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Suppose \(f\) is a differentiable function of one variable.
Show that all tangent planes to the surface \(z = xf(y/x)\)
intersect in a common point.
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Among all planes that are tangent to the surface \(xy^2z^2 = 1,\)
find the ones that are farthest from the origin.
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Putnam 1959
Find an equation of the smallest sphere
tangent to both lines:
\[
(x,y,z) = (t+1, 2t+4, -3t+5)
\qquad
(x,y,z) = (4t-12, -t+8, t+17)
\]
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Putnam & Beyond
Given \(n\) points in the plane,
suppose there is a unique line that minimizes the sum
of the distances from the points to the line.
Prove that the line passes through two of the points.
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Putnam & Beyond
Prove that for non-negative \(x,\) \(y,\) and \(z\)
such that \(x+y+z=1,\)
the following inequality holds:
\[0 \;\leq\; xy + yz + xz -2xyz \;\leq\; \frac{7}{27}\]
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Find the volume of the region of points \((x,y,z)\) such that
\[ (x^2 + y^2 + z^2 + 8)^2 \leq 36(x^2 + y^2). \]
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Let \(f\colon \mathbf{R}^2 \to \mathbf{R}\) be a function
such that \(f(x,y) + f(y,z) + f(z,x) = 0\)
for all real numbers \(x,\) \(y,\) and \(z.\)
Prove that there exists a function \(g\colon \mathbf{R} \to \mathbf{R}\)
such that \(f(x,y) = g(x) - g(y)\) for all real numbers \(x\) and \(y.\)
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Let \(f(x,y)\) be a continuous, real-valued function on \(\mathbf{R}^2\).
Suppose that, for every rectangular region \(R\) of area \(1,\)
the double integral of \(f(x,y)\) over \(R\) equals \(0.\)
Must \(f(x,y)\) be identically 0?
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Let \(f\) be a continuous real-valued function on \(\mathbf{R}^3.\)
Suppose that for every sphere \(S\) of radius \(1,\)
the integral of \(f(x,y,z)\) over the surface of \(S\) equals \(0.\)
Must \(f(x,y,z)\) be identically 0?
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Suppose that the function \(h\colon\mathbf{R}^2\to \mathbf{R}\)
has continuous partial derivatives and satisfies the equation
\[ h(x,y) = a \frac{\partial h}{\partial x}(x,y) +
b \frac{\partial h}{\partial y}(x,y) \]
for some constants \(a,b.\)
Prove that if there is a constant \(M\)
such that \(|h(x,y)|\leq M\) for all \((x,y) \in \mathbf{R}^2,\)
then \(h\) is identically zero.