Olympiad problems

1995 VJIMC, Problem 1

Discuss the solvability of the equations \begin{align*}\lambda x+y+z&=a\\x+\lambda y+z&=b\\x+y+\lambda z&=c\end{align*}for all numbers $\lambda,a,b,c\in\mathbb R$.

2017 Kazakhstan NMO, Problem 3

Tags: algebra , sequence

An infinite, strictly increasing sequence $\{a_n\}$ of positive integers satisfies the condition $a_{a_n}\le a_n + a_{n + 3}$ for all $n\ge 1$. Prove that there are infinitely many triples $(k, l, m)$ of positive integers such that $k <l <m$ and $a_k + a_m = 2a_l$.

LMT Accuracy Rounds, 2023 S Tie

Tags: algebra

Estimate the value of $$\sum^{2023}_{n=1} \left(1+ \frac{1}{n} \right)^n$$ to $3$ decimal places.

2008 Bulgarian Autumn Math Competition, Problem 9.3

Tags: algebra , number theory , natural number , divisor , polynomial

Let $n$ be a natural number. Prove that if $n^5+n^4+1$ has $6$ divisors then $n^3-n+1$ is a square of an integer.

1986 Balkan MO, 3

Tags: induction , algebra

Let $a,b,c$ be real numbers such that $ab\not= 0$ and $c>0$. Let $(a_{n})_{n\geq 1}$ be the sequence of real numbers defined by: $a_{1}=a, a_{2}=b$ and \[a_{n+1}=\frac{a_{n}^{2}+c}{a_{n-1}}\] for all $n\geq 2$. Show that all the terms of the sequence are integer numbers if and only if the numbers $a,b$ and $\frac{a^{2}+b^{2}+c}{ab}$ are integers.

2002 Kazakhstan National Olympiad, 6

Tags: algebra , polynomial

Find all polynomials $ P (x) $ with real coefficients that satisfy the identity $ P (x ^ 2) = P (x) P (x + 1) $.

1997 IMO Shortlist, 11

Tags: algebra , polynomial , coefficient

Let $ P(x)$ be a polynomial with real coefficients such that $ P(x) > 0$ for all $ x \geq 0.$ Prove that there exists a positive integer n such that $ (1 \plus{} x)^n \cdot P(x)$ is a polynomial with nonnegative coefficients.

2000 Bulgaria National Olympiad, 1

Tags: algebra , polynomial

Find all polynomials $P(x)$ with real coefficients such that \[P(x)P(x + 1) = P(x^2), \quad \forall x \in \mathbb R.\]

2018 Tajikistan Team Selection Test, 4

Tags: algebra , inequality

Problem 4. Let a,b be positive real numbers and let x,y be positive real numbers less than 1, such that: a/(1-x)+b/(1-y)=1 Prove that: ∛ay+∛bx≤1.

2020 Taiwan TST Round 1, 2

Tags: function , algebra , functional equation

Let $\mathbb{R}$ be the set of all real numbers. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for any $x,y\in \mathbb{R}$, there holds \[f(x+f(y))+f(xy)=yf(x)+f(y)+f(f(x)).\]

1986 India National Olympiad, 5

Tags: inequalities , algebra , polynomial

If $ P(x)$ is a polynomial with integer coefficients and $ a$, $ b$, $ c$, three distinct integers, then show that it is impossible to have $ P(a)\equal{}b$, $ P(b)\equal{}c$, $ P(c)\equal{}a$.

2023 BMT, Tie 2

Tags: algebra

The polynomial $P(x) = 3x^3 -2x^2 +ax+b$ has roots $\sin^2 \theta$, $\cos^2 \theta$, and $\sin \theta \cos\theta$ for some angle $\theta$. Compute $P(1)$.

2020 Jozsef Wildt International Math Competition, W31

Tags: polynomial , algebra

Let $P$ be a real polynomial with degree $n\ge1$ such that $$P(0),P(1),P(4),P(9),\ldots,P(n^2)$$ are in $\mathbb Z$. Prove that $\forall a\in\mathbb Z,P(a^2)\in\mathbb Z$. [i]Proposed by Moubinool Omarjee[/i]

1986 AMC 12/AHSME, 24

Tags: algebra , polynomial

Let $p(x) = x^{2} + bx + c$, where $b$ and $c$ are integers. If $p(x)$ is a factor of both \[x^{4} + 6x^{2} + 25\quad\text{and}\quad 3x^{4} + 4x^{2} + 28x + 5,\] what is $p(1)$? $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 8 $

1993 IMO Shortlist, 6

Tags: function , functional equation , algebra

Let $\mathbb{N} = \{1,2,3, \ldots\}$. Determine if there exists a strictly increasing function $f: \mathbb{N} \mapsto \mathbb{N}$ with the following properties: (i) $f(1) = 2$; (ii) $f(f(n)) = f(n) + n, (n \in \mathbb{N})$.

2019 Serbia National Math Olympiad, 6

Tags: sequence , algebra

Sequences $(a_n)_{n=0}^{\infty}$ and $(b_n)_{n=0}^{\infty}$ are defined with recurrent relations : $$a_0=0 , \;\;\; a_1=1, \;\;\;\; a_{n+1}=\frac{2018}{n} a_n+ a_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$ and $$b_0=0 , \;\;\; b_1=1, \;\;\;\; b_{n+1}=\frac{2020}{n} b_n+ b_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$ Prove that :$$\frac{a_{1010}}{1010}=\frac{b_{1009}}{1009}$$

2018 Korea National Olympiad, 7

Tags: algebra

Let there be a figure with $9$ disks and $11$ edges, as shown below. We will write a real number in each and every disk. Then, for each edge, we will write the square of the difference between the two real numbers written in the two disks that the edge connects. We must write $0$ in disk $A$, and $1$ in disk $I$. Find the minimum sum of all real numbers written in $11$ edges.

2014 Contests, 1

Tags: inequality , algebra

Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds. [i]UK - David Monk[/i]

2015 BMT Spring, P1

Tags: number theory , algebra

Find two disjoint sets $N_1$ and $N_2$ with $N_1\cup N_2=\mathbb N$, so that neither set contains an infinite arithmetic progression.

2019 Harvard-MIT Mathematics Tournament, 9

Tags: linear algebra , matrix , algebra , hmmt , combinatorics

Tessa the hyper-ant has a 2019-dimensional hypercube. For a real number $k$, she calls a placement of nonzero real numbers on the $2^{2019}$ vertices of the hypercube [i]$k$-harmonic[/i] if for any vertex, the sum of all 2019 numbers that are edge-adjacent to this vertex is equal to $k$ times the number on this vertex. Let $S$ be the set of all possible values of $k$ such that there exists a $k$-harmonic placement. Find $\sum_{k \in S} |k|$.

2005 MOP Homework, 2

Tags: algebra

The sequence of real numbers $\{a_n\}$, $n \in \mathbb{N}$ satisfies the following condition: $a_{n+1}=a_n(a_n+2)$ for any $n \in \mathbb{N}$. Find all possible values for $a_{2004}$.

2023 Bulgaria JBMO TST, 1

Tags: algebra , system of equations , algebraic manipulations

Determine all triples $(x,y,z)$ of real numbers such that $x^4 + y^3z = zx$, $y^4 + z^3x = xy$ and $z^4 + x^3y = yz$.

2018 Purple Comet Problems, 1

Tags: algebra

Find the positive integer $n$ such that $\frac12 \cdot \frac34 + \frac56 \cdot \frac78 + \frac{9}{10}\cdot \frac{11}{12 }= \frac{n}{1200}$ .

2020 Brazil Team Selection Test, 3

Tags: product , determinant , algebra

Let $x_1, x_2, \dots, x_n$ be different real numbers. Prove that \[\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}=\left\{\begin{array}{ll} 0, & \text { if } n \text { is even; } \\ 1, & \text { if } n \text { is odd. } \end{array}\right.\]

1993 AMC 12/AHSME, 26

Tags: parabola , algebra , conic , domain , function , inequalities , calculus

Find the largest positive value attained by the function \[ f(x)=\sqrt{8x-x^2}-\sqrt{14x-x^2-48}, \qquad x\ \text{a real number} \] $ \textbf{(A)}\ \sqrt{7}-1 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2\sqrt{3} \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{55}-\sqrt{5} $