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A lattice point in the plane is a point with integer coordinates.
For instance \((3,5)\) and \((0,-2)\) are lattice points,
but \((3/2,1)\) and \(\big(\sqrt{2},\sqrt{2}\big)\) are not.
Show that no three lattice points in the plane
can be the vertices of an equilateral triangle.
What about in three dimensions?
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If the tangent at a point \(P\) on the curve \(y = x^3\)
intersects the curve again at \(Q\), let \(A\) be the
area of the region bounded by the curve and the segment \(PQ\).
Let \(B\) be the area of the region
defined in the same way starting with \(Q\) instead of \(P.\)
What’s the relationship between \(A\) and \(B\)?
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A solid is generated by rotating about the \(x\)-axis
the region under the curve \(y = f(x)\),
where \(f\) is a positive function and \(x \geq 0\).
The volume generated by the part of the curve from
\(x = 0\) to \(x = b\) is \(b^2\) for all \(b \gt 0\).
Find the function \(f\).
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Find the least possible area of a convex set in the plane
that intersects both branches of the hyperbola \(xy = 1\)
and both branches of the hyperbola \(xy = —1.\)
(A set \(S\) in the plane is called convex
if for any two points in \(S\)
the line segment connecting them is contained in \(S.\))
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Show that the curve \(x^3 + 3xy + y^3 = 1\)
contains only one set of three distinct points, \(A,\) \(B,\) and \(C,\)
which are vertices of an equilateral triangle, and find its area.
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Find all values of \(\alpha\) for which the curves
\(y = \alpha x^2 + \alpha x+ \frac{1}{24}\) and
\(x = \alpha y^2 + \alpha y+ \frac{1}{24}\)
are tangent to each other.
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Determine all ordered pairs of real numbers \((a,b)\)
such that the line \(y = ax+b\)
intersects the curve \(y = \ln\left(1+x^2\right)\)
in exactly one point.
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Find, with explanation, the maximum value of \(f(x)=x^3-3x\)
on the set of all real numbers \(x\)
satisfying \(x^4+36\leq 13x^2.\)
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Denote by \(\mathbf{Z}^2\) the set of all points \((x,y)\)
in the plane with integer coordinates.
For each integer \(n \gt 0\), let \(P_n\) be the
subset of \(\mathbf{Z}^2\) consisting of the point \((0,0)\)
together with all points \((x,y)\) such that \(x^2 +y^2 = 2^k\)
for some integer \(k \lt n\).
Determine, as a function of \(n\),
the number of four-point subsets of \(P_n\),
whose elements are the vertices of a square.
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Can an arc of a parabola inside a circle of radius 1
have a length greater than 4?