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2730Submultisets With Size Divisible by ThreeFrom the multiset a,a,a,b,b,c,c,c,c , you may choose any submultiset. How many choices have total size divisible by 3?脑筋急转弯困难derivation未尝试面试订阅2731Coefficient in a Product of Short Generating FunctionsFind the coefficient of x 12 in (1+x+x 2) 4 (1+x 2+x 4) 2.脑筋急转弯中等derivation未尝试面试订阅2732Weak Compositions With Uniform Upper BoundsHow many weak compositions of 15 into 5 parts have every part at most 4? Solve it from the generating function ((1-x 5)/(1-x)) 5.脑筋急转弯中等derivation未尝试面试订阅2733Payment Combinations With an Odd Number of Nine-Unit NotesA cashier can use 2-unit, 5-unit, and 9-unit notes to make total payment 24. The number of 9-unit notes must be odd. Order does not matter. How many valid combinations exist?脑筋急转弯中等derivation未尝试面试订阅2734Weighted Sum With One Binary Heavy BlockHow many nonnegative integer solutions satisfy 2a+3b+4c+6d=20 if d is restricted to be either 0 or 1?脑筋急转弯困难derivation未尝试面试订阅2735Compositions Into Ones and TwosLet c n be the number of ordered compositions of n using only parts 1 and 2. Use a generating function to show c n = F n+1 , and then compute c 10.脑筋急转弯简单数值题未尝试面试订阅2736A Fibonacci Binomial Sum From Coefficient ExtractionEvaluate sum k>=0 C(12-k, k). You should explain why a generating function turns this into a Fibonacci-type coefficient.脑筋急转弯中等derivation未尝试面试订阅2737Inventory Paths With Holds InsertedA trader executes 4 buys, 4 sells, and 2 holds. Buys and sells each change inventory by plus or minus 1, holds leave inventory unchanged. The trader starts flat, ends flat, and inventory may never become negative at any intermediate time. How many execution sequences are possible?脑筋急转弯困难derivation未尝试面试订阅2738Flat Inventory Paths With Five Buys and Five SellsA flat-start inventory process consists of exactly 5 buys and 5 sells, each changing inventory by plus or minus 1. Inventory may never go negative. How many feasible paths are there?脑筋急转弯中等derivation未尝试面试订阅2739Catalan Generating Function for Balanced Trade SkeletonsLet C n count buy/sell skeletons of length 2n that start and end at inventory 0 and never go negative. Write the generating function C(x)=sum n>=0 C n x n, derive the quadratic equation it satisfies, and compute C 4.脑筋急转弯困难derivation未尝试面试订阅2740Coefficient in 1/((1-x^2)^2(1-x^3))Find the coefficient of x 10 in 1/((1-x 2) 2(1-x 3)).脑筋急转弯中等derivation未尝试面试订阅2741Bid Ceiling With a Fixed Outside OptionA trader values a data license at v. In a sealed first-price auction against one rival, winning with bid b occurs with probability p(b). If skipping the auction preserves an outside payoff of c, what inequality must b satisfy for participating with bid b to weakly dominate skipping?脑筋急转弯中等derivation未尝试面试订阅2742When a Rebate Makes Aggressive Bidding RationalIn a first-price auction, a bidder of value v receives a fixed rebate r if she loses, and nothing if she skips. If bidding b wins with probability p(b), derive the expected payoff of bidding b and compare it with skipping.脑筋急转弯中等derivation未尝试面试订阅2743Scaled Uniform Values and Win ProbabilityThree bidders have i.i.d. private values from Uniform (20,80). In a first-price auction, find the symmetric equilibrium bid of a bidder with value 56. Also compute that bidder's probability of winning under equilibrium play.脑筋急转弯中等derivation未尝试面试订阅2744First-Price Equilibrium Under a Quadratic CDFTwo bidders have i.i.d. private values on [0,1] with CDF F(v)=v 2. In a first-price auction, derive the symmetric equilibrium bid function and compute the bid for value 0.6.脑筋急转弯中等derivation未尝试面试订阅2745Reverse Auction With Two SuppliersA buyer runs a sealed-bid procurement auction with two risk-neutral suppliers. Each supplier's cost is i.i.d. Uniform (0,1), and the lowest bid wins and is paid its own bid. Derive the symmetric equilibrium bid function s(c), and compute the expected profit of a supplier with cost 0.2.脑筋急转弯中等derivation未尝试面试订阅2746Truthful Bidding Under a Capacity PenaltyIn a second-price auction for one extra inventory slot, paying above your true value v can backfire because if you win while your realized usage is low, you incur a deterministic penalty k. If the penalty is independent of the auction price and only applies when you win, what is the effective value that should replace v in the usual Vickrey truth-telling argument?脑筋急转弯困难derivation未尝试面试订阅2747All-Pay Contest Break-Even EffortTwo equally skilled quants enter an all-pay contest for a prize worth V. A contestant who spends effort e while the rival spends e 0 wins with probability e/(e+e 0), and everyone pays their own effort regardless of outcome. Holding e 0 fixed, write the payoff of effort e and solve the equation that defines break-even effort.脑筋急转弯简单derivation未尝试面试订阅2748Why Third-Price Auctions Are Not TruthfulShow by explicit counterexample that truthful bidding is not a dominant strategy in a third-price auction with three bidders, where the highest bidder wins but pays the third-highest bid.脑筋急转弯简单derivation未尝试面试订阅2749English Auction Dropout LogicFour bidders have independent private values 40, 55, 70, and 90. In an English ascending auction with no reserve, who wins, what price is paid, and why is the outcome equivalent to a second-price auction in this private-value setting?脑筋急转弯简单derivation未尝试面试订阅