Asian Pacific Mathematics Olympiad · 2011

Problems

Problem 1Let $a,b,c$ be positive integers. Prove that it is impossible for all three numbers $a^2+b+c$ , $b^2+c+a$ , and $c^2+a+b$ to be perfect squares.Solutions: 1
Problem 2Five points $A_1,A_2,A_3,A_4,A_5$ lie in the plane, with no three collinear. Determine the maximum possible value of the minimum angle $\angle A_iA_jA_k$ over all distinct $i,j,k$ .Solutions: 1
Problem 3Let $ABC$ be an acute triangle with $\angle BAC=30^\circ$ . The internal and external bisectors of $\angle ABC$ meet line $AC$ at $B_1,B_2$ , and those of $\angle ACB$ meet line $AB$ at $C_1,C_2$ . The circles with diameters $B_1B_2$ and $C_1C_2$ meet inside $ABC$ at $P$ . Prove that $\angle BPC=90^\circ$ .Solutions: 1
Problem 4Let $n$ be a fixed positive odd integer. Take $m+2$ distinct points $P_0,P_1,\ldots,P_{m+1}$ in the coordinate plane such that: (1) $P_0=(0,1)$ , $P_{m+1}=(n+1,n)$ , and for $1\le i\le m$ , both coordinates of $P_i$ are integers between $1$ and $n$ inclusive; (2) for $0\le i\le m$ , segment $P_iP_{i+1}$ is parallel to the $x$ -axis if $i$ is even and to the $y$ -axis if $i$ is odd; (3) for $0\le i<j\le m$ , segments $P_iP_{i+1}$ and $P_jP_{j+1}$ share at most one point. Determine the maximum possible value of $m$ .Solutions: 1
Problem 5Find all functions $f:\mathbb R\to\mathbb R$ that are bounded above and satisfy $f(xf(y))+yf(x)=xf(y)+f(xy)$ for all real $x,y$ .Solutions: 1