Individual Contest. English Version. Time limit: 90 minutes. Instructions:

Transcription

1 Elementry Mthemtics Interntionl Contest Instructions: Individul Contest Time limit: 90 minutes Do not turn to the first pge until you re told to do so. Write down your nme, your contestnt numer nd your tem's nme on the nswer sheet. Write down ll nswers on the nswer sheet. Only Aric NUMERICAL nswers re needed. Answer ll 5 prolems. Ech prolem is worth 0 points nd the totl is 50 points. For prolems involving more thn one nswer, full credit will e given only if ALL nswers re correct, no prtil credit will e given. There is no penlty for wrong nswer. Digrms shown my not e drwn to scle. No clcultor or clculting device is llowed. Answer the prolems with pencil, lue or lck ll pen. All ppers shll e collected t the end of this test. English Version

2 Elementry Mthemtics Interntionl Contest Individul Contest Time limit: 90 minutes 0 th July 0 Bli, Indonesi. For ny two numers nd, mens Clculte: L Suppose coconuts hve the sme cost s 4 pinepples, mngo hve the sme cost s pinepples, 0 mngo hve the sme cost s nns, nd 5 ornges hve the sme cost s nns. How mny coconuts hve the sme cost s ornges? A girl clcultes + + L + nd oy clcultes L +. Wht is the sum of their nswers? Wht is the first time etween 4:00 nd 5:00 tht the hour hnd nd the minute hnd re exctly 0 prt? 5. Two squirrels, Tim nd Kim, re dividing pile of hzelnuts. Tim strts y tking 5 hzelnuts. Therefter, they tke lternte turns, ech time tking more hzelnut thn the other in the preceding turn. If the numer of hzelnuts to e tken is lrger thn wht remins in the pile, then ll remining hzelnuts re tken. At the end, Tim hs tken 0 hzelnuts. Wht is the exct numer of hzelnuts t the eginning? 6. In how mny wys cn we py ill of $500 y comintion of $0, $0 nd $50 notes? 7. The lest common multiple of the numers 6, 50 nd A is 00. How mny positive integers A hve this property?

3 AM BN CP MQ NR PS 8. In the figure elow, = = = nd = = =. If the MB NC PA QN RP SM re of ABC is 60 cm, wht is the re of QRS, in cm? C P S R N Q A B M 9. In tle, there re 0 rectngles which consist of n even numer of unit squres. How mny rectngles re there in 6 9 tle which consist of n even numer of unit squres? 0. Find the smllest positive common multiple of 4 nd 6 such tht ech digit is either 4 or 6, there is t lest one 4 nd there is t lest one 6.

4 . We hve two kinds of isosceles tringles ech with two sides of length. The cute tringle hs 0 ngle etween the two equl sides, nd the right tringle hs right ngle etween the two equl sides. We plce sequence of isosceles tringles round point ccording to the following rules. The n-th isosceles tringle is right isosceles tringle if n is multiple of, nd n cute isosceles tringle if it is not. Moreover, the n-th nd (n+)-st isosceles tringles shre common side, s shown in the digrm elow. Wht is the smllest vlue of n>such tht the n-th isosceles tringle coincides with the -st one? When the digits of two-digit numer re reversed, the new numer is t lest times s lrge s the originl numer. How mny such two-digit numers re there?. In the qudrilterl ABCD, AB=CD, BCD=57, nd ADB + CBD = 80. Find the vlue of BAD. D A B C

5 4. Squres on n infinite chessord re eing pinted. As shown in the digrm elow, three squres (lightly shded) re initilly pinted. In the first step, we pint ll squres (drkly shded) which shre t lest one edge with squres lredy pinted. The sme rule pplies in ll susequent steps. Find the numer of pinted squres fter one hundred steps. First step 5. The rows of chessord re numered from to 0 from ottom to top, nd the columns from to 404 from left to right. A snil strts crwling from the cell on row nd column long row. Whenever it is out to crwl off the chessord or onto cell which it hs lredy visited, it will mke left turn nd then crwl forwrds in stright line. Thus it follows spirling pth until it hs visited every cell. Find the sum of the row numer nd the column numer of the cell where the pth ends. (The nswer is +=5 for 4 5 tle.) 4 4 5