Olympiad problems

2010 HMNT, 7

Tags: algebra

What is the remainder when $(1 + x)^{2010}$ is divided by $1 + x + x^2$?

2013 Junior Balkan Team Selection Tests - Romania, 4

Tags: algebra , sequence , combinatorics

For any sequence ($a_1,a_2,...,a_{2013}$) of integers, we call a triple ($i,j, k$) satisfying $1 \le i < j < k \le 2013$ to be [i]progressive [/i] if $a_k-a_j = a_j -a_i = 1$. Determine the maximum number of progressive triples that a sequence of $2013$ integers could have.

2012 Singapore Senior Math Olympiad, 5

Tags: inequalities , algebra

For $a,b,c,d \geq 0$ with $a + b = c + d = 2$, prove \[(a^2 + c^2)(a^2 + d^2)(b^2 + c^2)(b^2 + d^2) \leq 25\]

2023 Princeton University Math Competition, A4 / B6

Tags: algebra

The set of real values $a$ such that the equation $x^4-3ax^3+(2a^2+4a)x^2-5a^2x+3a^2$ has exactly two nonreal solutions is the set of real numbers between $x$ and $y,$ where $x<y.$ If $x+y$ can be written as $\tfrac{m}{n}$ for relatively prime positive integers $m,n,$ find $m+n.$

2022 BMT, 5

Tags: algebra

For real numbers $B,M,$ and $T,$ we have $B^2+M^2+T^2 =2022$ and $B+M+T =72.$ Compute the sum of the minimum and maximum possible values of $T.$

1989 Romania Team Selection Test, 2

Tags: coprime , number theory , system of equations , algebra

Let $a,b,c$ be coprime nonzero integers. Prove that for any coprime integers $u,v,w$ with $au+bv+cw = 0$ there exist integers $m,n, p$ such that $$\begin{cases} a = nw- pv \\ b = pu-mw \\ c = mv-nu \end{cases}$$

2014 BMT Spring, 4

Tags: polynomial , algebra

The function $f(x)=x^5-20x^4+ax^3+bx^2+cx+24$ has the interesting property that its roots can be arranged to form an arithmetic sequence. Determine $f(8)$.

2004 Argentina National Olympiad, 6

Tags: algebra , arithmetic progression , arithmetic sequence

Decide if it is possible to generate an infinite sequence of positive integers $a_n$ such that in the sequence there are no three terms that are in arithmetic progression and that for all $n$ $\left |a_n-n^2\right | <\frac{n}{2}$. Clarification: Three numbers $a$, $b$, $c$ are in arithmetic progression if and only if $2b=a+c$.

1995 Israel Mathematical Olympiad, 1

Tags: algebra , logarithm , system of equations

Solve the system $$\begin{cases} x+\log\left(x+\sqrt{x^2+1}\right)=y \\ y+\log\left(y+\sqrt{y^2+1}\right)=z \\ z+\log\left(z+\sqrt{z^2+1}\right)=x \end{cases}$$

2012 China Team Selection Test, 3

Tags: function , floor function , abstract algebra , group theory , algebra , induction

$n$ being a given integer, find all functions $f\colon \mathbb{Z} \to \mathbb{Z}$, such that for all integers $x,y$ we have $f\left( {x + y + f(y)} \right) = f(x) + ny$.

2015 Mexico National Olympiad, 3

Tags: algebra , functional equation , function

Let $\mathbb{N} =\{1, 2, 3, ...\}$ be the set of positive integers. Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be a function that gives a positive integer value, to every positive integer. Suppose that $f$ satisfies the following conditions: $f(1)=1$ $f(a+b+ab)=a+b+f(ab)$ Find the value of $f(2015)$ Proposed by Jose Antonio Gomez Ortega

2020 BMT Fall, 22

Tags: system of equations , algebra

Suppose that $x, y$, and $z$ are positive real numbers satisfying $$\begin{cases} x^2 + xy + y^2 = 64 \\ y^2 + yz + z^2 = 49 \\ z^2 + zx + x^2 = 57 \end{cases}$$ Then $\sqrt[3]{xyz}$ can be expressed as $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

1940 Moscow Mathematical Olympiad, 055

Tags: algebra , time

It takes a steamer $5$ days to go from Gorky to Astrakhan downstream the Volga river and $7$ days upstream from Astrakhan to Gorky. How long will it take for a raft to float downstream from Gorky to Astrakhan?

2010 Saudi Arabia BMO TST, 1

Tags: number theory , algebra

Find all pairs $(x, y)$ of positive integers such that $x^2 + y^2 + 33^2 =2010\sqrt{x-y}$.

2022 Latvia Baltic Way TST, P4

Tags: polynomial , algebra

Let $p(x)$ be a polynomial with real coefficients and $n$ be a positive integer. Prove that there exists a non-zero polynomial $q(x)$ with real coefficients such that the polynomial $p(x)\cdot q(x)$ has non-zero coefficients only by the powers which are multiples of $n$.

2013 China Team Selection Test, 3

Tags: algebra , modular arithmetic , induction

Find all positive real numbers $r<1$ such that there exists a set $\mathcal{S}$ with the given properties: i) For any real number $t$, exactly one of $t, t+r$ and $t+1$ belongs to $\mathcal{S}$; ii) For any real number $t$, exactly one of $t, t-r$ and $t-1$ belongs to $\mathcal{S}$.

1985 IMO Longlists, 70

Tags: algebra , function

Let $C$ be a class of functions $f : \mathbb N \to \mathbb N$ that contains the functions $S(x) = x + 1$ and $E(x) = x - [\sqrt x]^2$ for every $x \in \mathbb N$. ($[x]$ is the integer part of $x$.) If $C$ has the property that for every $f, g \in C, f + g, fg, f \circ g \in C$, show that the function $\max(f(x) - g(x), 0)$ is in $C$, for all $f; g \in C$.

2008 Dutch IMO TST, 1

Tags: algebra , functional equation , functional equation in n

Find all funtions $f : Z_{>0} \to Z_{>0}$ that satisfy $f(f(f(n))) + f(f(n)) + f(n) = 3n$ for all $n \in Z_{>0}$ .

2020 Baltic Way, 1

Tags: inequality , sequence , algebra

Let $a_0>0$ be a real number, and let $$a_n=\frac{a_{n-1}}{\sqrt{1+2020\cdot a_{n-1}^2}}, \quad \textrm{for } n=1,2,\ldots ,2020.$$ Show that $a_{2020}<\frac1{2020}$.

2005 Iran MO (3rd Round), 3

Tags: polynomial , algebra , number theory

$p(x)$ is an irreducible polynomial in $\mathbb Q[x]$ that $\mbox{deg}\ p$ is odd. $q(x),r(x)$ are polynomials with rational coefficients that $p(x)|q(x)^2+q(x).r(x)+r(x)^2$. Prove that \[p(x)^2|q(x)^2+q(x).r(x)+r(x)^2\]

1966 IMO Longlists, 58

Tags: system of equations , algebra , combinatorics

In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$ was twice the number who solved $C$. The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$. How many students solved only problem $B$?

2021 BMT, 14

Tags: algebra , complex number

Let $r_1, r_2, ..., r_{47}$ be the roots of $x^{47} - 1 = 0$. Compute $$\sum^{47}_{i=1}r^{2020}_i .$$

2011 Turkey Team Selection Test, 1

Tags: algorithm , functional equation , euclidean algorithm , algebra , number theory , function , induction

Let $\mathbb{Q^+}$ denote the set of positive rational numbers. Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$ that satisfy the conditions \[ f \left( \frac{x}{x+1}\right) = \frac{f(x)}{x+1} \qquad \text{and} \qquad f \left(\frac{1}{x}\right)=\frac{f(x)}{x^3}\] for all $x \in \mathbb{Q^+}.$

1991 Dutch Mathematical Olympiad, 3

Tags: function , algebra

A real function $ f$ satisfies $ 4f(f(x))\minus{}2f(x)\minus{}3x\equal{}0$ for all real numbers $ x$. Prove that $ f(0)\equal{}0$.

2023 Dutch BxMO TST, 2

Tags: functional equation , algebra , functional inequality

Find all functions $f : \mathbb R \to \mathbb R$ for which \[f(a - b) f(c - d) + f(a - d) f(b - c) \leq (a - c) f(b - d),\] for all real numbers $a, b, c$ and $d$. Note that there is only one occurrence of $f$ on the right hand side!