Algebra
A1. Let \(a,b\) be positive integers. Prove that \[ \frac{a^2}{b}+\frac{b^2}{a}\ge a+b, \] and determine when equality holds. S
A2. Let \(x_1,\dots,x_n\) be positive reals with \(x_1\cdots x_n=1\). Prove \[ \frac{1}{1+x_1}+\cdots+\frac{1}{1+x_n} \ge \frac{n}{2}. \] S
A3. Find all polynomials \(P(x)\in\mathbb{R}[x]\) such that for all \(x,y\in\mathbb{R}\) \[ P(x^2+1) + P(y^2+1) = 2P(xy+1). \] S
Combinatorics
C1. Let \(S\) be a finite set of integers. Show that there exists \(T\subseteq S\) such that \[ \left|\sum_{t\in T} t\right|\le\frac{1}{2}\sum_{s\in S}|s|. \] S
C2. An \(n\times n\) array of real numbers has all its row sums and column sums equal to 1. Show that every real entry lies between 0 and 1. S
C3. A finite sequence of integers is called *nice* if the difference between any two terms is at least the number of terms between them. Show that a sequence of 1995 distinct integers contains a nice subsequence of length 1995. S
Geometry
G1. Let \(ABC\) be a triangle with circumcircle \(\Gamma\). The tangent at \(A\) meets \(BC\) at \(D\). Show that \(\angle BAD = \angle ACB\). S
G2. In triangle \(ABC\), let \(D\) be the foot of the altitude from \(A\). If \(E\) is the reflection of \(D\) across the midpoint of \(BC\), show that \(\angle AED = \angle ABC\). S
Number Theory
N1. Show that there is no integer solution to \[ x^2+y^2=2^z \] for all integers \(x,y,z\) with \(z>1\). S
N2. Let \(p\) be an odd prime. Prove that \[ p^2\mid\binom{p}{1}^2+\binom{p}{2}^2+\cdots+\binom{p}{p-1}^2. \] S
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου