Let \(f(x)\) be a function such that \((1-x^2) f”(x) – 2x f'(x) + \alpha (\alpha+1) f(x) =0\)
for some \(\alpha \not\in \mathbb{N}\). Define \(P_n (x) = \frac{d^n}{dx^n} (x^2-1)^n\) for \(n =0,1,…\). Compute \(\int_{-1}^1 f(x) P_n(x) dx.\)
Prove or disprove that if lk(K, J) = 0, then there exist disjoint, compact, properly embedded, orientable surfaces F1, F2 ⊂ S3 × I such that
∂F1 = K × {1}
∂F2 = J × {1}.
Your solution should consist almost entirely of pictures. Each picture may have at most one short explanatory sentence.
(It turns out that the converse is also true.)
Let \(n\) be an odd positive integer, and let
\[
f:\{-1,1\}^n\to\{-1,1\}.
\]
Interpret \(x_i=1\) as voter \(i\) voting for candidate \(A\), and \(x_i=-1\) as voter \(i\) voting for candidate \(B\). The value \(f(x_1,\dots,x_n)\) is the choice.
Find all functions \(f\) satisfying the following properties:
1. Anonymity: for every permutation \(\sigma\in S_n\),
\[
f(x_1,\dots,x_n)=f(x_{\sigma(1)},\dots,x_{\sigma(n)}).
\]
2. Neutrality:
\[
f(-x_1,\dots,-x_n)=-f(x_1,\dots,x_n).
\]
3. Monotonicity: if \(x=(x_1,\dots,x_n)\) and \(y=(y_1,\dots,y_n)\) satisfy
\[
x_i\le y_i \qquad \text{for all } i=1,\dots,n,
\]
then
\[
f(x)\le f(y).
\]
Let \(V\) be the set of tuples \((a_1,…,a_5)\) such that \(a_1 \leq a_2 \leq \cdots \leq a_5 \) belong to \(\mathbb{R}\) and satisfy \[ \sum_{1\leq i\leq 5} a_i >0, \quad \sum_{1\leq i<j \leq 5} a_i a_j >0, \quad \sum_{1\leq i<j< k \leq 5} a_ia_ja_k >0.\]
What is the maximum number \(p\) such that there exists a tuple \((a_1,…,a_5) \) in \(V\) whose \(a_p\leq 0 \)?
Find all positive integer \( k \) satisfying the following statement: For any positive integers \( m \) and \( n \),
\[
\frac{((k+1)m)! ((k+1)n)!}{m! n! ((k-1)m + n)! (m + (k-1)n)!}
\]
is an integer.
The email account pow@mathsci.kaist.ac.kr was not working normally. It is now fixed.
POW2026-01 is revised to clarify the problem. (The revision is only for the clarification and there is essentially no change in the problem.)
We want to find the maximum area of two disjoint, simply connected, congruent tiles that can be packed inside a right triangle, one of whose angles is \( \pi/6 \) (30 degrees). What would be the maximal coverage of the right triangle by the tiles? (There is no restriction on the shape of the tiles, especially it does not need to be rectangular, as long as they are simply connected.) (4 points will be given to the one with the best answer, and 3 points for the next four best answers.)
Prove that for every positive integer \( k \) there exists a positive integer \( n \) such that
\[
\frac{(n+1)(n+2) \dots (2n-k)}{n(n-1) \dots (n-k+1)}
\]
is an integer and that \( k = o(n) \) for such \( n \).
Find all integers \( k \) such that the sequence \( (3n^2 + 3nk^2 + k^3 )_{n=1, 2, \dots} \) contains infinitely many squares.
Note: The numbering was wrong. It should be 2025-18.
Prove the following identity:
\[
\sum_{k=0}^{n-1} \binom{z}{k} \frac{x^{n-k}}{n-k} = \sum_{k=1}^n \binom{z-k}{n-k} \frac{(x+1)^k -1}{k}.
\]