The integer n is a number expressed as the sum of an even number of different positive integers less than or equal to 2000. 1+2+ · · · +2000 Find all of the following positive integers that cannot be the value of n.

Let $ABC$ be an acute triangle with circumcircle $\Omega$ and $\overline{AB} < \overline{AC}$. The angle bisector of $A$ meets $\Omega$ again at $D$, and the line through $D$, perpendicular to $BC$ meets $\Omega$ again at $E$. The circle centered at $A$, passing through $E$ meets the line $DE$ again at $F$. Let $K$ be the circumcircle of triangle $ADF$. Prove that $AK$ is perpendicular to $BC$.

The permutation $\sigma$ consisting of four words $A,B,C,D$ has $f_{AB}(\sigma)$, the sum of the number of $B$ placed rightside of every $A$. We can define $f_{BC}(\sigma)$,$f_{CD}(\sigma)$,$f_{DA}(\sigma)$ as the same way too. For example, $\sigma=ACBDBACDCBAD$, $f_{AB}(\sigma)=3+1+0=4$, $f_{BC}(\sigma)=4$,$f_{CD}(\sigma)=6$, $f_{DA}(\sigma)=3$ Find the maximal value of $f_{AB}(\sigma)+f_{BC}(\sigma)+f_{CD}(\sigma)+f_{DA}(\sigma)$, when $\sigma$ consists of $2020$ letters for each $A,B,C,D$

In an acute triangle $ABC$ with $\overline{AB} > \overline{AC}$, let $D, E, F$ be the feet of the altitudes from $A, B, C$, respectively. Let $P$ be an intersection of lines $EF$ and $BC$, and let $Q$ be a point on the segment $BD$ such that $\angle QFD = \angle EPC$. Let $O, H$ denote the circumcenter and the orthocenter of triangle $ABC$, respectively. Suppose that $OH$ is perpendicular to $AQ$. Prove that $P, O, H$ are collinear.

Let $a, b, c, d, e$ be real numbers satisfying the following conditions. \[a \le b \le c \le d \le e, \quad a+e=1, \quad b+c+d=3, \quad a^2+b^2+c^2+d^2+e^2=14\]Determine the maximum possible value of $ae$.

for a positive integer $n$, there are positive integers $a_1, a_2, ... a_n$ that satisfy these two. (1) $a_1=1, a_n=2020$ (2) for all integer $i$, $i$satisfies $2\leq i\leq n, a_i-a_{i-1}=-2$ or $3$. find the greatest $n$