Recent questions tagged integer

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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...
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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
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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
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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
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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 \(p(x) q(x)\)...
618
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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...
188
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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
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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
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Show that there exists a positive multiple of 1996 whose sum of digits is 1996.
272
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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
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Does there exist a base in which all the numbers \(10101,101010101,1010101010101\) ... are prime?
289
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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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1 answers
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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1 answers
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.
672
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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...
687
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1 answers
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...
391
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1 answers
Find the smallest integer \(n\) for which the sum of the integers from \(-25\) to \(n\) (including \(-25\) and \(n\) ) is at least \(26 .\)
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