Let $X_1,X_2,X_3$ be $\exp(1)$. Find the conditional distribution of $X_1|X_1+X_2+X_3=k$. $\textbf{(A)}~\operatorname{Uniform}(0,k)$ $\textbf{(B)}~\operatorname{Uniform}\left(0,\frac k3\right)$ $\textbf{(C)}~\operatorname{Uniform}\left(0,\frac{2k}3\right)$ $\textbf{(D)}~\text{None of the above}$

The number $2013$ has the property that its units digit is the sum of its other digits, that is $2+0+1=3$. How many integers less than $2013$ but greater than $1000$ share this property? ${ \textbf{(A)}\ 33 \qquad\textbf{(B)}\ 34 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}}\ 46\qquad\textbf{(E)}\ 58 $

Let the set $M =\{\frac{1998}{1999},\frac{1999}{2000} \}$. The set $M$ is completed with new fractions according to the rule: take two distinct fractions$ \frac{p_1}{q_1}$ and $\frac{p_2}{q_2}$ from $M$ thus there are no other numbers in $M$ located between them and a new fraction is formed, $\frac{p_1+p_2}{q_1+q_2}$ which is included in $M$, etc. Show that, after each procedure, the newly obtained fraction is irreducible and is different from the fractions previously included in $M$.

Let $ n\ge 5$ be a positive integer. Prove that the set $ \{1,2,\ldots,n\}$ can be partitioned into two non-zero subsets $ S_n$ and $ P_n$ such that the sum of elements in $ S_n$ is equal to the product of elements in $ P_n$.

Five persons wearing badges with numbers $1, 2, 3, 4, 5$ are seated on $5$ chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different)

How many real numbers $x$ are solutions to the following equation? \[ |x-1| = |x-2| + |x-3| \]

Determine the greatest positive integer $ n$ such that in three-dimensional space, there exist n points $ P_{1},P_{2},\cdots,P_{n},$ among $ n$ points no three points are collinear, and for arbitary $ 1\leq i < j < k\leq n$, $ P_{i}P_{j}P_{k}$ isn't obtuse triangle.

There are villager $0$, villager $1$, . . . , villager $2015$ i.e. $2016$ people in the village. You are villager $0$. Each villager is either honest or liar. You don’t know each villager is honest or liar, but you know you are honest and the number of liar is equal or smaller than integer $T$. The villagers became to write one letter without fail from one day. For integers $1 \le n \le 2015$, there are set integers $1 < k_n < 2015$. The letter villager $i$ wrote in day $n$ of the morning is delivered to villager $i + k_n$ if villager $i$ is honest, or villager $i - k_n$ if villager $i$ is liar in day $n$ of the evening. If $i - j$ is divisible by $2016$, villager $i$ and $j$ point same villager. Villagers don’t know $k_n$, but sender is told when letters are received. Villager can write anything on a letter, and each villager receives letters from any villagers a sufficient number of times after enough time. i.e. there are $n$ satisfying $k = k_n$ infinitely for each integer $1 \le k \le 2015$. You want to know the honest persons of this village. You can gather all villagers just once and instruct in one day of noon. The honest person obeys your instruction but the liar person not always obeys and he or she writes on a letter anything possible. One day from your instruction for a while, you could determine all honest persons of this village. Find the maximum value of $T$ such that it is possible to do this if you instruct appropriate regardless of the villagers who are honest or liar.

The square $DEFG$ is contained in equilateral triangle $ABC$, with $E$ on $\overline{AC}$, $G$ on $\overline{AD}$, and $F$ as the midpoint of $\overline{BC}$. Find $AD$ if $DE = 6$.

We call a set of positive integers [i]suitable [/i] if none of its elements is coprime to the sum of all elements of that set. Given a real number $\varepsilon \in (0,1)$, prove that, for all large enough positive integers $N$, there exists a suitable set of size at least $\varepsilon N$, each element of which is at most $N$.

A sequence of $N$ consecutive positive integers is called [i]good [/i] if it is possible to choose two of these numbers so that their product is divisible by the sum of the other $N-2$ numbers. For which $N$ do there exist infinitely many [i]good [/i] sequences?

A field is made of $2017 \times 2017$ unit squares. Luffy has $k$ gold detectors, which he places on some of the unit squares, then he leaves the area. Sanji then chooses a $1500 \times 1500$ area, then buries a gold coin on each unit square in this area and none other. When Luffy returns, a gold detector beeps if and only if there is a gold coin buried underneath the unit square it's on. It turns out that by an appropriate placement, Luffy will always be able to determine the $1500 \times 1500$ area containing the gold coins by observing the detectors, no matter how Sanji places the gold coins. Determine the minimum value of $k$ in which this is possible.

Let $ABCD$ be an isosceles trapezoid with $AB=AD=BC, AB//CD, AB>CD$. Let $E= AC \cap BD$ and $N$ symmetric to $B$ wrt $AC$. Prove that the quadrilateral $ANDE$ is cyclic.

Joe is given a permutation $p = (a_1, a_2, a_3, a_4, a_{5})$ of $(1, 2, 3, 4, 5)$. A [i]swap[/i] is an ordered pair $(i, j)$ with $1 \le i < j \le 5$, and this allows Joe to swap the positions $i$ and $j$ in the permutation. For example, if Joe starts with the permutation $(1, 2, 3, 4, 5)$, and uses the swaps $(1, 2)$ and $(1, 3)$, the permutation becomes \[(1, 2, 3, 4, 5) \rightarrow (2, 1, 3, 4, 5) \rightarrow (3, 1, 2, 4, 5). \]Out of all $\tbinom{5}{2} = 10$ swaps, Joe chooses $4$ of them to be in a set of swaps $S$. Joe notices that from $p$ he could reach any permutation of $(1, 2, 3, 4, 5)$ using only the swaps in $S$. How many different sets are possible? [i]Proposed by Yang Liu[/i]

Let $n$ be a positive integer and let $S$ be a set of $2^n+1$ elements. Let $f$ be a function from the set of two-element subsets of $S$ to $\{0, \dots, 2^{n-1}-1\}$. Assume that for any elements $x, y, z$ of $S$, one of $f(\{x,y\}), f(\{y,z\}), f(\{z, x\})$ is equal to the sum of the other two. Show that there exist $a, b, c$ in $S$ such that $f(\{a,b\}), f(\{b,c\}), f(\{c,a\})$ are all equal to 0.

Positive integer $q{}$ is $m-additive$ $(m\in\mathbb{N}, m\geq2)$ if there exist pairwise distinct positive integers $a_1,a_2,\ldots,a_m$ such that $q=a_1+a_2+\ldots+a_m$ and $a_i | a_{i+1}$ for $i=1,2,\ldots,m-1$. [b]a)[/b] Prove that $1993$ is $8$-additive, but $9$-additive. [b]b)[/b] Determine the greatest integer $m$ for which $2102$ is $m$-additive.