- James Stewart 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.
- James Stewart Among all planes that are tangent to the surface \(xy^2z^2 = 1,\) find the ones that are farthest from the origin.
- Putnam 1959 Find an equation of the smallest sphere tangent to both lines: \[ (x,y,z) = (t+1, 2t+4, -3t+5) \qquad (x,y,z) = (4t-12, -t+8, t+17) \]
- 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.
- Putnam & Beyond Prove that for non-negative \(x,\) \(y,\) and \(z\) such that \(x+y+z=1,\) the following inequality holds: \[0 \;\leq\; xy + yz + xz -2xyz \;\leq\; \frac{7}{27}\]
- Putnam 2006 A1 Find the volume of the region of points \((x,y,z)\) such that \[ (x^2 + y^2 + z^2 + 8)^2 \leq 36(x^2 + y^2). \]
- Putnam 2008 A1 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 \(y.\)
- Putnam 2012 A6 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?
- Putnam 2019 A4 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?
- Putnam 2010 A3 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.