# Recent questions tagged integer

109
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Prove the mean value theorem by first reducing to the case $$u(a)=u(b)=0$$ and then using the fact that $$u(x)$$ must take on a maximum or minimum value for some poin...
75
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Find a linear differential equation with constant coefficients satisfied by $$e^{2 x}, e^{-x}, e^{3 x}$$, and $$e^{5 x}$$.
68
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Find a linear differential equation with constant coefficients satisfied by $$e^{-2 x}$$ and $$e^{x} \cos 2 x$$.
552
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What is the power rule for differentiation?
639
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What is an integer?
507
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What is a whole number?
384
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Calculate the sum of the digits of $$2^{2015} \times 5^{2019}$$
363
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Prove that any polynomial of degree n has at most n roots.
219
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Let $$f(x)$$ be the polynomial $$\prod_{k=1}^{50}(x-(2 k-1))$$. Let $$c$$ be the coefficient of $$x^{48}$$ in $$f(x)$$. When $$c$$ is divided by 101 , what is the remaind...
278
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A mustache is created by taking the set of points $$(x, y)$$ in the $$x y$$ coordinate plane that satisfy $$4+4 \cos (\pi x / 24) \leq y \leq 6+6 \cos (\pi x / 24)$$ and ...
274
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Consider the base 27 number$n=A B C D E F G H I J K L M N O P Q R S T U V W X Y Z,$where each letter has the value of its position in the alphabet. What remainder do yo...
271
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For each integer from 1 through 2019, Tala calculated the product of its digits. Compute the sum of all 2019 of Tala's products.
341
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In the USA, standard letter-size paper is $$8.5$$ inches wide and 11 inches long. What is the largest integer that cannot be written as a sum of a whole number (possibly ...
563
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Let $$G$$ be the set of points $$(x, y)$$ such that $$x$$ and $$y$$ are positive integers less than or equal to 6 . A magic grid is an assignment of an integer to each po...
400
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The number $$734,851,474,594,578,436,096$$ is equal to $$n^6$$ for some positive integer $$n$$. What is the value of $$n$$ ?
414
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Let $$S$$ be the set of numbers of the form $$n^5-5 n^3+4 n$$, where $$n$$ is an integer that is not a multiple of 3 . What is the largest integer that is a divisor of ev...
267
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Determine the largest integer $$n$$ such that $$n<103$$ and $$n^3-1$$ is divisible by 103 .
199
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If $$\mathrm{A}$$ and $$\mathrm{B}$$ are subset $$\mathrm{s}$$ of $$\mathrm{X}=\{1,2,3,4,5\}$$ then find the probability such that $$\mathrm{n}(\mathrm{A} \cap \mathrm{B}... 241 views 0 answers If \(\mathrm{p}(\mathrm{x})$$ be a polynomial of degree three that has a local maximum value 8 at $$\mathrm{x}=1$$ and a local minimum value 4 at $$\mathrm{x}=2$$; then \...
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Let $$\alpha$$ and $$\beta$$ be the roots of the equation, $$5 \mathrm{x}^2+6 \mathrm{x}-2=0$$. If $$\mathrm{S}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}, \math... 587 views 0 answers Let \(\alpha>0, \beta>0$$ be such that $$\alpha^3+\beta^2=4$$. If the maximum value of the term independent of $$x$$ in the binomial expansion of $$\left(\alpha x^{1 / 9}... 760 views 1 answers Find all monic polynomials \(p(x)$$ with integer coefficients of degree two for which there exists a polynomial $$q(x)$$ with integer coefficients such that $$p(x) q(x)$$...
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For an integer $$n \geq 2$$, let $$D_n$$ be the set of points $$(x, y)$$ of the plane with integer coordinates such that $$-n \leq x, y \leq n$$.(a) Each of the points of...
223
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A natural number $$n$$ is called sensible if there is an integer $$r$$ with $$1<r<n-1$$ such that the representation of $$n$$ in base $$r$$ has all the digits equal. For ...
215
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A function $$f$$ is defined on the set $$\mathbb{N}$$ and satisfies(i) $$f(1)=1$$,(ii) $$f(2 n+1)=f(2 n)+1$$,(iii) $$f(2 n)=3 f(n)$$for all $$n \in \mathbb{N}$$. Find the...
604
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Let $$A$$ be the set of cubic polynomials $$f(x)$$ with the leading coefficient 1 having the following property: There exist a prime number $$p$$ not dividing 2004 and a ...
182
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Find all integers $$n$$ for which the polynomial $$p(x)=x^5-n x-n-2$$ can be represented as a product of two non-constant polynomials with integer coefficients.
389
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A function $$f: \mathbb{R} \rightarrow \mathbb{R}$$ is bounded and satisfies$f\left(x+\frac{1}{3}\right)+f\left(x+\frac{1}{2}\right)=(x)+f\left(x+\frac{5}{6}\right)$for...
384
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A function $$f: \mathbb{R} \rightarrow \mathbb{R}$$ is bounded and satisfies$f\left(x+\frac{1}{3}\right)+f\left(x+\frac{1}{2}\right)=(x)+f\left(x+\frac{5}{6}\right)$for...
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Nonnegative integers $$a$$ and $$b$$ are given. A soldier is walking on the infinite lattice $$\mathbb{Z} \times \mathbb{Z}$$ as follows. In each step, from a point $$(x,... 322 views 0 answers Let \(f(x)$$ be a polynomial with integer coefficients. Let us assume that there exists a positive integer $$k$$ and $$k$$ consecutive integers $$n, n+1, \ldots, n+k-1$$ ...
557
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Given a positive integer $$a$$, a function $$f$$ from $$\mathbb{N}$$ to $$\mathbb{R}$$ satisfies $$f(a)=f(1995)$$, $$f(a+1)=f(1996), f(a+2)=f(1997)$$ and$f(n+a)=\frac{f(... 316 views 1 answers Show that there exists a positive multiple of 1996 whose sum of digits is 1996. 272 views 0 answers Let $$n$$ be a positive integer. A child builds a wall along a line with $$n$$ identical cubes. He lays the first cube on the line and at each subsequent step he lays the... 193 views 0 answers Does there exist a base in which all the numbers $$10101,101010101,1010101010101$$ ... are prime? 289 views 1 answers If $$x$$ and $$y$$ are two positive numbers less than 1 , prove that\[\frac{1}{1-x^2}+\frac{1}{1-y^2} \geq \frac{2}{1-x y}$
372
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State the power rule of differentiation with integer exponents
471
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T/F: Let $$f$$ be a position function. The average rate of change on $$[a, b]$$ is the slope of the line through the points $$(a, f(a))$$ and $$(b, f(b))$$
571
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What is an integer?
355
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What is a multiple of a number?
323
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Let $$\mathrm{n} \geq 5$$ be an integer. If $$9^{n}-8 n-1=64 \alpha$$ and $$6^{n}-5 n-1=25 \beta$$, then $$\alpha-\beta$$ is equal to:
852
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If sum of coefficients in $$({x}-2 {y}+3 {z})^{{n}}$$ is 128 then greatest coefficient in $$(1+x)^{n}$$ is
559
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Prove that every integer $$n \geq 2$$ is a product of prime numbers.
360
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Find the cubic polynomial, in $$x$$, with integer coefficients that has $$\cos 20^{\circ}$$ as a root. The coefficient of $$x^{3}$$ should be positive, and the coefficien...
469
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When the expression $$\left(2^{1}\right)\left(2^{2}\right)\left(2^{3}\right) \cdots\left(2^{99}\right)\left(2^{100}\right)$$ is written as an integer, what is the product...
553
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The number of toy cars that Ray has is a multiple of 6 . When he loses two of them, the number of cars that he has left is a multiple of $$n$$. If $$n$$ is a positive eve...
561
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Ray will choose at random an integer $$Q$$, such that $$34<Q<43$$. What is the probability that Ray will choose a prime number? Express your answer as a common fraction.
A positive integer $$X$$ is 2 more than a multiple of 3 . Its units digit is the same as the units digit of a number that is 4 more than a multiple of 5 . What is the sma...
For each integer $$n$$, let $$f(n)$$ be the sum of the elements of the $$n$$th row (i.e. the row with $$n+1$$ elements) of Pascal's triangle minus the sum of all the elem...
Find the smallest integer $$n$$ for which the sum of the integers from $$-25$$ to $$n$$ (including $$-25$$ and $$n$$ ) is at least $$26 .$$