People solved this too ... How legendary wowow

\begin{problem}

Attachments:
Problem 23 v2.pdf (50kb)

is the above solution correct?

It seems fine to me unless i am also making a mistake. But $10^6 -1,4,10$ seem far

We will show that $10^6$ points lie on the plane in a way, such that $3$ of them are collinear. Either we cannot construct any "nearly convex" polygons. Else we can construct at least $10^6-1$ ''nearly convex'' polygons. Consider there is a concave polygon $A_1A_2\hdots A_{10^6}.$ Where $\widehat {A_{10^6}A_1A_2}>180^{\circ}. $ Call it the vertices of the polygon in a counterclockwise order.$(\clubsuit) $ Claim- $10^6-1$ ''nearly convex'' polygons $T_1,T_2,\hdots,T_{10^6-1}.$ At first, build a polygon $(T)=A_2A_3\hdots A_{10^6}.$ We will prove that $T $ is convex and $A_1$ lies inside $T. $ Proof- Note, $T_1$ is a concave polygon $A_1A_2\hdots A_{10^6}.$ If $T_1$ is called nearly convex. $\widehat {A_{10^6}A_1A_2}> 180^{\circ} $ among the interior angles of $T_1.$ Therefore, $\forall i\in\{2,3,\hdots,10^6-1\}. $ Points lie on a half-plane of $A_iA_{i+1} (\heartsuit). $ Suppose, $A_3, A_4,\cdots,A_i $ all points lie on a half-plane containing $A_3, $ of $A_2A_{10^6} ,$ and $A_{i+1} $ lies on the remaining on half-plane of $A_2A_{10^6}. $ $ \widehat {A_{i-1}A_iA_{i+1}}>180^{\circ}. $ $\widehat {A_{10^6}A_1A_2}>180^{\circ}\implies\begin {cases}i-1&=10^6\\i&=1\\i+1&=2\end {cases} $ Thus, we cannot find $i. $ $\widehat{A_{i-1}A_1A_{i+1}}\ne\widehat {A_{10^6}A_1A_2} > 180^{\circ}.$ A contradiction $T_1$ has one interior angle $>180^{\circ}. $ All points lie on the half-plane of $A_2A_{10^6}. $ Combining $(\heartsuit) $ we get $T $ is convex. Suppose $A_1$ lies outside $T. $ $A_1A_2$ intersects $T. $ Else there will be $3$ collinear points. $A_1$ lies on $T. $ $T_i \left(i\in\{2,3,\hdots,10^6-1\}\right), $ removes the the segment $A_iA_{i+1} $ of $T $ and connect $A_{i+1} $ with $A_1, A_i $ with $A_iA_{i+1}.$ $A_1$ lies inside $T.$ Let $\omega >180^{\circ}$ be an angle of $A_1.$ Rest angles of $T<180^{\circ}. $ Hence, $T_1$ is a nearly-convex. We cannot construct any nearly-convex polygons or we construct at least $10^6-1$ nearly-convex polygons. If it is not possible that there exists precisely $10$ such polygons.