Four positive integers $m,n,p,q$ with $p\lt m$ and $q\lt n$ are given. Consider $A(0,0)$, $B(p,0)$, $C(m,q)$, $D(m,n)$. Paths $f:A\to D$ and $g:B\to C$ consist of unit steps right or up. Let $S$ be the number of ordered pairs $(f,g)$ with no common points. Prove
$$S=\binom{m+n}{n}\binom{m+q-p}{q}-\binom{m+q}{q}\binom{m+n-p}{n}.$$
Let $A_0,B_0,C_0$ respectively be the midpoints of the altitudes $AH,BK,CL$ of a non-equilateral triangle $ABC$. Let $O$ and $I$ be its circumcenter and incenter. Its incircle touches $BC,CA,AB$ at $D,E,F$, respectively. Prove that $A_0D,B_0E,C_0F,OI$ are concurrent.
A function satisfies $f(0,0)=5^{2003}$, $f(0,n)=0$ for all $n\in\mathbb N$, and, for all integers $m\gt 0$ and $n$,
$$f(m,n)=f(m-1,n)-2\left\lfloor\frac{f(m-1,n)}2\right\rfloor+\left\lfloor\frac{f(m-1,n-1)}2\right\rfloor+\left\lfloor\frac{f(m-1,n+1)}2\right\rfloor.$$
Show that some positive integer $M$ satisfies $f(M,n)=1$ for $|n|\le(5^{2003}-1)/2$ and $f(M,n)=0$ otherwise.
Editorial clarification: the printed condition $n\in\mathbb N$ is preserved above. It omits negative initial indices and, if zero belongs to $\mathbb N$, conflicts with $f(0,0)$. The intended well-posed initial condition used for the proof is $f(0,n)=0$ for every integer $n\ne0$. The bracket notation in the source is read as floor.
Let $M,N,P$ be the midpoints of $BC,CA,AB$ respectively in triangle $ABC$. Let $M_1,N_1,P_1$ be points on its perimeter such that each of $MM_1,NN_1,PP_1$ bisects the perimeter. (a) Prove that these lines have a common point $K$. (b) Show that at least one of $KA/BC$, $KB/CA$, $KC/AB$ is not less than $1/\sqrt3$.
Here “bisects the perimeter” means that the two boundary arcs between a pair of endpoints have equal length.
Let $\mathcal A$ be the set of permutations of $1,2,\ldots,2003$ that fix no proper subset of $\{1,\ldots,2003\}$. For $a=(a_1,\ldots,a_{2003})\in\mathcal A$, let
$$d(a)=\sum_{i=1}^{2003}(a_i-i)^2.$$
(a) Find its minimum $d_0$. (b) Find all permutations attaining $d_0$.
Editorial clarification: “fix no proper subset” means leave no nonempty proper subset invariant as a set under $i\mapsto a_i$. The empty set is always invariant and must be excluded. Thus the permitted permutations are exactly single cycles of length $2003$.
Write $\mathbb R$ for the set of real numbers. A function $f:\mathbb R\to\mathbb R$ satisfies
$$f(\cot x)=\cos2x+\sin2x$$
for every $0\lt x\lt \pi$. Determine $g(x)=f(x)f(1-x)$ for $-1\le x\le1$. Find the maximum and minimum values of $g(x)$ on $[-1,1]$.
Two circles $C_1,C_2$ touch externally at $M$, and the radius of $C_2$ is greater than that of $C_1$. A point $A$ moves on $C_2$ and is not on the line through the two centers. Points $B,C$ on $C_1$ are chosen so that $AB,AC$ are tangent to $C_1$. Lines $BM,CM$ meet $C_2$ again at $E,F$, respectively. Let $D$ be the intersection of the tangent at $A$ and the line $EF$. Prove that the locus of $D$ as $A$ varies is a straight line.
Let $S_n$ be the number of permutations $(a_1,a_2,\ldots,a_n)$ of $(1,2,\ldots,n)$ satisfying $1\le|a_k-k|\le2$ for every $k$. Prove that
$$\frac74S_{n-1}\lt S_n\lt 2S_{n-1}$$
for every integer $n\gt 6$.
Find the largest positive integer $n$ for which the following equations have integer solutions $x,y_1,y_2,\ldots,y_n$:
$$(x+1)^2+y_1^2=(x+2)^2+y_2^2=\cdots=(x+n)^2+y_n^2.$$
Define the polynomials
$$p(x)=4x^3-2x^2-15x+9,\qquad q(x)=12x^3+6x^2-7x+1.$$
Prove that each polynomial has at least three distinct real roots. Let $A$ be the largest root of $p(x)$ and $B$ the largest root of $q(x)$. Prove that $A^2+3B^2=4$.
Write $\mathbb R^+$ for the positive real numbers. Let $\mathcal F$ be the set of all functions $f:\mathbb R^+\to\mathbb R^+$ satisfying
$$f(3x)\ge f(f(2x))+x$$
for every $x\gt 0$. Find the largest real constant $\alpha$ such that $f(x)\ge\alpha x$ for every $f\in\mathcal F$ and every $x\in\mathbb R^+$.