Olympiad problems

1989 IMO Longlists, 55

The set $ \{a_0, a_1, \ldots, a_n\}$ of real numbers satisfies the following conditions: [b](i)[/b] $ a_0 \equal{} a_n \equal{} 0,$ [b](ii)[/b] for $ 1 \leq k \leq n \minus{} 1,$ \[ a_k \equal{} c \plus{} \sum^{n\minus{}1}_{i\equal{}k} a_{i\minus{}k} \cdot \left(a_i \plus{} a_{i\plus{}1} \right)\] Prove that $ c \leq \frac{1}{4n}.$

1998 Polish MO Finals, 1

Tags: algebra

Find all solutions in positive integers to: \begin{eqnarray*} a + b + c = xyz \\ x + y + z = abc \end{eqnarray*}

2021 Brazil EGMO TST, 1

Tags: number theory , algebra

Let $x_0,x_1,x_2,\dots$ be a infinite sequence of real numbers, such that the following three equalities are true: I- $x_{2k}=(4x_{2k-1}-x_{2k-2})^2$, for $k\geq 1$ II- $x_{2k+1}=|\frac{x_{2k}}{4}-k^2|$, for $k\geq 0$ III- $x_0=1$ a) Determine the value of $x_{2022}$ b) Prove that there are infinite many positive integers $k$, such that $2021|x_{2k+1}$

2006 MOP Homework, 5

Tags: algebra

Let $\{a_n\}^{\inf}_{n=1}$ and $\{b_n\}^{\inf}_{n=1}$ be two sequences of real numbers such that $a_{n+1}=2b_n-a_n$ and $b_{n+1}=2a_n-b_n$ for every positive integer $n$. Prove that $a_n>0$ for all $n$, then $a_1=b_1$.

2019 AMC 12/AHSME, 17

Tags: algebra , polynomial

Let $s_k$ denote the sum of the $\textit{k}$th powers of the roots of the polynomial $x^3-5x^2+8x-13$. In particular, $s_0=3$, $s_1=5$, and $s_2=9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a \, s_k + b \, s_{k-1} + c \, s_{k-2}$ for $k = 2$, $3$, $....$ What is $a+b+c$? $\textbf{(A)} \; -6 \qquad \textbf{(B)} \; 0 \qquad \textbf{(C)} \; 6 \qquad \textbf{(D)} \; 10 \qquad \textbf{(E)} \; 26$

2010 Contests, 2

Tags: inequalities , analytic geometry , parabola , algebra , polynomial , vieta , conic

Let $ a\geq 2$ be a real number; with the roots $ x_{1}$ and $ x_{2}$ of the equation $ x^2\minus{}ax\plus{}1\equal{}0$ we build the sequence with $ S_{n}\equal{}x_{1}^n \plus{} x_{2}^n$. [b]a)[/b]Prove that the sequence $ \frac{S_{n}}{S_{n\plus{}1}}$, where $ n$ takes value from $ 1$ up to infinity, is strictly non increasing. [b]b)[/b]Find all value of $ a$ for the which this inequality hold for all natural values of $ n$ $ \frac{S_{1}}{S_{2}}\plus{}\cdots \plus{}\frac{S_{n}}{S_{n\plus{}1}}>n\minus{}1$

2014 Regional Competition For Advanced Students, 2

Tags: algebra , system of equations

You can determine all 4-ples $(a,b, c,d)$ of real numbers, which solve the following equation system $\begin{cases} ab + ac = 3b + 3c \\ bc + bd = 5c + 5d \\ ac + cd = 7a + 7d \\ ad + bd = 9a + 9b \end{cases} $

2001 Spain Mathematical Olympiad, Problem 1

Tags: polynomial , algebra , geometry

Prove that the graph of the polynomial $P(x)$ is symmetric in respect to point $A(a,b)$ if and only if there exists a polynomial $Q(x)$ such that: $P(x) = b + (x-a)Q((x-a)^2)).$

1915 Eotvos Mathematical Competition, 1

Tags: algebra , number theory , inequalities

Let $A, B, C$ be any three real numbers. Prove that there exists a number $\nu$ such that $$An^2 + Bn+ < n!$$ for every natural number $n > \nu.$

VII Soros Olympiad 2000 - 01, 9.5

Tags: parameterization , algebra , cubic , equation

For all valid values of $a$ and $b$, solve the equation $$\frac{x^3}{(x-a) (x-b)} +\frac{a^3}{(a-b) (a-x)} + \frac{b^3}{ (b-x) (b-a)}= x^2 + a + b$$

2017 Greece Junior Math Olympiad, 2

Tags: function , algebra , inequalities

Let $x,y,z$ is positive. Solve: $\begin{cases}{x\left( {6 - y} \right) = 9}\\ {y\left( {6 - z} \right) = 9}\\ {z\left( {6 - x} \right) = 9}\end{cases}$

2001 Tuymaada Olympiad, 7

Tags: integer , algebra , fractional part

Several rational numbers were written on the blackboard. Dima wrote off their fractional parts on paper. Then all the numbers on the board squared, and Dima wrote off another paper with fractional parts of the resulting numbers. It turned out that on Dima's papers were written the same sets of numbers (maybe in different order). Prove that the original numbers on the board were integers. (The fractional part of a number $x$ is such a number $\{x\}, 0 \le \{x\} <1$, that $x-\{x\}$ is an integer.)

2011 Morocco National Olympiad, 2

Tags: linear algebra , matrix , inequalities , polynomial , algebra

Solve in $(\mathbb{R}_{+}^{*})^{4}$ the following system : $\left\{\begin{matrix} x+y+z+t=4\\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{t}=5-\frac{1}{xyzt} \end{matrix}\right.$

2021 Romania Team Selection Test, 3

Tags: algebra , sequence

Let $\alpha$ be a real number in the interval $(0,1).$ Prove that there exists a sequence $(\varepsilon_n)_{n\geq 1}$ where each term is either $0$ or $1$ such that the sequence $(s_n)_{n\geq 1}$ \[s_n=\frac{\varepsilon_1}{n(n+1)}+\frac{\varepsilon_2}{(n+1)(n+2)}+...+\frac{\varepsilon_n}{(2n-1)2n}\]verifies the inequality \[0\leq \alpha-2ns_n\leq\frac{2}{n+1}\] for any $n\geq 2.$

2016 Iran Team Selection Test, 4

Tags: algebra

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

1995 China National Olympiad, 2

Tags: function , algebra , functional equation

Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying the following conditions: (1) $f(1)=1$; (2) $\forall n\in \mathbb{N}$, $3f(n) f(2n+1) =f(2n) ( 1+3f(n) )$; (3) $\forall n\in \mathbb{N}$, $f(2n) < 6 f(n)$. Find all solutions of equation $f(k) +f(l)=293$, where $k<l$. ($\mathbb{N}$ denotes the set of all natural numbers).

2021 JHMT HS, 10

Tags: algebra , polynomial , calculus , derivative

A polynomial $P(x)$ of some degree $d$ satisfies $P(n) = n^3 + 10n^2 - 12$ and $P'(n) = 3n^2 + 20n - 1$ for $n = -2, -1, 0, 1, 2.$ Also, $P$ has $d$ distinct (not necessarily real) roots $r_1, r_2, \ldots, r_d.$ The value of \[ \sum_{k=1}^{d}\frac{1}{4 - r_k^2} \] can be expressed as a common fraction $\tfrac{p}{q}.$ What is the value of $p + q?$

1999 IMO, 2

Tags: inequalities , algebra , maximization

Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such the inequality \[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C \left(\sum_{i}x_{i} \right)^4\] holds for any $x_{1}, \ldots ,x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.

2014 Contests, 2

Tags: algebra , system of equations

You can determine all 4-ples $(a,b, c,d)$ of real numbers, which solve the following equation system $\begin{cases} ab + ac = 3b + 3c \\ bc + bd = 5c + 5d \\ ac + cd = 7a + 7d \\ ad + bd = 9a + 9b \end{cases} $

1966 IMO Shortlist, 40

Tags: parametric equation , equation , algebra

For a positive real number $p$, find all real solutions to the equation \[\sqrt{x^2 + 2px - p^2} -\sqrt{x^2 - 2px - p^2} =1.\]

1982 Poland - Second Round, 1

Tags: algebra , polynomial

Prove that if $ c, d $ are integers with $ c \neq d $, $ d > 0 $ then the equation $$ x^3 - 3cx^2 - dx + c = 0$$ has no more than one rational root.

VI Soros Olympiad 1999 - 2000 (Russia), 9.2

Tags: algebra , number theory , fractional part , floor function

Solve the equation $[x]\{x\} = 1999x$, where $[x]$ denotes the largest integer less than or equal to $x$, and $\{x\} = x -[x] $

2023 UMD Math Competition Part I, #10

Tags: algebra

There are $100$ people in a room. Some are [i]wise[/i] and some are [i]optimists[/i]. $\quad \bullet~$ A [i]wise[/i] person can look at someone and know if they are wise or if they are an optimist. $\quad \bullet~$ An [i]optimist[/i] thinks everyone is wise (including themselves). Everyone in the room writes down what they think is the number of wise people in the room. What is the smallest possible value for the average? $$ \mathrm a. ~ 10\qquad \mathrm b.~25\qquad \mathrm c. ~50 \qquad \mathrm d. ~75 \qquad \mathrm e. ~100 $$

1970 Bulgaria National Olympiad, Problem 2

Tags: rates , algebra

Two bicyclists traveled the distance from $A$ to $B$, which is $100$ km, with speed $30$ km/h and it is known that the first started $30$ minutes before the second. $20$ minutes after the start of the first bicyclist from $A$, there is a control car started whose speed is $90$ km/h and it is known that the car is reached the first bicyclist and is driving together with him for $10$ minutes, went back to the second and was driving for $10$ minutes with him and after that the car is started again to the first bicyclist with speed $90$ km/h and etc. to the end of the distance. How many times will the car drive together with the first bicyclist? [i]K. Dochev[/i]

2019 Purple Comet Problems, 9

Tags: algebra , logarithm

Find the positive integer $n$ such that $32$ is the product of the real number solutions of $x^{\log_2(x^3)-n} = 13$