MathDB

5

Part of 2013 Putnam

Problems(2)

Putnam 2013 A5

Source:

12/9/2013
For m≥3,m\ge 3, a list of (m3)\binom m3 real numbers aijka_{ijk} (1≤i<j<k≤m)(1\le i<j<k\le m) is said to be area definite for Rn\mathbb{R}^n if the inequality ∑1≤i<j<k≤maijk⋅Area(△AiAjAk)≥0\sum_{1\le i<j<k\le m}a_{ijk}\cdot\text{Area}(\triangle A_iA_jA_k)\ge0 holds for every choice of mm points A1,…,AmA_1,\dots,A_m in Rn.\mathbb{R}^n. For example, the list of four numbers a123=a124=a134=1,a234=−1a_{123}=a_{124}=a_{134}=1, a_{234}=-1 is area definite for R2.\mathbb{R}^2. Prove that if a list of (m3)\binom m3 numbers is area definite for R2,\mathbb{R}^2, then it is area definite for R3.\mathbb{R}^3.
Putnamgeometryinequalitiessymmetryvectorintegrationgeometric transformation
Putnam 2013 B5

Source:

12/9/2013
Let X={1,2,…,n},X=\{1,2,\dots,n\}, and let k∈X.k\in X. Show that there are exactly k⋅nn−1k\cdot n^{n-1} functions f:X→Xf:X\to X such that for every x∈Xx\in X there is a j≥0j\ge 0 such that f(j)(x)≤k.f^{(j)}(x)\le k.
[Here f(j)f^{(j)} denotes the jjth iterate of f,f, so that f(0)(x)=xf^{(0)}(x)=x and f(j+1)(x)=f(f(j)(x)).f^{(j+1)}(x)=f\left(f^{(j)}(x)\right).]
Putnamfunctioninductionprobabilitylinear algebramatrixalgebra