一些比较难的英文数学题(追加20分)1.Apolynomialp(x)satisfiesp(5−x)=p(5+x)forallrealx.Supposep(x)=0hasfourdistinctrealroots.Findthesumoftheseroots.2.Apersoncashesacheckatabank.Bymistakethetellerpa

Question

一些比较难的英文数学题(追加20分)
1.A polynomial p(x) satisfies p(5 − x) = p(5 + x) for all real x.Suppose p(x) = 0 has four
distinct real roots.Find the sum of these roots.
2.A person cashes a check at a bank.By mistake the teller pays the number of cents as
dollars and dollars as cents.The person spends $3.50 before noticing,then later finds that the remaining money is exactly double the amount of the check.What was the amount?
3.Let a,b,c be integers such that 2005 divides both ab+9b+81 and bc+9c+81.Prove that 2005 also divides ca + 9a + 81.
4.Two thieves stole an open chain with 2k white beads and 2m black beads.They want to share the loot equally,by cutting the chain in pieces in such a way that each one gets k white beads and m black beads.What is the minimum number of cuts that is sufficient?Describe an algorithm to make the cuts.
5.Let ABCD be a unit square and mark off points E,F,G,H successively in sides AB,BC,CD,DA
so that AE = BF = CG = DH = 2011/2012 .Prove that the region that is the intersection oftriangles AGB,BHC,CED,DFA is a square with area
1/(20112 + 20122)^2 .

Answer 1

既然你问到这么难的问题,我就直接用英文了,名词什么的我想你也应该懂的.
1.since p(5 − x) = p(5 + x) ,we can tell that p(x) is an even function,and is symmetric around y axis.From the properties of even function we know that sum of roots of p(x) =0.
2.Let D be the number of dollars,and C the number of cents the check is
made out for.The true value of the check then is 100*D + C cents.
The teller gives her D + 100*C cents instead.After she
spends 5 cents,she has D + 100*C - 350,which is twice 100*D + C.
This gives you one equation:
D + 100*C - 350 = 2*(100*D + C),
which can be rearranged into the form
98*C = 199*D + 350.
There is a hidden condition that you will also need:
0