Let $x$ be a positive number. Denote by $[x]$ the integer part of $x$ and by $\{x\}$ the decimal part of $x$. Find the sum of all positive numbers satisfying $5\{x\} + 0.2[x] = 25$.
A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have $3$ rows of small congruent equilateral triangles, with $5$ small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of $2003$ small equilateral triangles?
Find a polynomial with integral coefficients whose zeros include $\sqrt{2}+\sqrt{5}$.
Let $p(x)$ be a polynomial with integer coefficients. Assume that $p(a) = p(b) = p(c) = -1$, where $a, b, c$ are three different integers. Prove that $p(x)$ has no integral zeros.
Prove that the sum $$\sqrt{1001^2 + 1} + \sqrt{1002^2 + 1} + \cdots + \sqrt{2000^2 + 1}$$ is irrational.
If $P(x)$ denotes a polynomial of degree $n$ such that $P(k) = k/(k +1)$ for $k = 0, 1, 2, \dots n$, determine $P(n + 1)$.
The product of two of the four zeros of the quartic equation $$x^4 - 18x^3 + kx^2 + 200x - 1984 = 0$$ is $-32$. Find $k$.
Let $n$ be an even positive integer, and let $p(x)$ be an $n$-degree polynomial such that $p(-k) = p(k)$ for $k = 1, 2, \dots , n$. Prove that there is a polynomial $q(x)$ such that $p(x) = q(x^2)$.
Let $a, b, c$ be distinct integers. Can the polynomial $(x - a)(x - b)(x - c) - 1$ be factored into the product of two polynomials with integer coefficients?
Let $p_1, p_2, \cdots, p_n$ be distinct integers and let $f(x)$ be the polynomial of degree $n$ given by $$f(x) = (x - p_1)(x - p_2)\cdots (x -p_n)$$ Prove that the polynomial $g(x) = (f(x))^2 + 1$ cannot be expressed as the product of two non-constant polynomials with integral coefficients.
Find the remainder when you divide $(x^{81} + x^{49} + x^{25} + x^9 + x)$ by $(x^3 - x)$.
Does there exist a polynomial $f(x)$ for which $xf(x - 1) = (x + 1)f(x)$
Is it possible to write the polynomial $f(x) = x^{105}-9$ as the product of two polynomials of degree less than 105 with integer coefficients?
Find all prime numbers $p$ that can be written $p = x^4 + 4y^4$, where $x, y$ are positive integers.
Is $4^{545} + 545^{4}$ a prime?
Prove that if $n>1$, then $(n^4 + 4^n)$ is a composite number.
Compute $$\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}$$
Find the largest prime divisor of $25^2+72^2$
Calculate the value of $$\dfrac{2014^4+4 \times 2013^4}{2013^2+4027^2}-\dfrac{2012^4+4 \times 2013^4}{2013^2+4025^2}$$
Let $P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ be a polynomial with integral coefficients. Suppose that there exist four distinct integers $a, b, c, d$ with $P(a) = P(b) = P(c) = P(d) = 5$. Prove that there is no integer $k$ satisfying $P(k) = 8$.
Show that $(1 + x + \cdots + x^n)^2 - x^n$ is the product of two polynomials.
Determine all polynomials such that $P(0) = 0$ and $P(x^2 + 1) = P(x)^2 + 1$.
Consider the lines that meet the graph $y = 2x^4 + 7x^3 + 3x - 5$ in four distinct points $P_i = (x_i, y_i), i = 1, 2, 3, 4$. Prove that
$$\frac{x_1 + x_2 + x_3 + x_44}{4}$$
is independent of the line, and compute its value.
Find the value of $(2 + \sqrt{5})^{1/3} - (-2 + \sqrt{5})^{1/3}$.
Find the largest integer not exceeding $1 + \frac{1}{\sqrt{2}}+ \frac{1}{\sqrt{3}} + \cdots + + \frac{1}{\sqrt{10000}}$