Help me in my Olympiad preparation

Hi everybody, I solved this paper just now. I want to check my answer. Please try these questions and tell me the answer with complete logic. Thanks

5 years, 1 month ago

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Problem 1. We are given $2\le a\le b\le c\le d\le16,$$(d-1)^2=(a-1)^2+(b-1)^2+(c-1)^2,\text{ and}$$(d+1)^2+(a+1)^2=(b+1)^2+(c+1)^2$and we want to check if there is one unique integer solution. Except there is not, as $$(a,b,c,d)=(3,6,15,16),(3,7,10,12)$$ are both solutions.

Problem 3. $$c=6$$. Clearly $$y=\pm1$$ cannot have a finite solution. For $$y=0$$, $$x=-6$$ is a solution. For $$x\ge5$$, $$1<|y|<2$$, so it suffices just to check $$1\le x\le4$$. This gives only the solutions $$(1,7)$$ and $$(4,-2)$$, for a total of 3 solutions as desired.

- 5 years, 1 month ago

I have found it to be 10 @ (4,2) (2,3) and (1,11), is it correct?? And what have you found in Questions 1, 2, 4 and 5?

- 5 years, 1 month ago

Multiple solutions exist. It just asked you to find a solution. Also, 10 is incorrect, as it passes through (-10,0). Also, I'm posting solutions at my leisure.

- 5 years, 1 month ago

For question 3, it says "in positive integers". So 6 doesn't work, and 10 does.

Staff - 5 years, 1 month ago

Yes!! This is exactly what I was thinking. Thanks I found out 18 to be okay. Is there any sequence of numbers which satisfy the given conditions?

- 5 years, 1 month ago

nupur, you are 21 years old and you are allowed to sit for olympiads?? which olympiad by the way??

- 5 years, 1 month ago

I am training my Brother, 15, so I am posting on behalf of him. Good observation.....

- 5 years, 1 month ago

wew.... I think i cant reach this level.. I wish I can...

- 5 years, 1 month ago

Never mind...

- 5 years, 1 month ago

I think you should buy "mathematical olympiad primer" if you want to prepare for BMO1. I have it and this BMO1 paper is included there with complete solutions.

- 5 years, 1 month ago

What is BMO1?

- 5 years, 1 month ago

- 5 years, 1 month ago

No, I am Indian and I am going to give RMO(Indian exam similar to BMO) but just for practice I solved a BMO paper.

- 5 years, 1 month ago

Problem 3 (Yes I know Cody already did a solution to this problem. Bad Cody.)

We claim that $$c = 10$$ works.

Manipulating the given equation, we get \begin{aligned} xy^2-y^2-x+y &= 10 \\ (xy^2-x) - (y^2-y) &= 10 \\ x(y-1)(y+1) - y(y-1) &= 10 \\ (y-1)(xy+x-y) &= 10 \end{aligned} Since $$y \ge 1,$$ we must have $$y - 1 \in \{1, 2, 5, 10\}.$$

Case 1: $$y - 1 = 1$$ and $$xy+x-y = 10$$

From the first EQ, we get $$y = 2.$$ substituting into the second EQ and solving gives $$x = 4,$$ giving one solution in this case.

Case 2: $$y - 1 = 2$$ and $$xy+x-y = 5$$

From the first EQ, we get $$y = 3.$$ substituting into the second EQ and solving gives $$x = 2,$$ giving one solution in this case.

Case 3: $$y - 1 = 5$$ and $$xy+x-y = 2$$

From the first EQ, we get $$y = 6.$$ substituting into the second EQ and solving gives $$x = \dfrac87,$$ which is not an integer; thus there are no solutions in this case.

Case 4: $$y - 1 = 10$$ and $$xy+x-y = 1$$

From the first EQ, we get $$y = 11.$$ substituting into the second EQ and solving gives $$x = 1,$$ giving one solution in this case.

In conclusion, setting $$c = 10$$ gives three solutions, as requested. $$\square$$

- 5 years, 1 month ago

This is exactly how I solved.

- 5 years, 1 month ago

Can anybody give me the hint how I should proceed to solve this question? Q. In a very hotly fought battle, at least 70% of the soldiers lost an eye, at least 75% lost an ear ,at least 80% lost an arm ,at least 85% lost a leg.How many lost all four limbs?

- 5 years, 1 month ago

It can't be determined exactly...

- 5 years, 1 month ago

Hints: 1) Bounding. 2) Similar triangles. In fact, first find an expression for tan(<APX)/tan(<PAX) to see what you need to prove is constant. 3) Experiment: First see if $c=p$ for a prime $p$ works. Then try $c=pq$ (p,q primes), etc. You will find $(p,q)=(2,5)$ works. 4) Bound! It boils down to showing that such $r$ exists for all $n\le 27$ (if i remember correctly) 5) What can $f(1)$ be? Once you've got that sorted, you'll need a clever insight, but once you find it the problem is just computation.

- 5 years, 1 month ago

Which book are you referring for RMO? I need help for my olympiad.

- 5 years, 1 month ago