Practice With Solutions

2822

Find $8$ prime numbers, not necessarily distinct such that the sum of the squares of these numbers is $992$ less than $4$ times of the product of these numbers.

2824

Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected?

2828

Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $100$. For how many values of $n$ is $q+r$ divisible by $11$?

2831

Solve in positive integers the equation $3^x + 4^y = 5^z$ .

2832

Solve in positive integers the equation $8^x + 15^y = 17^z$.

2833

Find the number of ordered pairs of positive integer solutions $(m, n)$ to the equation $20m + 16n = 2016$

2835

Find a polynomial with integral coefficients whose zeros include $\sqrt{2}+\sqrt{5}$.

2836

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.

2837

Prove that the sum $$\sqrt{1001^2 + 1} + \sqrt{1002^2 + 1} + \cdots + \sqrt{2000^2 + 1}$$ is irrational.

2839

The product of two of the four zeros of the quartic equation $$x^4 - 18x^3 + kx^2 + 200x - 1984 = 0$$ is $-32$. Find $k$.

2840

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 $p(x) = q(x^2)$.

2841

Let $P(x)$ be a polynomial with integer coefficients satisfying that both $P(0)$ and $P(1)$ are odd. Show that $P(x)$ has no integer zeros.

2842

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?

2843

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.

2844

Find the remainder when you divide $(x^{81} + x^{49} + x^{25} + x^9 + x)$ by $(x^3 - x)$.

2845

Does there exist a polynomial $f(x)$ for which $xf(x - 1) = (x + 1)f(x)$

2846

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?

2847

Find all prime numbers $p$ that can be written $p = x^4 + 4y^4$, where $x, y$ are positive integers.

2848

Is $4^{545} + 545^{4}$ a prime?

2849

Prove that if $n>1$, then $(n^4 + 4^n)$ is a composite number.

2850

Compute $$\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}$$

2851

Find the largest prime divisor of $25^2+72^2$

2852

Calculate the value of $$\dfrac{2014^4+4 \times 2013^4}{2013^2+4027^2}-\dfrac{2012^4+4 \times 2013^4}{2013^2+4025^2}$$

2853

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 $P(k) = 8$.

2854

Show that $(1 + x + \cdots + x^n)^2 - x^n$ is the product of two polynomials.

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