CN-TST 2015

The circle $\Gamma$ through $A$ of triangle $ABC$ meets sides $AB,AC$ at $E$ , $F$ respectively, and circumcircle of $ABC$ at $P$ . Prove: Reflection of $P$ across $EF$ is on $BC$ if and only if $\Gamma$ passes through $O$ (the circumcentre of $ABC$ ).

Fix positive integers $k,n$ . A candy vending machine has many different colours of candy, where there are $2n$ candies of each colour. A couple of kids each buys from the vending machine $2$ candies of different colours. Given that for any $k+1$ kids there are two kids who have at least one colour of candy in common, find the maximum number of kids.

FIx positive integer $n$ . Prove: For any positive integers $a,b,c$ not exceeding $3n^2+4n$ , there exist integers $x,y,z$ with absolute value not exceeding $2n$ and not all $0$ , such that $ax+by+cz=0$

There are some players in a Ping Pong tournament, where every $2$ players play with each other at most once. Given: (1) Each player wins at least $a$ players, and loses to at least $b$ players. ( $a,b\geq 1$ ) (2) For any two players $A,B$ , there exist some players $P_1,...,P_k$ ( $k\geq 2$ ) (where $P_1=A$ , $P_k=B$ ), such that $P_i$ wins $P_{i+1}$ ( $i=1,2...,k-1$ ). Prove that there exist $a+b+1$ distinct players $Q_1,...Q_{a+b+1}$ , such that $Q_i$ wins $Q_{i+1}$ ( $i=1,...,a+b$ )

For a positive integer $n$ , and a non empty subset $A$ of $\{1,2,...,2n\}$ , call $A$ good if the set $\{u\pm v|u,v\in A\}$ does not contain the set $\{1,2,...,n\}$ . Find the smallest real number $c$ , such that for any positive integer $n$ , and any good subset $A$ of $\{1,2,...,2n\}$ , $|A|\leq cn$ .

Let $n$ be a positive integer, let $f_1(x),\ldots,f_n(x)$ be $n$ bounded real functions, and let $a_1,\ldots,a_n$ be $n$ distinct reals. Show that there exists a real number $x$ such that $\sum^n_{i=1}f_i(x)-\sum^n_{i=1}f_i(x-a_i)<1$ .

Set $S$ to be a subset of size $68$ of $\{1,2,...,2015\}$ . Prove that there exist $3$ pairwise disjoint, non-empty subsets $A,B,C$ such that $|A|=|B|=|C|$ and $\sum_{a\in A}a=\sum_{b\in B}b=\sum_{c\in C}c$

$\triangle{ABC}$ is isosceles with $AB = AC >BC$ . Let $D$ be a point in its interior such that $DA = DB+DC$ . Suppose that the perpendicular bisector of $AB$ meets the external angle bisector of $\angle{ADB}$ at $P$ , and let $Q$ be the intersection of the perpendicular bisector of $AC$ and the external angle bisector of $\angle{ADC}$ . Prove that $B,C,P,Q$ are concyclic.

Let $X$ be a non-empty and finite set, $A_1,...,A_k$ $k$ subsets of $X$ , satisying:

(1) $|A_i|\leq 3,i=1,2,...,k$ (2) Any element of $X$ is an element of at least $4$ sets among $A_1,....,A_k$ .

Show that one can select $[\frac{3k}{7}]$ sets from $A_1,...,A_k$ such that their union is $X$ .

Let $a,b$ be two integers such that their gcd has at least two prime factors. Let $S = \{ x \mid x \in \mathbb{N}, x \equiv a \pmod b \}$ and call $y \in S$ irreducible if it cannot be expressed as product of two or more elements of $S$ (not necessarily distinct). Show there exists $t$ such that any element of $S$ can be expressed as product of at most $t$ irreducible elements.

Let $G$ be the complete graph on $2015$ vertices. Each edge of $G$ is dyed red, blue or white. For a subset $V$ of vertices of $G$ , and a pair of vertices $(u,v)$ , define $$L(u,v) = \{ u,v \} \cup \{ w | w \in V \ni \triangle{uvw} \text{ has exactly 2 red sides} \}$$Prove that, for any choice of $V$ , there exist at least $120$ distinct values of $L(u,v)$ .