Algebra
A1. Let \(f:\mathbb{R}\to\mathbb{R}\) satisfy \[ f(x+y)=f(x)f(y)-f(xy)+1 \] for all real \(x,y\). Find all such functions. S
A2. Let \(x_1,x_2,\dots,x_n>0\). Prove \[ \frac{1}{1+x_1} + \frac{1}{1+x_2} + \cdots + \frac{1}{1+x_n} \geq \frac{n}{1+\sqrt[n]{x_1x_2\cdots x_n}}. \] S
A3. Let \(a,b,c\) be positive integers such that \(abc+1\) divides \((a+1)(b+1)(c+1)\). Prove that \[ \frac{(a+1)(b+1)(c+1)}{abc+1} \] is a square. S
Combinatorics
C1. 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 length 1992 contains a nice subsequence of length 1992. S
C2. 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
C3. A tile consists of 5 unit squares. For a \(5\times n\) rectangle let \(T_n\) be the number of tilings. Show that \(T_{2n}=2\cdot3^{n-1}\) and \(T_{2n+1}=0\). S
Geometry
G1. In triangle \(ABC\), let \(H\) be the orthocenter and \(O\) the circumcenter. Show that \[ AH^2+BH^2+CH^2=9OG^2, \] where \(G\) is the centroid. S
G2. Let \(ABCD\) be a convex quadrilateral with perpendicular diagonals. Let \(P\) be the foot from \(B\) to \(AD\). Show that a certain circle through \(P\) has special properties (see the original). S
G3. In an acute triangle \(ABC\), show that certain angle relations hold between altitude feet and original triangle vertices. S
Number Theory
N1. Prove that there are infinitely many primes \(p\) such that \(p\equiv 1\pmod{4}\). S
N2. For any positive integer \(n\) not divisible by 3, show there exists a multiple of \(n\) whose digit sum is 1992. S
N3. Let \(a_n\) be defined by \(a_0=0,a_1=1\) and \[ a_{n+2}=na_{n+1}+(n+1)a_n. \] Find \(a_{1992}\). S
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