Olympiad problems

2007 Romania National Olympiad, 2

Let $f: \mathbb{R}\to\mathbb{R}$ be a continuous function, and $a<b$ be two points in the image of $f$ (that is, there exists $x,y$ such that $f(x)=a$ and $f(y)=b$). Show that there is an interval $I$ such that $f(I)=[a,b]$.

1992 Baltic Way, 9

Tags: polynomial , algebra

A polynomial $ f(x)\equal{}x^3\plus{}ax^2\plus{}bx\plus{}c$ is such that $ b<0$ and $ ab\equal{}9c$. Prove that the polynomial $ f$ has three different real roots.

PEN Q Problems, 6

Tags: polynomial , calculus , derivative , binomial theorem , binomial coefficient , algebra

Prove that for a prime $p$, $x^{p-1}+x^{p-2}+ \cdots +x+1$ is irreducible in $\mathbb{Q}[x]$.

PEN B Problems, 7

Tags: algebra , modular arithmetic , polynomial , primitive root

Suppose that $p>3$ is prime. Prove that the products of the primitive roots of $p$ between $1$ and $p-1$ is congruent to $1$ modulo $p$.

2011 IFYM, Sozopol, 1

Tags: floor function , algebra , polynomial

Let $n$ be a positive integer. Find the number of all polynomials $P$ with coefficients from the set $\{0,1,2,3\}$ and for which $P(2)=n$.

2018 China Team Selection Test, 1

Tags: algebra , rational root theorem , sequence , polynomial

Define the polymonial sequence $\left \{ f_n\left ( x \right ) \right \}_{n\ge 1}$ with $f_1\left ( x \right )=1$, $$f_{2n}\left ( x \right )=xf_n\left ( x \right ), \; f_{2n+1}\left ( x \right ) = f_n\left ( x \right )+ f_{n+1} \left ( x \right ), \; n\ge 1.$$ Look for all the rational number $a$ which is a root of certain $f_n\left ( x \right ).$

2021 Iran Team Selection Test, 4

Tags: algebra , number theory , polynomial , prime number , combi , function

Assume $\Omega(n),\omega(n)$ be the biggest and smallest prime factors of $n$ respectively . Alireza and Amin decided to play a game. First Alireza chooses $1400$ polynomials with integer coefficients. Now Amin chooses $700$ of them, the set of polynomials of Alireza and Amin are $B,A$ respectively . Amin wins if for all $n$ we have : $$\max_{P \in A}(\Omega(P(n))) \ge \min_{P \in B}(\omega(P(n)))$$ Who has the winning strategy. Proposed by [i]Alireza Haghi[/i]

2022 Belarusian National Olympiad, 10.7

Tags: algebra , polynomial

Find all positive integers $a$ for which there exists a polynomial $p(x)$ with integer coefficients such that $p(\sqrt{2}+1)=2-\sqrt{2}$ and $p(\sqrt{2}+2)=a$

2010 Iran Team Selection Test, 12

Tags: function , algebra , number theory , polynomial

Prove that for each natural number $m$, there is a natural number $N$ such that for each $b$ that $2\leq b\leq1389$ sum of digits of $N$ in base $b$ is larger than $m$.

2009 Albania Team Selection Test, 2

Tags: function , polynomial , induction , calculus , derivative , limit , algebra

Find all the functions $ f :\mathbb{R}\mapsto\mathbb{R} $ with the following property: $ \forall x$ $f(x)= f(x/2) + (x/2)f'(x)$

2018 Singapore Senior Math Olympiad, 3

Tags: algebra , polynomial

Determine the largest positive integer $n$ such that the following statement is true: There exists $n$ real polynomials, $P_1(x),\ldots,P_n(x)$ such that the sum of any two of them have no real roots but the sum of any three does.

2012 NIMO Problems, 6

Tags: trigonometry , algebra , polynomial

The polynomial $P(x) = x^3 + \sqrt{6} x^2 - \sqrt{2} x - \sqrt{3}$ has three distinct real roots. Compute the sum of all $0 \le \theta < 360$ such that $P(\tan \theta^\circ) = 0$. [i]Proposed by Lewis Chen[/i]

2014 Costa Rica - Final Round, 6

Tags: algebra , polynomial , recurrence relation , sequence

The sequences $a_n$, $b_n$ and $c_n$ are defined recursively in the following way: $a_0 = 1/6$, $b_0 = 1/2$, $c_0 = 1/3,$ $$a_{n+1}= \frac{(a_n + b_n)(a_n + c_n)}{(a_n - b_n)(a_n - c_n)},\,\, b_{n+1}= \frac{(b_n + a_n)(b_n + c_n)}{(b_n - a_n)(b_n - c_n)},\,\, c_{n+1}= \frac{(c_n + a_n)(c_n + b_n)}{(c_n - a_n)(c_n - b_n)}$$ For each natural number $N$, the following polynomials are defined: $A_n(x) =a_o+a_1 x+ ...+ a_{2N}x^{2N}$ $B_n(x) =b_o+a_1 x+ ...+ a_{2N}x^{2N}$ $C_n(x) =a_o+a_1 x+ ...+ a_{2N}x^{2N}$ Assume the sequences are well defined. Show that there is no real $c$ such that $A_N(c) = B_N(c) = C_N(c) = 0$.

2005 IberoAmerican Olympiad For University Students, 1

Tags: analytic geometry , induction , algebra , polynomial , limit , ratio

Let $P(x,y)=(x^2y^3,x^3y^5)$, $P^1=P$ and $P^{n+1}=P\circ P^n$. Also, let $p_n(x)$ be the first coordinate of $P^n(x,x)$, and $f(n)$ be the degree of $p_n(x)$. Find \[\lim_{n\to\infty}f(n)^{1/n}\]

1997 Canada National Olympiad, 5

Tags: algebra , polynomial , ratio

Write the sum $\sum_{i=0}^{n}{\frac{(-1)^i\cdot\binom{n}{i}}{i^3 +9i^2 +26i +24}}$ as the ratio of two explicitly defined polynomials with integer coefficients.

2011 Iran MO (3rd Round), 5

Tags: trigonometry , algebra , polynomial

$f(x)$ is a monic polynomial of degree $2$ with integer coefficients such that $f(x)$ doesn't have any real roots and also $f(0)$ is a square-free integer (and is not $1$ or $-1$). Prove that for every integer $n$ the polynomial $f(x^n)$ is irreducible over $\mathbb Z[x]$. [i]proposed by Mohammadmahdi Yazdi[/i]

1993 Dutch Mathematical Olympiad, 3

Tags: algebra , limit , polynomial

A sequence of numbers is defined by $ u_1\equal{}a, u_2\equal{}b$ and $ u_{n\plus{}1}\equal{}\frac{u_n\plus{}u_{n\minus{}1}}{2}$ for $ n \ge 2$. Prove that $ \displaystyle\lim_{n\to\infty}u_n$ exists and express its value in terms of $ a$ and $ b$.

1953 Putnam, B2

Tags: integer , polynomial

Let $a_0 ,a_1 , \ldots, a_n$ be real numbers and let $f(x) =a_n x^n +\ldots +a_1 x +a_0.$ Suppose that $f(i)$ is an integer for all $i.$ Prove that $n! \cdot a_k$ is an integer for each $k.$

MIPT student olimpiad autumn 2022, 2

Tags: geometry , polynomial

Let $n \geq 3$ be an integer. Find the minimum degree of one algebraic (polynomial) equation that defines the set of vertices of the correct $n$-gon on plane $R^2$.

2009 Hungary-Israel Binational, 1

Tags: relatively prime , algebra , modular arithmetic , polynomial , number theory

For a given prime $ p > 2$ and positive integer $ k$ let \[ S_k \equal{} 1^k \plus{} 2^k \plus{} \ldots \plus{} (p \minus{} 1)^k\] Find those values of $ k$ for which $ p \, |\, S_k$.

2001 Romania National Olympiad, 2

Tags: algebra , polynomial , linear algebra , matrix , complex number

We consider a matrix $A\in M_n(\textbf{C})$ with rank $r$, where $n\ge 2$ and $1\le r\le n-1$. a) Show that there exist $B\in M_{n,r}(\textbf{C}), C\in M_{r,n}(\textbf{C})$, with $%Error. "rank" is a bad command. B=%Error. "rank" is a bad command. C = r$, such that $A=BC$. b) Show that the matrix $A$ verifies a polynomial equation of degree $r+1$, with complex coefficients.

2018 Vietnam Team Selection Test, 3

Tags: algebra , polynomial

For every positive integer $n\ge 3$, let $\phi_n$ be the set of all positive integers less than and coprime to $n$. Consider the polynomial: $$P_n(x)=\sum_{k\in\phi_n} {x^{k-1}}.$$ a. Prove that $P_n(x)=(x^{r_n}+1)Q_n(x)$ for some positive integer $r_n$ and polynomial $Q_n(x)\in\mathbb{Z}[x]$ (not necessary non-constant polynomial). b. Find all $n$ such that $P_n(x)$ is irreducible over $\mathbb{Z}[x]$.