Olympiad problems

2017 CMIMC Algebra, 8

Tags: algebra

Suppose $a_1$, $a_2$, $\ldots$, $a_{10}$ are nonnegative integers such that \[\sum_{k=1}^{10}a_k=15\qquad\text{and}\qquad \sum_{k=1}^{10}ka_k = 80.\] Let $M$ and $m$ denote the maximum and minimum respectively of $\sum_{k=1}^{10}k^2a_k$. Compute $M-m$.

2021 USA IMO Team Selection Test, 3

Tags: algebra , functional equation

Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfy the inequality \[ f(y) - \left(\frac{z-y}{z-x} f(x) + \frac{y-x}{z-x}f(z)\right) \leq f\left(\frac{x+z}{2}\right) - \frac{f(x)+f(z)}{2} \] for all real numbers $x < y < z$. [i]Proposed by Gabriel Carroll[/i]

2017 Iran MO (2nd Round), 4

Tags: algebra , 2-variable inequality , inequality , inequalities

Let $x,y$ be two positive real numbers such that $x^4-y^4=x-y$. Prove that $$\frac{x-y}{x^6-y^6}\leq \frac{4}{3}(x+y).$$

2023 LMT Spring, 10

Tags: algebra

The sequence $a_0,a_1,a_2,...$ is defined such that $a_0 = 2+ \sqrt3$, $a_1 =\sqrt{5-2\sqrt5}$, and $$a_n a_{n-1}a_{n-2} - a_n + a_{n-1} + a_{n-2} = 0.$$ Find the least positive integer $n$ such that $a_n = 1$.

2008 Baltic Way, 5

Tags: algebra , symmetry , tetrahedron

Suppose that Romeo and Juliet each have a regular tetrahedron to the vertices of which some positive real numbers are assigned. They associate each edge of their tetrahedra with the product of the two numbers assigned to its end points. Then they write on each face of their tetrahedra the sum of the three numbers associated to its three edges. The four numbers written on the faces of Romeo's tetrahedron turn out to coincide with the four numbers written on Juliet's tetrahedron. Does it follow that the four numbers assigned to the vertices of Romeo's tetrahedron are identical to the four numbers assigned to the vertices of Juliet's tetrahedron?

1981 Swedish Mathematical Competition, 3

Tags: algebra , polynomial , divisible

Find all polynomials $p(x)$ of degree $5$ such that $p(x) + 1$ is divisible by $(x-1)^3$ and $p(x) - 1$ is divisible by $(x+1)^3$.

1950 Moscow Mathematical Olympiad, 180

Tags: equation , radical , algebra

Solve the equation $\sqrt {x + 3 - 4 \sqrt{x -1}} +\sqrt{x + 8 - 6 \sqrt{x - 1}}= 1$.

1969 IMO Shortlist, 65

Tags: algebraic identities , algebra

$(USS 2)$ Prove that for $a > b^2,$ the identity ${\sqrt{a-b\sqrt{a+b\sqrt{a-b\sqrt{a+\cdots}}}}=\sqrt{a-\frac{3}{4}b^2}-\frac{1}{2}b}$

2016 Miklós Schweitzer, 4

Tags: sequence , algebra , function

Prove that there exists a sequence $a(1),a(2),\dots,a(n),\dots$ of real numbers such that \[ a(n+m)\le a(n)+a(m)+\frac{n+m}{\log (n+m)} \] for all integers $m,n\ge 1$, and such that the set $\{a(n)/n:n\ge 1\}$ is everywhere dense on the real line. [i]Remark.[/i] A theorem of de Bruijn and Erdős states that if the inequality above holds with $f(n + m)$ in place of the last term on the right-hand side, where $f(n)\ge 0$ is nondecreasing and $\sum_{n=2}^\infty f(n)/n^2<\infty$, then $a(n)/n$ converges or tends to $(-\infty)$.

2021 CMIMC, 2.2

Tags: algebra

Suppose $a,b$ are positive real numbers such that $a+a^2 = 1$ and $b^2+b^4=1$. Compute $a^2+b^2$. [i]Proposed by Thomas Lam[/i]

2024 Iran MO (3rd Round), 1

Tags: algebra

For positive real numbers $a,b,c,d$ such that $$ \dfrac{a^2}{b+c+d} + \dfrac{b^2}{a+c+d} + \dfrac{c^2}{a+b+d} = \dfrac{3d^2}{a+b+c} $$ prove that $$ \dfrac{3}{a}+ \dfrac{3}{b} + \dfrac{3}{c}+ \dfrac{3}{d} \geq \dfrac{16}{a+b+2d} + \dfrac{16}{b+c+2d} + \dfrac{16}{a+c+2d}. $$ Proposed by [i]Mojtaba Zare[/i]

1976 Poland - Second Round, 5

Tags: irrational , rational , trigonometry , algebra

Prove that if $ \cos \pi x =\frac{1}{3} $ then $ x $ is an irrational number.

Russian TST 2019, P2

Tags: algebra , sequence , maximum value

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

2019 PUMaC Algebra B, 6

Tags: algebra , function

Let $\mathbb N_0$ be the set of non-negative integers. There is a triple $(f,a,b)$, where $f$ is a function from $\mathbb N_0$ to $\mathbb N_0$ and $a,b\in\mathbb N_0$ that satisfies the following conditions: [list] [*]$f(1)=2$ [*]$f(a)+f(b)\leq 2\sqrt{f(a)}$ [*]For all $n>0$, we have $f(n)=f(n-1)f(b)+2n-f(b)$ [/list] Find the sum of all possible values of $f(b+100)$.

1989 Romania Team Selection Test, 2

Tags: recurrence relation , sequence , algebra , number theory

The sequence ($a_n$) is defined by $a_1 = a_2 = 1, a_3 = 199$ and $a_{n+1} =\frac{1989+a_na_{n-1}}{a_{n-2}}$ for all $n \ge 3$. Prove that all terms of the sequence are positive integers

2014 APMO, 1

Tags: algebra

For a positive integer $m$ denote by $S(m)$ and $P(m)$ the sum and product, respectively, of the digits of $m$. Show that for each positive integer $n$, there exist positive integers $a_1, a_2, \ldots, a_n$ satisfying the following conditions: \[ S(a_1) < S(a_2) < \cdots < S(a_n) \text{ and } S(a_i) = P(a_{i+1}) \quad (i=1,2,\ldots,n). \] (We let $a_{n+1} = a_1$.) [i]Problem Committee of the Japan Mathematical Olympiad Foundation[/i]

2013 IFYM, Sozopol, 1

Tags: divisibility , sequence , algebra

Let $u_1=1,u_2=2,u_3=24,$ and $u_{n+1}=\frac{6u_n^2 u_{n-2}-8u_nu_{n-1}^2}{u_{n-1}u_{n-2}}, n \geq 3.$ Prove that the elements of the sequence are natural numbers and that $n\mid u_n$ for all $n$.

1999 Harvard-MIT Mathematics Tournament, 1

Tags: quadratic , function , algebra

If $a@b=\dfrac{a^3-b^3}{a-b}$, for how many real values of $a$ does $a@1=0$?

2015 Saint Petersburg Mathematical Olympiad, 4

Tags: algebra , inequalities

Positive numbers $x, y, z$ satisfy the condition $$xy + yz + zx + 2xyz = 1.$$ Prove that $4x + y + z \ge 2.$ [i]A. Khrabrov[/i]

2004 Nordic, 1

Tags: algebra

Twenty-seven balls labelled from $1$ to $27$ are distributed in three bowls: red, blue, and yellow. What are the possible values of the number of balls in the red bowl if the average labels in the red, blue and yellow bowl are $15$, $3$, and $18$, respectively?

2011 QEDMO 10th, 1

Tags: functional , algebra , functional equation

Find all functions $f: R\to R$ with the property that $xf (y) + yf (x) = (x + y) f (xy)$ for all $x, y \in R$.

2023 Ukraine National Mathematical Olympiad, 8.5

Tags: equality , algebra

Do there exist $10$ real numbers, not all of which are equal, each of which is equal to the square of the sum of the remaining $9$ numbers? [i]Proposed by Bogdan Rublov[/i]

2011 District Olympiad, 4

Tags: logarithm , algebra , a implies b

[b]a)[/b] Show that , if $ a,b>1 $ are two distinct real numbers, then $ \log_a\log_a b >\log_b\log_a b. $ [b]b)[/b] Show that if $ a_1>a_2>\cdots >a_n>1 $ are $ n\ge 2 $ real numbers, then $$ \log_{a_1}\log_{a_1} a_2 +\log_{a_2}\log_{a_2} a_3 +\cdots +\log_{a_{n-1}}\log_{a_{n-1}} a_n +\log_{a_n}\log_{a_n} a_1 >0. $$

2020 BMT Fall, 10

Tags: algebra , floor function

For $k\ge 1$, define $a_k=2^k$. Let $$S=\sum_{k=1}^{\infty}\cos^{-1}\left(\frac{2a_k^2-6a_k+5}{\sqrt{(a_k^2-4a_k+5)(4a_k^2-8a_k+5)}}\right).$$ Compute $\lfloor 100S\rfloor$.

2018 AMC 10, 16

Tags: sum of cubes , algebra

Let $a_1,a_2,\dots,a_{2018}$ be a strictly increasing sequence of positive integers such that $$a_1+a_2+\cdots+a_{2018}=2018^{2018}.$$ What is the remainder when $a_1^3+a_2^3+\cdots+a_{2018}^3$ is divided by $6$? $\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$