Olympiad problems

2006 China Team Selection Test, 1

Tags: inequalities , limit , algebra , ratio , induction

Two positive valued sequences $\{ a_{n}\}$ and $\{ b_{n}\}$ satisfy: (a): $a_{0}=1 \geq a_{1}$, $a_{n}(b_{n+1}+b_{n-1})=a_{n-1}b_{n-1}+a_{n+1}b_{n+1}$, $n \geq 1$. (b): $\sum_{i=1}^{n}b_{i}\leq n^{\frac{3}{2}}$, $n \geq 1$. Find the general term of $\{ a_{n}\}$.

1976 IMO Shortlist, 8

Tags: algebra , coefficient , polynomial

Let $P$ be a polynomial with real coefficients such that $P(x) > 0$ if $x > 0$. Prove that there exist polynomials $Q$ and $R$ with nonnegative coefficients such that $P(x) = \frac{Q(x)}{R(x)}$ if $x > 0.$

2011 AMC 10, 12

Tags: algebra , system of equations

The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws did they make? $\textbf{(A)}\,13 \qquad\textbf{(B)}\,14 \qquad\textbf{(C)}\,15 \qquad\textbf{(D)}\,16 \qquad\textbf{(E)}\,17$

2003 District Olympiad, 1

Tags: algebra , number theory

Find the disjoint sets $B$ and $C$ such that $B \cup C = \{1,2,..., 10\}$ and the product of the elements of $C$ equals the sum of elements of $B$.

2024 Vietnam Team Selection Test, 1

Tags: function , functional equation , polynomial , algebra

Let $P(x) \in \mathbb{R}[x]$ be a monic, non-constant polynomial. Determine all continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(P(x))+y+2023f(y))=P(x)+2024f(y),$$ for all reals $x,y$.

1993 Balkan MO, 1

Tags: algebra

Let $a,b,c,d,e,f$ be six real numbers with sum 10, such that \[ (a-1)^2+(b-1)^2+(c-1)^2+(d-1)^2+(e-1)^2+(f-1)^2 = 6. \] Find the maximum possible value of $f$. [i]Cyprus[/i]

2024 Greece National Olympiad, 1

Tags: quadratic , algebra

Let $a, b, c$ be reals such that some two of them have difference greater than $\frac{1}{2 \sqrt{2}}$. Prove that there exists an integer $x$, such that $$x^2-4(a+b+c)x+12(ab+bc+ca)<0.$$

2004 Bulgaria Team Selection Test, 1

Tags: algebra , number theory , induction , polynomial

Let $n$ be a positive integer. Find all positive integers $m$ for which there exists a polynomial $f(x) = a_{0} + \cdots + a_{n}x^{n} \in \mathbb{Z}[X]$ ($a_{n} \not= 0$) such that $\gcd(a_{0},a_{1},\cdots,a_{n},m)=1$ and $m|f(k)$ for each $k \in \mathbb{Z}$.

2021 Regional Olympiad of Mexico West, 1

Tags: inequalities , algebra

Let $a$ and $b$ be positive real numbers such that $a+b = 1$. Prove that $$\frac12 \le \frac{a^3+b^3}{a^2+b^2} \le 1$$

2023 Grand Duchy of Lithuania, 1

Tags: algebra , functional equation

Given a non-zero real number $a$. Find all functions $f : R \to R$, such that $$f(f(x + y)) = f(x + y) + f(x)f(y) + axy$$ for all $x, y \in R$.

2023 ELMO Shortlist, A3

Tags: algebra

Does there exist an infinite sequence of integers $a_0$, $a_1$, $a_2$, $\ldots$ such that $a_0\ne0$ and, for any integer $n\ge0$, the polynomial \[P_n(x)=\sum_{k=0}^na_kx^k\] has $n$ distinct real roots? [i]Proposed by Amol Rama and Espen Slettnes[/i]

2003 China Team Selection Test, 3

Tags: binomial theorem , modular arithmetic , induction , algebra

Let $ \left(x_{n}\right)$ be a real sequence satisfying $ x_{0}=0$, $ x_{2}=\sqrt[3]{2}x_{1}$, and $ x_{n+1}=\frac{1}{\sqrt[3]{4}}x_{n}+\sqrt[3]{4}x_{n-1}+\frac{1}{2}x_{n-2}$ for every integer $ n\geq 2$, and such that $ x_{3}$ is a positive integer. Find the minimal number of integers belonging to this sequence.

2010 District Olympiad, 4

Tags: algebra , function , search

Consider the sequence $ a_n\equal{}\left|z^n\plus{}\frac{1}{z^n}\right|\ ,\ n\ge 1$, where $ z\in \mathbb{C}^*$ is given. i) Prove that if $ a_1>2$, then: \[ a_{n\plus{}1}<\frac{a_n\plus{}a_{n\plus{}2}}{2}\ ,\ (\forall)n\in \mathbb{N}^*\] ii) Prove that if there is a $ k\in \mathbb{N}^*$ such that $ a_k\le 2$, then $ a_1\le 2$.

1983 IMO Longlists, 68

Tags: function , trigonometry , algebra

Three of the roots of the equation $x^4 -px^3 +qx^2 -rx+s = 0$ are $\tan A, \tan B$, and $\tan C$, where $A, B$, and $C$ are angles of a triangle. Determine the fourth root as a function only of $p, q, r$, and $s.$

1984 USAMO, 4

Tags: algebra , combinatorics , system of equations

A difficult mathematical competition consisted of a Part I and a Part II with a combined total of $28$ problems. Each contestant solved $7$ problems altogether. For each pair of problems, there were exactly two contestants who solved both of them. Prove that there was a contestant who, in Part I, solved either no problems or at least four problems.

II Soros Olympiad 1995 - 96 (Russia), 11.3

Tags: trigonometry , algebra , system of equations

Solve the system of equations $$\begin{cases} \sin \frac{\pi}{2}xy =z \\ \sin \frac{\pi}{2}yz =x \\ \sin \frac{\pi}{2}zx =y \end{cases} \,\,\, ?$$

2021 BMT, 3

Tags: algebra

Compute $\log_2 6 \cdot \log_3 72 - \log_2 9 - \log_3 8$.

2002 Indonesia MO, 5

Tags: modular arithmetic , algebra

Nine of the numbers $4, 5, 6, 7, 8, 12, 13, 16, 18, 19$ are going to be inputted to the empty cells in the following table: $\begin{array} {|c|c|c|} \cline{1-3} 10 & & \\ \cline{1-3} & & 9 \\ \cline{1-3} & 3 & \\ \cline{1-3} 11 & & 17 \\ \cline{1-3} & 20 & \\ \cline{1-3} \end{array}$ such that each row sums to the same number, and each column sums to the same number. Determine all possible arrangements.

2011 BAMO, 3

Tags: set , arithmetic sequence , distance , algebra

Let $S$ be a finite, nonempty set of real numbers such that the distance between any two distinct points in $S$ is an element of $S$. In other words, $|x-y|$ is in $S$ whenever $x \ne y$ and $x$ and $y$ are both in $S$. Prove that the elements of $S$ may be arranged in an arithmetic progression. This means that there are numbers $a$ and $d$ such that $S = \{a, a+d, a+2d, a+3d, ..., a+kd, ...\}$.

2017 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: equation , root , algebra

If $a$ is real number such that $x_1$ and $x_2$, $x_1\neq x_2$ , are real numbers and roots of equation $x_2-x+a=0$. Prove that $\mid {x_1}^2-{x_2}^2 \mid =1$ iff $\mid {x_1}^3-{x_2}^3 \mid =1$

II Soros Olympiad 1995 - 96 (Russia), 11.2

Tags: algebra , trigonometry

Solve the equation $$arc \sin (\sin x) + arc \cos (\cos x)=0$$

2007 German National Olympiad, 5

Tags: arithmetic sequence , algebra , real number , set , finite

Determine all finite sets $M$ of real numbers such that $M$ contains at least $2$ numbers and any two elements of $M$ belong to an arithmetic progression of elements of $M$ with three terms.

2013 Iran MO (3rd Round), 4

Tags: function , functional equation , algebra

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(0) \in \mathbb Q$ and \[f(x+f(y)^2 ) = {f(x+y)}^2.\] (25 points)

2013 Peru IMO TST, 2

Tags: absolute value , algebra , polynomial

Let $a \geq 3$ be a real number, and $P$ a polynomial of degree $n$ and having real coefficients. Prove that at least one of the following numbers is greater than or equal to $1:$ $$|a^0- P(0)|, \ |a^1-P(1)| , \ |a^2-P(2)|, \cdots, |a^{n + 1}-P(n + 1)|.$$

2011 Romania National Olympiad, 3

Tags: function , cyclic function , algebra

Let be three positive real numbers $ a,b,c. $ Show that the function $ f:\mathbb{R}\longrightarrow\mathbb{R} , $ $$ f(x)=\frac{a^x}{b^x+c^x} +\frac{b^x}{a^x+c^x} +\frac{c^x}{a^x+b^x} , $$ is nondecresing on the interval $ \left[ 0,\infty \right) $ and nonincreasing on the interval $ \left( -\infty ,0 \right] . $