None of the preceding} $

2000 Baltic Way, 11

Tags: algebra

A sequence of positive integers $a_1,a_2,\ldots $ is such that for each $m$ and $n$ the following holds: if $m$ is a divisor of $n$ and $m<n$, then $a_m$ is a divisor of $a_n$ and $a_m<a_n$. Find the least possible value of $a_{2000}$.

2017 HMNT, 9

Tags: algebra

Find the minimum value of $\sqrt{58-42x}+\sqrt{149-140\sqrt{1-x^2}}$ where $-1 \le x \le 1$.

2013 Greece Junior Math Olympiad, 1

Tags: algebra

(a) Write $A = k^4 + 4$, where $k$ is a positive integer, as a product of two factors each of them is sum of two squares of integers. (b) Simplify the expression$$K=\frac{(2^4+\frac14)(4^4+\frac14)...((2n)^4+\frac14)}{(1^4+\frac14)(3^4+\frac14)...((2n-1)^4+\frac14)}$$and write it as sum of squares of two consecutive positive integers

2017 Saint Petersburg Mathematical Olympiad, 1

Tags: algebra

It’s allowed to replace any of three coefficients of quadratic trinomial by its discriminant. Is it true that from any quadratic trinomial that does not have real roots, we can perform such operation several times to get a quadratic trinomial that have real roots?

2010 Pan African, 3

Tags: domain , function , induction , algebra

Does there exist a function $f:\mathbb{Z}\to\mathbb{Z}$ such that $f(x+f(y))=f(x)-y$ for all integers $x$ and $y$?

2021 Peru Cono Sur TST., P7

Tags: algebra

Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$. [i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]

2021 Nordic, 2

Tags: algebra , functional equation

Find all functions $f:R->R$ satisfying that for every $x$ (real number): $f(x)(1+|f(x)|)\geq x \geq f(x(1+|x|))$

2010 Malaysia National Olympiad, 6

Tags: algebra

Find the number of different pairs of positive integers $(a,b)$ for which $a+b\le100$ and \[\dfrac{a+\frac{1}{b}}{\frac{1}{a}+b}=10\]

2011 Germany Team Selection Test, 1

Tags: algebra , inequality , sequence , get the smallest

A sequence $x_1, x_2, \ldots$ is defined by $x_1 = 1$ and $x_{2k}=-x_k, x_{2k-1} = (-1)^{k+1}x_k$ for all $k \geq 1.$ Prove that $\forall n \geq 1$ $x_1 + x_2 + \ldots + x_n \geq 0.$ [i]Proposed by Gerhard Wöginger, Austria[/i]

2010 Cuba MO, 9

Tags: inequalities , number theory , algebra

Let $A$ be the subset of the natural numbers such that the sum of Its digits are multiples of$ 2009$. Find $x, y \in A$ such that $y - x > 0$ is minimum and $x$ is also minimum.

1959 AMC 12/AHSME, 45

Tags: algebra , logarithm

If $\left(\log_3 x\right)\left(\log_x 2x\right)\left( \log_{2x} y\right)=\log_{x}x^2$, then $y$ equals: $ \textbf{(A)}\ \frac92\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 81 $

2006 Regional Competition For Advanced Students, 2

Tags: algebra , system of equations

Let $ n>1$ be a positive integer an $ a$ a real number. Determine all real solutions $ (x_1,x_2,\dots,x_n)$ to following system of equations: $ x_1\plus{}ax_2\equal{}0$ $ x_2\plus{}a^2x_3\equal{}0$ … $ x_k\plus{}a^kx_{k\plus{}1}\equal{}0$ … $ x_n\plus{}a^nx_1\equal{}0$

XMO (China) 2-15 - geometry, 6.2

Tags: inequalities , algebra , geometric inequality , complex number

Assume that complex numbers $z_1,z_2,...,z_n$ satisfy $|z_i-z_j| \le 1$ for any $1 \le i <j \le n$. Let $$S= \sum_{1 \le i <j \le n} |z_i-z_j|^2.$$ (1) If $n = 6063$, find the maximum value of $S$. (2) If $n= 2021$, find the maximum value of $S$.

2018 Brazil Team Selection Test, 2

Tags: algebra

Let $f(x)$ and $g(x)$ be given by $f(x) = \frac{1}{x} + \frac{1}{x-2} + \frac{1}{x-4} + \cdots + \frac{1}{x-2018}$ $g(x) = \frac{1}{x-1} + \frac{1}{x-3} + \frac{1}{x-5} + \cdots + \frac{1}{x-2017}$. Prove that $|f(x)-g(x)| >2$ for any non-integer real number $x$ satisfying $0 < x < 2018$.

2020 Greece JBMO TST, 4

Tags: inequalities , algebra , sum , product , min

Let $A$ and $B$ be two non-empty subsets of $X = \{1, 2, . . . , 8 \}$ with $A \cup B = X$ and $A \cap B = \emptyset$. Let $P_A$ be the product of all elements of $A$ and let $P_B$ be the product of all elements of $B$. Find the minimum possible value of sum $P_A +P_B$. PS. It is a variation of [url=https://artofproblemsolving.com/community/c6h2267998p17621980]JBMO Shortlist 2019 A3 [/url]

2021 BMT, 7

Tags: algebra

Ditty can bench $80$ pounds today. Every week, the amount he benches increases by the largest prime factor of the weight he benched in the previous week. For example, since he started benching $80$ pounds, next week he would bench $85$ pounds. What is the minimum number of weeks from today it takes for Ditty to bench at least $2021$ pounds?

2016 Lusophon Mathematical Olympiad, 3

Tags: polynomial , algebra , integer polynomial , sum , roots of the equation

Suppose a real number $a$ is a root of a polynomial with integer coefficients $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$. Let $G=|a_n|+|a_{n-1}|+...+|a_1|+|a_0|$. We say that $G$ is a [i]gingado [/i] of $a$. For example, as $2$ is root of $P(x)=x^2-x-2$, $G=|1|+|-1|+|-2|=4$, we say that $4$ is a [i]gingado[/i] of $2$. What is the fourth largest real number $a$ such that $3$ is a [i]gingado [/i] of $a$?

2012 AIME Problems, 8

Tags: algebra , quadratic , complex number , quadratic formula

The complex numbers $z$ and $w$ satisfy the system \begin{align*}z+\frac{20i}{w}&=5+i,\\w+\frac{12i}{z}&=-4+10i.\end{align*} Find the smallest possible value of $|zw|^2$.

1990 Austrian-Polish Competition, 9

Tags: combinatorics , algebra , sum , partition

$a_1, a_2, ... , a_n$ is a sequence of integers such that every non-empty subsequence has non-zero sum. Show that we can partition the positive integers into a finite number of sets such that if $x_i$ all belong to the same set, then $a_1x_1 + a_2x_2 + ... + a_nx_n$ is non-zero.

2008 ISI B.Math Entrance Exam, 3

Tags: algebra , vector , complex number

Let $z$ be a complex number such that $z,z^2,z^3$ are all collinear in the complex plane . Show that $z$ is a real number .

1967 IMO Shortlist, 1

Tags: sequence , algebraic identities , algebra

Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let \[ c_n = \sum^8_{k=1} a^n_k\] for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$