Measure concentration for the sphere $S^n$, is a nice problem if the use of a theorem (Brun-Minkowski inequality) counts as a trick.

**Problem:** Let $\mu$ be the normalized measure in the unit sphere $S^n$. Let $X$ be a measurable subset such that $\mu(X)=\frac12$, and $X_{\delta}$ be the set of all $x\in S^n$ for which there is $x'\in X$ such that $||x-x'||=\delta$. Prove that $\mu(X_{\delta})\geq 1-2e^{-\frac{n\delta^2}{4}}$. (So increasing $X$ by just a little gives almost the entire sphere.)

On a similar line stands the "Classical Isoperimetric Inequality". (Among all bodies of the same volume, the ball has the least surface area.)

Another fact that comes to mind (with the trick being a compactness argument) is Bunt-Motzkin theorem:

**Problem:** Show that a simple polytope $P\subset \mathbb{R}^n$ is convex iff for every point $p$ not in $P$, there exists a unique point in $P$ that's closest to $p$.