Olympiad problems

2000 Poland - Second Round, 6

Polynomial $w(x)$ of second degree with integer coefficients takes for integer arguments values, which are squares of integers. Prove that polynomial $w(x)$ is a square of a polynomial.

2015 Harvard-MIT Mathematics Tournament, 3

Tags: algebra , limit

Let $p$ be a real number and $c\neq 0$ such that \[c-0.1<x^p\left(\dfrac{1-(1+x)^{10}}{1+(1+x)^{10}}\right)<c+0.1\] for all (positive) real numbers $x$ with $0<x<10^{-100}$. (The exact value $10^{-100}$ is not important. You could replace it with any "sufficiently small number".) Find the ordered pair $(p,c)$.

2004 Romania National Olympiad, 2

Tags: function , polynomial , inequalities , calculus , algebra , logarithm , integration

Let $P(n)$ be the number of functions $f: \mathbb{R} \to \mathbb{R}$, $f(x)=a x^2 + b x + c$, with $a,b,c \in \{1,2,\ldots,n\}$ and that have the property that $f(x)=0$ has only integer solutions. Prove that $n<P(n)<n^2$, for all $n \geq 4$. [i]Laurentiu Panaitopol[/i]

2005 USA Team Selection Test, 4

Tags: polynomial , algebra

Consider the polynomials \[f(x) =\sum_{k=1}^{n}a_{k}x^{k}\quad\text{and}\quad g(x) =\sum_{k=1}^{n}\frac{a_{k}}{2^{k}-1}x^{k},\] where $a_{1},a_{2},\ldots,a_{n}$ are real numbers and $n$ is a positive integer. Show that if 1 and $2^{n+1}$ are zeros of $g$ then $f$ has a positive zero less than $2^{n}$.

2023-24 IOQM India, 10

Tags: algebra

The Sequence $\{a_{n}\}_{n \geqslant 0}$ is defined by $a_{0}=1, a_{1}=-4$ and $a_{n+2}=-4a_{n+1}-7a_{n}$ , for $n \geqslant 0$. Find the number of positive integer divisors of $a^2_{50}-a_{49}a_{51}$.

2005 Estonia National Olympiad, 4

Tags: algebra , equation , trinomial

Find all pairs of real numbers $(x, y)$ that satisfy the equation $(x + y)^2 = (x + 3) (y - 3)$.

2007 Austria Beginners' Competition, 3

Tags: inequalities , algebra

For real numbers $x \ge 0$ and $y \ge 0$, write $A= \frac{x+y}{2}$ for the arithmetic mean and $G=\sqrt{xy}$ for the geometric mean of $x$ and $y$. Furthermore, let $W= \frac{\sqrt{x}+\sqrt{y}}{2}$ be the arithmetic mean of $\sqrt{x}$ and $\sqrt{y}$. Prove that $$G\le W^2 \le A.$$ Determine all $x$ and $y$ such that $G= W^2 = A$

2021 Winter Stars of Mathematics, 4

Tags: algebra , number theory

Let $a_0 = 1, \ a_1 = 2,$ and $a_2 = 10,$ and define $a_{k+2} = a_{k+1}^3+a_k^2+a_{k-1}$ for all positive integers $k.$ Is it possible for some $a_x$ to be divisible by $2021^{2021}?$ [i]Flavian Georgescu[/i]

2016 Mexico National Olmypiad, 3

Tags: inequalities , algebra , floor function

Find the minimum real $x$ that satisfies $$\lfloor x \rfloor <\lfloor x^2 \rfloor <\lfloor x^3 \rfloor < \cdots < \lfloor x^n \rfloor < \lfloor x^{n+1} \rfloor < \cdots$$

2012 India PRMO, 15

Tags: algebra , function , floor function

How many non-negative integral values of $x$ satisfy the equation $ \lfloor \frac{x}{5}\rfloor = \lfloor \frac{x}{7}\rfloor $

2023 India National Olympiad, 2

Tags: inequalities , polynomial , algebra

Suppose $a_0,\ldots, a_{100}$ are positive reals. Consider the following polynomial for each $k$ in $\{0,1,\ldots, 100\}$: $$a_{100+k}x^{100}+100a_{99+k}x^{99}+a_{98+k}x^{98}+a_{97+k}x^{97}+\dots+a_{2+k}x^2+a_{1+k}x+a_k,$$where indices are taken modulo $101$, [i]i.e.[/i], $a_{100+i}=a_{i-1}$ for any $i$ in $\{1,2,\dots, 100\}$. Show that it is impossible that each of these $101$ polynomials has all its roots real. [i]Proposed by Prithwijit De[/i]

2019 All-Russian Olympiad, 2

Tags: trigonometry , algebra

Is it true, that for all pairs of non-negative integers $a$ and $b$ , the system \begin{align*} \tan{13x} \tan{ay} =& 1 \\ \tan{21x} \tan{by}= & 1 \end{align*} has at least one solution?

2000 Vietnam Team Selection Test, 2

Tags: algebra , function

Let $a > 1$ and $r > 1$ be real numbers. (a) Prove that if $f : \mathbb{R}^{+}\to\mathbb{ R}^{+}$ is a function satisfying the conditions (i) $f (x)^{2}\leq ax^{r}f (\frac{x}{a})$ for all $x > 0$, (ii) $f (x) < 2^{2000}$ for all $x < \frac{1}{2^{2000}}$, then $f (x) \leq x^{r}a^{1-r}$ for all $x > 0$. (b) Construct a function $f : \mathbb{R}^{+}\to\mathbb{ R}^{+}$ satisfying condition (i) such that for all $x > 0, f (x) > x^{r}a^{1-r}$ .

2019 Bosnia and Herzegovina Junior BMO TST, 1

Tags: algebra

Let $x,y,z$ be real numbers ( $x \ne y$, $y\ne z$, $x\ne z$) different from $0$. If $\frac{x^2-yz}{x(1-yz)}=\frac{y^2-xz}{y(1-xz)}$, prove that the following relation holds: $$x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$$

1987 USAMO, 1

Tags: algebra , diophantine equation , equation

Determine all solutions in non-zero integers $a$ and $b$ of the equation \[(a^2+b)(a+b^2) = (a-b)^3.\]

1959 AMC 12/AHSME, 29

Tags: function , algebra

On a examination of $n$ questions a student answers correctly $15$ of the first $20$. Of the remaining questions he answers one third correctly. All the questions have the same credit. If the student's mark is $50\%$, how many different values of $n$ can there be? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ \text{the problem cannot be solved} $

2009 China Team Selection Test, 3

Tags: polynomial , algebra , induction , modular arithmetic , binomial coefficient

Let $ f(x)$ be a $ n \minus{}$degree polynomial all of whose coefficients are equal to $ \pm 1$, and having $ x \equal{} 1$ as its $ m$ multiple root. If $ m\ge 2^k (k\ge 2,k\in N)$, then $ n\ge 2^{k \plus{} 1} \minus{} 1.$

1999 Mongolian Mathematical Olympiad, Problem 3

Tags: inequalities , algebra , sequence

Let $(a_n)^\infty_{n=1}$ be a non-decreasing sequence of natural numbers with $a_{20}=100$. A sequence $(b_n)$ is defined by $b_m=\min\{n|an\ge m\}$. Find the maximum value of $a_1+a_2+\ldots+a_{20}+b_1+b_2+\ldots+b_{100}$ over all such sequences $(a_n)$.