-
If \(P(x)\) is a quadratic polynomial and
\(P(3) = aP(0) + bP(1) + cP(2),\)
what is the value of \(a-b+c\)?
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Turkish College Entrance Exam
Suppose \(P\) is a fourth-degree polynomial
for which \(P(x) \geq x\) for all real numbers \(x\),
and such that \(P(1)=1,\) \({P(2)=4},\) and \({P(3)=3}.\)
What must the value of \(P(4)\) be?
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Nick’s Mathematical Puzzles ☆
For distinct integers \(a,\) \(b,\) and \(c\)
prove that there is no polynomial \(P\) with integer coefficients such that
\(P(a) = b\) and \(P(b) = c\) and \(P(c) = a\) simultaneously.
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Putnam and Beyond
Let \(P(x)\) be a (monic?) polynomial with integer coefficients
all of whose roots are real and lie in the interval \((0,3)\).
Prove that the roots of this polynomial must be members of the set
\[ \left\{ 1, 2, \frac{3-\sqrt{5}}{2}, \frac{3+\sqrt{5}}{2} \right\}\,. \]
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Putnam and Beyond
Find all polynomials \(P(x)\) with integer coefficients
satisfying \(P\bigl(P'(x)\bigr) = P'\bigl(P(x)\bigr)\)
for all \(x \in \mathbf{R}.\)
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Let \(k\) be a fixed positive integer.
The \(n\)th derivative of \(\frac{1}{x^k-1}\) has the form
\[ \frac{P_n(x)}{\left(x^k-1\right)^{n+1}} \]
where \(P_n(x)\) is a polynomial. Find \(P_n(1)\).
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Find a nonzero polynomial \(P(x,y)\)
such that \(P(\lfloor a \rfloor, \lfloor 2a \rfloor) = 0\)
for all real numbers \(a\).
(Note: \(\lfloor \nu \rfloor\) is the greatest integer less than or equal to \(\nu\).)
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Is there an infinite sequence \(a_0,a_1,a_2,\dotsc\)
of nonzero real numbers such that for each non-negative integer \(n\)
the polynomial \(p_n(x) = a_0 + a_1x + a_2x^2 + \dotsb + a_nx^n\)
has exactly \(n\) distinct real roots?
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Let \(f\) be a [non-constant] polynomial with positive integer coefficients.
Prove that if \(n\) is a positive integer,
then \(f(n)\) divides \(f\big(f(n)+1\big)\) if and only if \(n=1\).
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Let
\[ P_n(x) = (x+n)(x+n-1)\dotsb(x+1) - (x-1)(x-2)\dotsb(x-n) \,. \]
Show that all the roots of \(P_n\) are purely imaginary,
i.e., have real part zero.
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Putnam and Beyond
The zeros of the nth-degree polynomial \(P(x)\)
are all real and distinct.
Prove that the zeros of the polynomial
\(G(x) = nP(x)P''(x)-(n-1)\bigl(P'(x)\bigr)^2\) are all complex.
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Find the smallest positive integer \(j\)
such that for every polynomial \(p(x)\) with integer coefficients
and for every integer \(k\), the integer
\[
p^{(j)}(k) = \left. \frac{\mathrm{d}^j}{\mathrm{d}x^j} p(x) \right|_{x=k}
\]
(the \(j\)-th derivative of \(p(x)\) at \(k\)) is divisible by 2016.
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Let \(n\) be an even positive integer.
Let \(p\) be a monic, real polynomial of degree \(2n;\)
that is to say, \(p(x) = x^{2n} + a_{2n-1} x^{2n-1} + \dotsb + a_1 x + a_0\)
for some real coefficients \(a_0, \dotsc, a_{2n-1}.\)
Suppose that \(p(1/k) = k^2\) for all integers \(k\)
such that \(1 \leq |k| \leq n.\)
Find all other real numbers \(x\) for which \(p(1/x) = x^2.\)
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Suppose that \(P(x) = a_1 x + a_2 x^2 + \cdots + a_n x^n\)
is a polynomial with integer coefficients, with \(a_1\) odd.
Suppose that \(\mathrm{e}^{P(x)} = b_0 + b_1 x + b_2 x^2 + \cdots\) for all \(x\).
Prove that \(b_k\) is nonzero for all \(k \geq 0\).
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The non-constant polynomials \(P(z)\) and \(Q(z)\) with complex coefficients
have the same set of numbers for their zeros
but possibly different multiplicities.
The same is true of the polynomials \(P(z)+1\) and \(Q(z) + 1\).
Prove that \(P(z) = Q(z)\).
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Find polynomials \(f(x)\), \(g(x)\), and \(h(x)\), if they exist,
such that for all \(x,\)
\[
|f(x)|-|g(x)|+h(x) = \begin{cases} -1 & \text{if } x \lt -1
\\ 3x+2 & \text{if } -1 \leq x \leq 0
\\ -2x+2 & \text{if } x \gt 0\,.
\end{cases}
\]