Olympiad problems

2016 Mathematical Talent Reward Programme, MCQ: P 13

Tags: quadratic , algebra

Let $P(x)=x^2+bx+c$. Suppose $P(P(1))=P(P(-2))=0$ and $P(1)\neq P(-2)$. Then $P(0)=$ [list=1] [*] $-\frac{5}{2}$ [*] $-\frac{3}{2}$ [*] $-\frac{7}{4}$ [*] $\frac{6}{7}$ [/list]

2017 Serbia Team Selection Test, 3

Tags: algebra , function

A function $f:\mathbb{N} \rightarrow \mathbb{N} $ is called nice if $f^a(b)=f(a+b-1)$, where $f^a(b)$ denotes $a$ times applied function $f$. Let $g$ be a nice function, and an integer $A$ exists such that $g(A+2018)=g(A)+1$. a) Prove that $g(n+2017^{2017})=g(n)$ for all $n \geq A+2$. b) If $g(A+1) \neq g(A+1+2017^{2017})$ find $g(n)$ for $n <A$.

2015 IFYM, Sozopol, 1

Tags: functional equation , algebra

Determine all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ which satisfy the following equations: a) $f(f(n))=4n+3$ $\forall$ $n \in \mathbb{Z}$; b) $f(f(n)-n)=2n+3$ $\forall$ $n \in \mathbb{Z}$.

2011 South africa National Olympiad, 2

Tags: algebra , system of equations

Suppose that $x$ and $y$ are real numbers that satisfy the system of equations $2^x-2^y=1$ $4^x-4^y=\frac{5}{3}$ Determine $x-y$

2024 Ukraine National Mathematical Olympiad, Problem 4

Tags: algebra , quadratic

The board contains $20$ non-constant linear functions, not necessarily distinct. For each pair $(f, g)$ of these functions ($190$ pairs in total), Victor writes on the board a quadratic function $f(x)\cdot g(x) - 2$, and Solomiya writes on the board a quadratic function $f(x)g(x)-1$. Victor calculated that exactly $V$ of his quadratic functions have a root, and Solomiya calculated that exactly $S$ of her quadratic functions have a root. Find the largest possible value of $S-V$. [i]Remarks.[/i] A linear function $y = kx+b$ is called non-constant if $k\neq 0$. [i]Proposed by Oleksiy Masalitin[/i]

1989 Swedish Mathematical Competition, 2

Tags: functional equation , algebra , continuous , functional

Find all continuous functions $f$ such that $f(x)+ f(x^2) = 0$ for all real numbers $x$.

1998 APMO, 3

Tags: algebra , inequalities , trigonometry

Let $a$, $b$, $c$ be positive real numbers. Prove that \[ \biggl(1+\frac{a}{b}\biggr) \biggl(1+\frac{b}{c}\biggr) \biggl(1+\frac{c}{a}\biggr) \ge 2 \biggl(1+\frac{a+b+c}{\sqrt[3]{abc}}\biggr). \]

2018 Hanoi Open Mathematics Competitions, 14

Tags: polynomial , algebra

Let $P(x)$ be a polynomial with degree $2017$ such that $P(k) =\frac{k}{k + 1}$, $\forall k = 0, 1, 2, ..., 2017$ . Calculate $P(2018)$.

2004 Denmark MO - Mohr Contest, 3

Tags: algebra

The digits from $1$ to $9$ are placed in the figure below with one digit in each square. The sum of three numbers placed in the same horizontal or vertical line is $13$. Show that the marked place says $4$. [img]https://cdn.artofproblemsolving.com/attachments/a/f/517b644caf59bbc57701662f21d57465855dc1.png[/img]

2023 BMT, Tie 1

Tags: algebra

Wen finds $17$ consecutive positive integers that sum to $2023$. Compute the smallest of these integers.

1992 Tournament Of Towns, (332) 4

Tags: algebra , combinatorics

$10$ numbers are placed on a circle. Their sum is equal to $100$. A sum of any three neighbouring numbers is no less than $29$. Find the minimal number $A$ such that for any such set of 10 numbers none of them is greater than $A$. Prove that this value for $A$ is really minimal. (A. Tolpygo, Kiev)

2022 AMC 12/AHSME, 21

Tags: algebra , polynomial

Let $P(x) = x^{2022} + x^{1011} + 1$. Which of the following polynomials divides $P(x)$? $\textbf{(A)}~x^2 - x + 1\qquad\textbf{(B)}~x^2 + x + 1\qquad\textbf{(C)}~x^4 + 1\qquad\textbf{(D)}~x^6 - x^3 + 1\qquad\textbf{(E)}~x^6 + x^3 + 1$

2013 AIME Problems, 8

Tags: trapezoid , algebra , circumcircle , quadratic , trigonometry , geometry

A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$.

2020 JBMO TST of France, 4

Tags: algebra , inequality

$a, b, c$ are real positive numbers for which $a+b+c=3$. Prove that $a^{12}+b^{12}+c^{12}+8(ab+bc+ca) \geq 27$

2004 IMC, 4

Tags: polynomial , matrix , algebra , linear algebra , vector

For $n\geq 1$ let $M$ be an $n\times n$ complex array with distinct eigenvalues $\lambda_1,\lambda_2,\ldots,\lambda_k$, with multiplicities $m_1,m_2,\ldots,m_k$ respectively. Consider the linear operator $L_M$ defined by $L_MX=MX+XM^T$, for any complex $n\times n$ array $X$. Find its eigenvalues and their multiplicities. ($M^T$ denotes the transpose matrix of $M$).

2020 Mediterranean Mathematics Olympiad, 3

Tags: algebra , inequalities

Prove that all postive real numbers $a,b,c$ with $a+b+c=4$ satisfy the inequality $$\frac{ab}{\sqrt[4]{3c^2+16}}+ \frac{bc}{\sqrt[4]{3a^2+16}}+ \frac{ca}{\sqrt[4]{3b^2+16}} \le\frac43 \sqrt[4]{12}$$

2017 Argentina National Olympiad, 2

Tags: number theory , sum , consecutive , algebra

In a row there are $51$ written positive integers. Their sum is $100$ . An integer is [i]representable [/i] if it can be expressed as the sum of several consecutive numbers in a row of $51$ integers. Show that for every $k$ , with $1\le k \le 100$ , one of the numbers $k$ and $100-k$ is representable.

2024 Belarus Team Selection Test, 1.3

Tags: algebra , inequality

Prove that for any real numbers $a,b,c,d \geq \frac{1}{3}$ the following inequality holds: $$\sqrt{\frac{a^6}{b^4+c^3}+\frac{b^6}{c^4+d^3}+\frac{c^6}{d^4+a^3}+\frac{d^6}{a^4+b^3}}\geq \frac{a+b+c+d}{4}$$ [i]D. Zmiaikou[/i]

PEN M Problems, 1

Tags: induction , algebra , polynomial , recursive sequences

Let $P(x)$ be a nonzero polynomial with integer coefficients. Let $a_{0}=0$ and for $i \ge 0$ define $a_{i+1}=P(a_{i})$. Show that $\gcd ( a_{m}, a_{n})=a_{ \gcd (m, n)}$ for all $m, n \in \mathbb{N}$.

2006 Junior Balkan Team Selection Tests - Moldova, 1

Tags: algebra

Let the number $x$. Using multiply and division operations of any 2 given or already given numbers we can obtain powers with natural exponent of the number $x$ (for example, $x\cdot x=x^{2}$, $x^{2}\cdot x^{2}=x^{4}$, $x^{4}: x=x^{3}$, etc). Determine the minimal number of operations needed for calculating $x^{2006}$.

1964 Czech and Slovak Olympiad III A, 3

Tags: algebra , parameterization , quadratic , quadratic polynomial

Determine all values of parameter $\alpha\in [0,2\pi]$ such that the equation $$(2\cos\alpha-1)x^2+4x+4\cos\alpha+2=0$$ has 1) a positive root $x_1$, 2) if a second root $x_2$ exists and if $x_2\neq x_1$, the $x_2\leq 0$.

2019 Moldova Team Selection Test, 7

Tags: polynomial , algebra

Let $P(X)=a_{2n+1}X^{2n+1}+a_{2n}X^{2n}+...+a_1X+a_0$ be a polynomial with all positive coefficients. Prove that there exists a permutation $(b_{2n+1},b_{2n},...,b_1,b_0)$ of numbers $(a_{2n+1},a_{2n},...,a_1,a_0)$ such that the polynomial $Q(X)=b_{2n+1}X^{2n+1}+b_{2n}X^{2n}+...+b_1X+b_0$ has exactly one real root.

1992 All Soviet Union Mathematical Olympiad, 570

Tags: number theory , sequence , integer sequence , perfect square , algebra

Define the sequence $a_1 = 1, a_2, a_3, ...$ by $$a_{n+1} = a_1^2 + a_2 ^2 + a_3^2 + ... + a_n^2 + n$$ Show that $1$ is the only square in the sequence.

2009 Brazil Team Selection Test, 1

Tags: equation , number theory , algebra

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

2021 Iran Team Selection Test, 5

Tags: algebra , arithmetic sequence , sequence

Call a triple of numbers [b]Nice[/b] if one of them is the average of the other two. Assume that we have $2k+1$ distinct real numbers with $k^2$ [b] Nice[/b] triples. Prove that these numbers can be devided into two arithmetic progressions with equal ratios Proposed by [i]Morteza Saghafian[/i]