Lesson Math Olympiad level problem on caves and bats
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<H2>Math Olympiad level problem on caves and bats</H2> <H3>Problem 1</H3>There is a mountain with 30 bat caves in a row that contain 340 bats, in all. Any 7 caves in a row contain exactly 77 bats. Suppose the first cave has 7 times as many bats as the last cave. How many bats are in the 29th cave? <B>Solution</B> Let me start my solution from the reference to an <U>Internet dictionary</U> "in a row" means "in succession". In this problem, it is the only meanings of this term. <pre> 1. Since any 7 caves in a row contain exactly 77 bats, we have, in particular {{{a[1]+a[2]+a[3]+a[4]+a[5]+a[6]+a[7]}}} = 77, (1) {{{a[2]+a[3]+a[4]+a[5]+a[6]+a[7]+a[8]}}} = 77. (2) Subtract equation (1) from equation (2). The terms from {{{a[2]}}} to {{{a[6]}}} will cancel each other, and you will get {{{a[8]}}} - {{{a[1]}}} = 0, or {{{a[1]}}} = {{{a[8]}}}. Making the same step by step, you will get {{{a[2]}}} = {{{a[9]}}}, {{{a[3]}}} = {{{a[10]}}}, {{{a[4]}}} = {{{a[11]}}}, {{{a[5]}}} = {{{a[12]}}}, {{{a[6]}}} = {{{a[13]}}}, {{{a[7]}}} = {{{a[14]}}}, {{{a[8]}}} = {{{a[15]}}} = {{{a[1]}}} and so on. It means that the sequence {{{a[1]}}}, {{{a[2]}}}, {{{a[3]}}}, . . . , {{{a[30]}}} is <U>PERIODICAL</U> with the period 7. It is the <U>FIRST IDEA</U> of the solution. 2. Its consequence is that {{{a[1]}}} = {{{a[29]}}}. Therefore, from the condition, we have this equation {{{a[1]}}} = {{{a[29]}}} = {{{7*a[30]}}}. (3) It is the <U>SECOND IDEA</U> of the solution. 3. From the condition, we have {{{a[1]+a[2]+a[3]+a[4]+a[5]+a[6]+a[7]}}} = 77, {{{a[8]+a[9]+a[10]+a[11]+a[12]+a[13]+a[14]}}} = 77, {{{a[15]+a[16]+a[17]+a[18]+a[19]+a[20]+a[21]}}} = 77, {{{a[22]+a[22]+a[24]+a[25]+a[26]+a[27]+a[28]}}} = 77. since any 7 caves in a row contain exactly 77 bats. Add these 4 equation (both sides). You will get {{{a[1]}}} + {{{a[2]}}} + {{{a[3]}}} + . . . + {{{a[28]}}} = 4*77 = 308. It means that {{{a[29]}}} + {{{a[30]}}} = 340 - 308 = 32. (4) It is the <U>THIRD IDEA</U> of the solution. 4. Now, from equations (3) and (4) we have {{{a[30]}}} + {{{7*a[30]}}} = 32. It implies {{{8*a[30]}}} = 32, or {{{a[30]}}} = 32/8 = 4. Hence, {{{a[29]}}} = {{{7*a[30]}}} = 7*4 = 28. <U>ANSWER</U>. {{{a[29]}}} = 28. 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