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2d

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Game on the tree
I think it is not determined who wins if you join together two first person wins so that one has an even size and the other has an odd size, so one has to subdivide the games into more than those categories. 
2d

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Game on the tree
The question doesn't need to say that there is a difference between a usable algorithm and an unusable one like brute force search. It is obvious that brute force takes a finite but ridiculously large amount of time. Writing programming in all caps repeatedly is just strange, not convincing. This is a mathematical problem to determine the structure of this family of games so that a fast algorithm can be produced. If someone asks how to factor a 200 digit number, would you say "Here is trial division. Now it is a PROGRAMMING problem?" That is what you have done here. 
Aug 28 
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Game on the tree
You commented that this problem is easy and straightforward, but you have only shown a brute force approach that is like trial division for factorization. That's not progress. That's not helpful. Asking for a translation of this game into known parts of combinatorial game theory is a very reasonable mathematical question. Coding up a very slow brute force search is neither an answer nor mathematics. If you can use your code to find some patterns and make some conjectures about the structure of the problem, that would be mathematics. 
Aug 28 
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Game on the tree
If you really think this is offtopic, you shouldn't post a supposed answer. However, there are many differences between this problem and an IMO problem. It shouldn't be dismissed as trivial and straightforward just because particular cases are finite and you see that a brute force algorithm with a horrible running time exists. 
Aug 28 
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Game on the tree
Is this just a brute force search with memoization? Whether something runs in $n \log n$ time or $2^n$ time is not just a programming problem, nor is discovering the faster algorithm. Not all families of finite problems are mathematically trivial. "PRIMES is in P" was published in the Annals of Mathematics even though testing whether any particular number is prime is a finite calculation. Figuring out a reasonably fast algorithm for playing this game is a math problem, not just a programming problem. 
Aug 26 
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Minimum number of transitive paths in tournament
Is "transitive path" standard notation? I think "transitive subtournament" would be more natural. 
Aug 26 
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An inequality for moments of a random variable
What do you mean by a class of random variables? It seems easy to construct sets of examples. 
Aug 26 
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Probability of one binomial variable being greater than another.
@CommuSoft: I used the assumption that $p \gt q$. As far as I can see, this is a correct application of Hoeffding's inequality on random variables in $[1,1]$. Perhaps what happens if you reverse that assumption indicates that Hoeffding's inequality can't be sharp. 
Aug 23 
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Game on the tree
Suppose you classify games with $\lbrace 0,1 \rbrace$ values by the parity of the number of moves left, who wins if Pieguy moves first, and who wins if Piegirl moves first. If you can classify the results of joining pairs of games of these types, this gives a fast recursive solution (as opposed to the naive exponential recursive algorithms). So, are any of the pairs undetermined? 
Aug 23 
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Game on the tree
The operations respect collapses of the linear order, so if you can solve the problem for values in $\lbrace 0,1\rbrace$ then you can determine whether the value should be greater than $c$ or not, and use binary search to find the fair value. 
Aug 21 
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ErdősSzekeres empty pseudoconvex $k$gons
There must be some compatibility conditions between the usual geometry of the plane and the pseudolines. Otherwise, you can send any $n$ points and pseudolines through them to any other $n$ points and a combinatorially equivalent set of pseudolines, so there is nothing special about a given set of points, and you can choose the pseudolines so that the points are in convex position. This is not the case if you do something like say that the pseudolines have to be rotated graphs of functions. I don't know if that's a good condition but it's invariant under affine transformations. 
Aug 21 
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Where to find (personal) motivation
Some time ago, I looked up the CEOs of $5$ of the $10$ largest companies on Wikipedia. $4$ out of $5$ had engineering degrees, the last had a degree in agricultural economics. 
Aug 20 
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Practical use of estimates for the Gauss Circle Problem
The title doesn't seem to match the body of your question. It seems like you are looking for effective estimates, not practical uses. 
Aug 18 
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Steady state expectation of dynamic system of urns & balls
You have to solve for $x$. It is not a free parameter. If you choose $N=10, C=10, B=50$ then this determines $x$ by the property that there is a geometric series $C+Cr+Cr^2+...+Cr^N = B$. This determines $r=0.823679$, and then $x=Cr^N=1.43741$. 
Aug 18 
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How to solve such an optimization problem
130.44.194.100/mcom/200170236/S002557180001262X/… mentions that the GaussLobatto points maximize the Vandermonde determinant on $[1,1]$, citing another paper: Fej´er, L., Bestimmung derjenigen Abszissen eines Intervalles f¨ur welche die Quadratsumme der Grundfunktionen der Lagrangeschen Interpolation im Intervalle [−1, 1] ein m¨oglichst kleines Maximum besitzt, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mt. Ser. II, 1, 263–276, 1932. 
Aug 17 
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How to solve such an optimization problem
For the Vandermonde determinant itself, the points which maximize it might be called logarithmic Fekete points. I haven't found a description of them directly for the interval, but they might be related to the roots of some orthogonal polynomials. 
Aug 17 
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How to solve such an optimization problem
If you use the range $[1,1]$, then for $N=K=6$, the optimum seems to occur at the roots of $21x^635x^4+15x^21.$ 
Aug 17 
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Steady state expectation of dynamic system of urns & balls
There is a deterministic continuous dynamical system which might be helpful. To fill $U_E$ with $Cx$ balls, the contents of the other urns decrease by a factor of $ \frac{BC}{Bx}$. This has a fixed point in which the urn's counts form a geometric series connecting $C$ to $x$ so that the sum is $B$. I think it might help to analyze the stability of this fixed point, to see if this acts as a potential well near which the process gets stuck for a long time, in which case the expected value of $U_E$ will be close to $x$. 
Aug 17 
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Steady state expectation of dynamic system of urns & balls
That would be the case if the counts started out equal, but that's not stable, and I think the stable distribution is concentrated about a fixed point of a continuous analogue. 
Aug 17 
revised 
Set of distinct real numbers such that all combination of sums are distinct
Added Anthony Quas's simpler condition. 