Olympiad problems

1982 All Soviet Union Mathematical Olympiad, 331

Once upon a time, three boys visited a library for the first time. The first decided to visit the library every second day. The second decided to visit the library every third day. The third decided to visit the library every fourth day. The librarian noticed, that the library doesn't work on Wednesdays. The boys decided to visit library on Thursdays, if they have to do it on Wednesdays, but to restart the day counting in these cases. They strictly obeyed these rules. Some Monday later I met them all in that library. What day of week was when they visited a library for the first time?

2008 Alexandru Myller, 2

Tags: algebra , polynomial

Find all natural numbers $ n\ge 3 $ and real numbers $ a $ which have the property that the polynomial $ X^n-aX-1 $ admits a monic quadratic integer polynomial. [i]Mihai Bălună[/i]

II Soros Olympiad 1995 - 96 (Russia), 11.4

Tags: algebra , analytic geometry

Draw on the coordinate plane a set of points $M(a, b)$ such that the equation $x^4+ax+b=0$ has a unique root satisfying the condition $0 \le x \le 1$.

2016 Nigerian Senior MO Round 2, Problem 7

Tags: induction , irrational number , algebra

Prove that $(2+\sqrt{3})^{2n}+(2-\sqrt{3})^{2n}$ is an even integer and that $(2+\sqrt{3})^{2n}-(2-\sqrt{3})^{2n}=w\sqrt{3}$ for some positive integer $w$, for all integers $n \geq 1$.

2017 Harvard-MIT Mathematics Tournament, 10

Tags: algebra

[b]D[/b]enote $\phi=\frac{\sqrt{5}+1}{2}$ and consider the set of all finite binary strings without leading zeroes. Each string $S$ has a “base-$\phi$” value $p(S)$. For example, $p(1101)=\phi^3+\phi^2+1$. For any positive integer n, let $f(n)$ be the number of such strings S that satisfy $p(S) =\frac{\phi^{48n}-1}{\phi^{48}-1}$. The sequence of fractions $\frac{f(n+1)}{f(n)}$ approaches a real number $c$ as $n$ goes to infinity. Determine the value of $c$.

1976 AMC 12/AHSME, 7

Tags: algebra , function , polynomial

If $x$ is a real number, then the quantity $(1-|x|)(1+x)$ is positive if and only if $\textbf{(A) }|x|<1\qquad\textbf{(B) }|x|>1\qquad\textbf{(C) }x<-1\text{ or }-1<x<1\qquad$ $\textbf{(D) }x<1\qquad \textbf{(E) }x<-1$

2008 Greece National Olympiad, 1

Tags: algebra

A computer generates all pairs of real numbers $x, y \in (0, 1)$ for which the numbers $a = x+my$ and $b = y+mx$ are both integers, where $m$ is a given positive integer. Finding one such pair $(x, y)$ takes $5$ seconds. Find $m$ if the computer needs $595$ seconds to find all possible ordered pairs $(x, y)$.

2008 USA Team Selection Test, 9

Tags: algorithm , algebra , function , geometry , modular arithmetic , 3d geometry , polynomial

Let $ n$ be a positive integer. Given an integer coefficient polynomial $ f(x)$, define its [i]signature modulo $ n$[/i] to be the (ordered) sequence $ f(1), \ldots , f(n)$ modulo $ n$. Of the $ n^n$ such $ n$-term sequences of integers modulo $ n$, how many are the signature of some polynomial $ f(x)$ if a) $ n$ is a positive integer not divisible by the square of a prime. b) $ n$ is a positive integer not divisible by the cube of a prime.

KoMaL A Problems 2018/2019, A. 738

Tags: algebra , number theory , recurrence relation

Consider the following sequence: $a_1 = 1$, $a_2 = 2$, $a_3 = 3$, and \[a_{n+3} = \frac{a_{n+1}^2 + a_{n+2}^2 - 2}{a_n}\] for all integers $n \ge 1$. Prove that every term of the sequence is a positive integer.

2008 Indonesia TST, 4

Tags: algebra , inequalities

Let $a, b, c$ be positive reals. Prove that $$\left(\frac{a}{a+b}\right)^2+\left(\frac{b}{b+c}\right)^2+\left(\frac{c}{c+a}\right)^2\ge \frac34$$

2006 CHKMO, 3

Tags: algebra , function , inequalities

Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=1$. Prove that\[ 6(a^3+b^3+c^3+d^3)\ge(a^2+b^2+c^2+d^2)+\frac{1}{8} \]

2014 Contests, 3

Tags: function , algebra , quadratic , inequalities , quadratic formula

Let $ x,y,z $ be three non-negative real numbers such that \[x^2+y^2+z^2=2(xy+yz+zx). \] Prove that \[\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.\]

2011 Olympic Revenge, 1

Tags: algebra , geometry , trigonometry , parallelogram

Let $p, q, r, s, t \in \mathbb{R}^{*}_{+}$ satisfying: i) $p^2 + pq + q^2 = s^2$ ii) $q^2 + qr + r^2 = t^2$ iii) $r^2 + rp + p^2 = s^2 - st + t^2$ Prove that \[\frac{s^2 - st + t^2}{s^2t^2} = \frac{r^2}{q^2t^2} + \frac{p^2}{q^2s^2} - \frac{pr}{q^2ts}\]

2016 NIMO Problems, 6

Tags: algebra

Let $S$ be the sum of all positive integers that can be expressed in the form $2^a \cdot 3^b \cdot 5^c$, where $a$, $b$, $c$ are positive integers that satisfy $a+b+c=10$. Find the remainder when $S$ is divided by $1001$. [i]Proposed by Michael Ren[/i]

2002 All-Russian Olympiad, 1

Tags: functional equation , algebra , equation , polynomial , quadratic , ring theory

The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.

2017 Tuymaada Olympiad, 1

Tags: algebra , function

Functions $f$ and $g$ are defined on the set of all integers in the interval $[-100; 100]$ and take integral values. Prove that for some integral $k$ the number of solutions of the equation $f(x)-g(y)=k$ is odd.\\ ( A. Golovanov)

2003 Romania Team Selection Test, 5

Tags: algebra , function , calculus , polynomial , integration , complex number , absolute value

Let $f\in\mathbb{Z}[X]$ be an irreducible polynomial over the ring of integer polynomials, such that $|f(0)|$ is not a perfect square. Prove that if the leading coefficient of $f$ is 1 (the coefficient of the term having the highest degree in $f$) then $f(X^2)$ is also irreducible in the ring of integer polynomials. [i]Mihai Piticari[/i]

2000 Hungary-Israel Binational, 2

Tags: algebra , binomial theorem , number theory

Prove or disprove: For any positive integer $k$ there exists an integer $n > 1$ such that the binomial coeffcient $\binom{n}{i}$ is divisible by $k$ for any $1 \leq i \leq n-1.$

2001 India National Olympiad, 2

Tags: modular arithmetic , algebra , number theory , arithmetic sequence , polynomial

Show that the equation $x^2 + y^2 + z^2 = ( x-y)(y-z)(z-x)$ has infintely many solutions in integers $x,y,z$.

2022 ELMO Revenge, 5

Tags: algebra

Let $f(x)=x+3x^{\frac 23}, g(x)=x+x^{\frac 13}$. Call a sequence $\{a_i\}_{i\ge 0}$ satisfactory if for all $i\ge 1, a_i\in \{f(a_{i-1}), g(a_{i-1})\}$. Find all pairs of real numbers $(x,y)$ such that there exist satisfactory sequences $(a_i)_{i\ge 0}, (b_i)_{i\ge 0}$ and positive integers $m$ and $n$, such that $a_0 =x$, $b_0 = y$, and $$|a_m-b_n|<1$$

2003 AMC 12-AHSME, 25

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

Let $ f(x)\equal{}\sqrt{ax^2\plus{}bx}$. For how many real values of $ a$ is there at least one positive value of $ b$ for which the domain of $ f$ and the range of $ f$ are the same set? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \text{infinitely many}$

2011 AMC 12/AHSME, 21

Tags: function , domain , algebra

Let $f_1(x)=\sqrt{1-x}$, and for integers $n \ge 2$, let $f_n(x)=f_{n-1}(\sqrt{n^2-x})$. If $N$ is the largest value of $n$ for which the domain of $f_n$ is nonempty, the domain of $f_N$ is ${c}$. What is $N+c$? $ \textbf{(A)}\ -226 \qquad \textbf{(B)}\ -144 \qquad \textbf{(C)}\ -20 \qquad \textbf{(D)}\ 20 \qquad \textbf{(E)}\ 144$

2021 Polish MO Finals, 4

Tags: algebra , inequality

Prove that for every pair of positive real numbers $a, b$ and for every positive integer $n$, $$(a+b)^n-a^n-b^n \ge \frac{2^n-2}{2^{n-2}} \cdot ab(a+b)^{n-2}.$$

2009 Belarus Team Selection Test, 3

Tags: algebra , equation , number theory

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

2013 District Olympiad, 2

Tags: linear algebra , matrix , algebra

Let the matrices of order 2 with the real elements $A$ and $B$ so that $AB={{A}^{2}}{{B}^{2}}-{{\left( AB \right)}^{2}}$ and $\det \left( B \right)=2$. a) Prove that the matrix $A$ is not invertible. b) Calculate $\det \left( A+2B \right)-\det \left( B+2A \right)$.