仙都峰居士

静雯 Yesterday I solved it finally and got the correct answer.

希望牛娃们在拜六取得好成绩！

静雯

(Selected from RI Mathselection test)  转述，并非原题。若有出入，请版亲指正。

CuteNumber
If a square could be cut into n pieces of at most 2 smaller different squares, n  is a Cute Number.

For example:

n=4,                                n =10 are Cute Numbers.               

Prove that everyinteger greater than 5 is a Cute Number.

以下是我给出的证明，若有不妥，请不吝赐教，谢谢！

Proof:

That means odd number whichgreater than or equal to 7 is a Cute Number.

Therefore, everyinteger greater than 5 is a Cute Number.     Proved.

davidbin

(1)Every even number more than 4 is a cute number
                  2*n
       o---o---o---o---o---o---o
       |       2*n-2          |    |2
       o                         o---o
       |                         |    |2
       o                         o---o
       |2*n-2                 |    |2
2*n o                         o---o
       |                         |    |2
       o                         o---o
       |                         |    |2
       o---o---o---o---o---o---o
       |    |    |    |    |    |    |2
       o---o---o---o---o---o---o
         2    2    2   2    2   2

This cuts a square  into 2*n smaller squares for any number n >= 2.

(2) Now 2*n (n>=2) is a cute number, so is 2*n + 3. Just cut the
large square into quarters (reduce one big square, increase 4 smaller one)

This is enough to show that all numbers greater than 5 are cute number.

davidbin

十台面包机，每台每次生产一公斤面包一只。由于程序故障，其中一台面包机生产的面包重量从一公斤变成了二公斤但外形不变。给你一台电子秤，一次找出哪台出了故障。

静雯

（以前做过一道类似的题。）（红色为更正部分）
取第一台生产的面包 1 个，
   第二台生产的面包 2 个，
   第三台生产的面包 3 个，
  ... ...
   第十台生产的面包10 个，
一起放到电子秤上称，
在没有面包机出故障的情况下，电子秤应显示55kg，
如果是第一台面包机出现了故障，电子秤显示55+1 kg,
如果是第二台面包机出现了故障，电子秤显示55+2 kg,
如果是第三台面包机出现了故障，电子秤显示55+3 kg,
... ...
如果是第十台面包机出现了故障，电子秤显示55+10 kg,
所以根据电子秤显示的数据，我们可以判断出是哪一台出了故障。

（如果是小型电子秤，显示不了那么大的数字，就将面包各10等分，再各自取1 份，2 份，... ，10份）

海云天

想请教楼主，有什么奥数练习题的参考书可以推荐吗？popular买的的SAP的那套全做了。

静雯

davidbin  如果是第N台面包机出现了故障，电子秤显示55+N kg

谢谢davidbin的更正。

静雯

今天是来问题的。可论坛断了好一段时间。大家都还在吧？（困扰了好多天了，快想想办法吧！）

In triangle ABC, D is the midpoint of AB. Let E and F be on CA and CB extend such that DE=DF. Construct perpendiculars from E and F to CE and CF respectively, and denote P the intersection of the two perpendiculars. Show that angle PAE= angle PBF.

想到一个方法了。说一个大致思路。
Let M and N be the midpoints of PB and PA.  Join FM, MD, ND and NE.
We can get MF=MB=ND， NE=NA=MD,  
Triangle MFD is congruent to triangle NDE.
angle FMD=angle DNE.
(we also can get angle BMD = angle DNA)
so angle FMB= angle ENA.
then we can get angle PAE= angle PBF.

静雯

One more question:

In triangle ABC, angle ABC = twice of angle C. Let D be on BC such that AD bisects angle BAC. Let M be the midpoint of BC and let the perpendicular of M to AD intersect AD extended at F and AB extended at E. Prove that BE = half of BD.

frekiwang

证明完了，慢慢打答案中。。。

1. Draw the perpendicular from C to AD, intersecting AC extend at G, intersecting AB extend at H.

2. Since both ME and CH are perpendicular to AD, ME is parallel to CH.

3. In triangle BCH, M is the mid-point of BC, ME is parallel to CH, we can conclude E is the mid-point of BH. (Either by similar triangles or by mid-point theorem). BE is half of BH.

4. In triangle AHC, AG is the angular bisector as well as the height. Therefore, AHC is isoceles, G is the mid-point of CH, angle AHC=angle ACH.

5. Now in triangle DHC, DG is the perpendicular bisector of CH. Therefore, DHC is isoceles, angle DHC = angle DCH.

6.  Now angle AHD=angle AHC - angle DHC = angle ACH - angle DCH = angle ACB, so angle BHD = angle ACB.

7. angle ABC is given as 2 * angle ACB = 2 * angle BHD.

8. On the other hand, as an exterior angle of HBD, angle ABC = angle BHD + angle BDH

9. From 7&8, 2 * angle BHD = angle BHD + angle BDH, thus angle BHD = angle BDH, triangle BHD is isoceles.

10. From 3 and 9. BE = half of BH = half of BD. (Shown)

疯狂的题目，感觉难度不低于SMO决赛了。