Algebra
A1. Let \(a,b,c\) be real numbers. Prove that \[ (a^2-2ab+b^2)(b^2-2bc+c^2)(c^2-2ca+a^2)\le\frac{1}{8}(a^2+b^2+c^2)^3. \] S
A2. Let \(x_1,x_2,\dots,x_n>0\). Prove the inequality \[ \frac{1}{1+x_1}+\frac{1}{1+x_2}+\cdots+\frac{1}{1+x_n} \ge\frac{n}{1+\sqrt[n]{x_1x_2\cdots x_n}}. \] 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. A tile is made of five unit squares. For a \(5\times n\) rectangle let \(T_n\) be the number of tilings. Show that \(T_{2n}=2\cdot 3^{n-1}\) and \(T_{2n+1}=0\). S
Geometry
G1. In triangle \(ABC\), show that \[ \sin^2\alpha+\sin^2\beta+\sin^2\gamma\ge 2+2\cos\alpha\cos\beta\cos\gamma, \] where \(\alpha,\beta,\gamma\) are the angles of \(ABC\). S
G2. Let \(ABCD\) be a convex quadrilateral with perpendicular diagonals. Let \(P\) lie on \(BC\) such that \[ \frac{AB}{AC}=\frac{PB}{PC}. \] Show that \(AP\perp CD\). S
Number Theory
N1. Show that there are infinitely many primes \(p\) such that \(p\equiv 1\pmod{4}\). S
N2. Let \(n\) be a positive integer not divisible by 3. Prove that there exists a multiple of \(n\) whose digit sum equals 1988. S
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