Problems from or inspired by mathematics competitions. Questions regarding mathematics competitions.

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0
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1answer
128 views

Do degenerate triangles count? (2014 AMC 12B #12)

The problem is this: A set S consists of triangles whose sides have integer lengths less than 5, and no two elements of S are congruent or similar. What is the largest number of elements that S can ...
2
votes
3answers
273 views

2014 AMC 12 B problem 25

What is the sum of all positive real solutions $x$ to the following equation? $$2\cos(2x)\left( \cos(2x) - \cos{\left(\frac{2014\pi^2}{x^2}\right)} \right) = \cos(4x) - 1 $$
1
vote
2answers
140 views

A tricky question from the AMC test (American Mathematics competitions)

A man walks into a store with just enough money to buy exactly 30 balloons, he then he discovers the store has a buy 1 get, one 1/3 off, sale. (a rather ridiculous sale if I do say so myself) how many ...
2
votes
1answer
51 views

Functional equation of non-negative function

Find all $ f:[0,\infty)\rightarrow [0,\infty) $ such that $ f (2)=0 $, $ f (x)\not= 0 $ for $ x\in [0, 2) $ and $$ f (xf (y)) f (y)=f (x+y) $$ for all $ x, y\ge 0 $. I tried plugging in values ...
5
votes
3answers
145 views

An inequality for sides of a triangle

Let $ a, b, c $ be sides of a triangle and $ ab+bc+ca=1 $. Show $$(a+1)(b+1)(c+1)<4 $$ I tried Ravi substitution and got a close bound, but don't know how to make it all the way to $4 $. I am ...
20
votes
4answers
2k views

Sum of four squares not a prime

Let $ a, b, c, d $ be natural numbers such that $ ab=cd $. Prove that $ a^2+b^2+c^2+d^2 $ is not a prime. I am clueless on this one. I tried contradiction, but didn't get anywhere. Can you help? ...
18
votes
2answers
844 views

Tough contest problem

I found this problem in a collection of contest problems of a Russian competition in 1995 and wasn't able to solve it. Solve for real $x$: $$ \cos (\cos (\cos (\cos(x))))=\sin (\sin (\sin (\sin ...
1
vote
2answers
137 views

A counting problem using Burnside's lemma

Suppose we have 12 objects (say, 6 indistinguishable black ones and 6 indistinguishable white ones). How many seatings at a round table can we form from them? The answer is $80$, but how could this ...
7
votes
8answers
246 views

Proof of Divisibility of $n(n^2+20)$ by 48.

This is a question from Bangladesh National Math Olympiad 2013 - Junior Category that still haunts me a lot. I want to find an answer to this question. Please prove this. If $n$ is an even ...
3
votes
2answers
117 views

Eliminate numbers from $1,2,3. . .30$ such that the remaining sequence does not contain both $x$ and $2x$

BdMO 2014 nationals From the sequence 1,2,3. . . .30,pick another sequence of numbers such that if x is in our new sequence,then 2x is not there(or vice versa).What is the maximum number of terms ...
1
vote
0answers
84 views

Proving that $\sqrt{4ab-1}=m^2$ is equivalent to $a=b$. where $a$ and $b$ are non zero integers

So the original question was to prove that if $4ab-1$ divides $4a^2-1)^2$, then $a=b$ where $a$ and $b$ are non zero integers. (IMO 2007) I proceed this way: $(4a²-1)²/(4ab-1)=q$ where $q$ is ...
5
votes
4answers
787 views

Solve without a calculator: What is the possible value of 2*((1+1/100)^100)?

What is the possible value of $2·((1+\tfrac{1}{100})^{100})$? Google will give $2·((1+\tfrac{1}{100})^{100}) = 5.40962765884$. How can I find the possible value without Google or a calculator? How ...
3
votes
1answer
110 views

I am looking for a proof of the “ begonia theorem”.

Let $D$, $E$, $F$ be points on respective (extended) sides $\overleftrightarrow{BC}$, $\overleftrightarrow{CA}$, $\overleftrightarrow{AB}$ of $\triangle ABC$, such that $\overleftrightarrow{AD}$, ...
4
votes
1answer
129 views

If one eats $100$ chocolates in $58$ days,then he must be eating exactly 15 chocolates in some consecutive days

BdMO 2014 Nationals $X$ eats 100 chocolates in 58 days,eating at least 1 chocolate per day.Prove that,in some consecutive days,she ate exactly 15 chocolates. I tried using the pigeonhole ...
2
votes
1answer
90 views

Area of triangle inside triangle

In triangle $ABC$ we choose 3 points $D,E,F$, such that $\overline{AD} = \frac 13 \overline{AB}, \overline{BE} = \frac 13 \overline{BC}, \overline{CF} = \frac 13 \overline{CA}$. Draw segments ...
2
votes
2answers
165 views

Solve exponential-polynomial equation

Solve the equation in $\mathbb{R}$ $$10^{-3}x^{\log_{10}x} + x(\log_{10}^2x - 2\log_{10} x) = x^2 + 3x$$ To be fair I wasn't able to make any progress. I tried using substitution for the ...
7
votes
0answers
171 views

Most famous competition problems? [closed]

When I've attended math competition discussions, I've often heard people remark "oh, this is a famous problem" or say that it's similar to one. Most of them I've actually never heard of before. ...
4
votes
2answers
121 views

Find all functions ${\rm f} :{ \mathbb R}_{+}\to{ \mathbb R}_{+}$

Find all functions ${\rm f}:{\mathbb R}_{+} \to {\mathbb R}_{+}$ , such that $\forall\ x,y \in \mathbb R_+$ the equation $$ \left[1 + y{\rm f}\left(x\right)\right]\left[1 - y{\rm f}\left(x + ...
2
votes
1answer
15 views

i do not how to prove this degenerate polygon

A polygon is called degenerate if one of its vertices falls on a line that joins its neighboring two vertices. In a pentagon ABCDE, AB = AE, BC = DE. P and Q are midpoints of AE and AB. PQ||CD, BD is ...
1
vote
1answer
165 views

Maximizing the perimeter of a triangle inside a square

BdMO 2014: We have a square $ABCD$ of side length 5.We take a point $E$ on $AD$ and $F$ on $AB$ so that $\angle FCE=45^\circ$. What can be the maximum perimeter of $\triangle AEF$? I can ...
0
votes
0answers
30 views

Completing sets of numbers solely with trigonometric functions and an initial zero?

Last week an extra-curricular math academy I attend gave us this question as a challenge: You start with $0$, and the only functions you can do are $\sin, \cos, \tan, \sin^{-1}, \cos^{-1}, \tan^{-1}$ ...
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vote
1answer
44 views

For a prime $p$, $6p\mid a^p+1$ for no $a$ or infinitely many $a$

BdMO Nationals Secondary: Show that for any prime $p$, there are either infinitely many or no positive integer $a$, so that $6p$ divides $a^p+1$ . Find all those primes for which there exists no ...
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4answers
178 views

Odd one out questions

These are two questions given to a grade 5 student. I couldn't get a conclusive and compelling answer to any.
3
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1answer
140 views

Answering a flawed Mathletes question (finding $x^2 + y^2 = p$ given $p$ for large $p$)

There was a mathletes meet today (high school) and this was one of the questions: "-Some background on Fermat's 4k+1 sum of square theorem- One such prime is $367369$. What integers $x, y$ satisfy ...
2
votes
1answer
104 views

Fifteen pennies lie on the table in the shape of a triangle

Fifteen pennies lie on the table in the shape of a triangle, with five pennies on each side. For some reason, the pennies are painted either black or white. Prove that there exist three pennies of ...
2
votes
1answer
208 views

A problem in elementary calculus

Let $P(x), Q(x)$ be two polynomials with real coefficients and set $F(x) = \frac{P(x)}{Q(x)}$. Consider a table which has the function $e^{\int_0^x F \, dx}$ on it. The table has the set of rules that ...
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votes
1answer
49 views

What is a right way to think when going about trying to solve a math problem? [closed]

What is good step-by-step method to deconstruct a math problem.
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0answers
242 views

Finding the number of arrangement of N people of different height such that K of them are visible from front

Moderator Note: This is a current contest question on codechef.com. [Initially, I had asked this question in stackoverflow, but someone suggested to post it here, and hence this question is ...
2
votes
1answer
124 views

IMO problem 4, $1998$

Determine all pairs $(a, b)$ of positive integers such that $ab^{2} + b + 7$ divides $a^{2}b + a + b$. I really have no idea where to start with this. This is the first IMO problem that I attempted, ...
2
votes
1answer
46 views

AIME 1986:different sequences of coin tosses

AIME 1986 Problem-13 In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We ...
1
vote
1answer
62 views

Choosing $2n-1$ points from $n\times n$ grid such that $3$ points always form a right triangle

NOTE: Looking for a hint,not the whole solution. BdMO 2012 Nationals Secondary Consider a $n×n$ grid of points. Prove that no matter how we choose $2n-1$ points from these, there will always ...
1
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1answer
38 views

The second triangle?

BdMO National 2013 Junior Q. 2 Two isosceles triangles are possible with 120 square unit area of each and length of edges are integers. Such one is with 17, 17 and 16 unit edges. Determine the ...
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2answers
120 views

algebra , JEE-IIT entrance test sample questions [closed]

$x$ is a real number. If $x^3+1/x^3=52$, find the value of $x^5+1/x^5$.
1
vote
2answers
2k views

Diophantine Equation (2014 AMC 12A)

There are exactly $N$ distinct rational numbers $k$ such that $|k| < 200$ and $$5x^2 + kx + 12 = 0 $$ has at least one integer solution for $x$. What is $N$? (My idea was to consider the equation ...
4
votes
1answer
143 views

how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y|\le 1000$

The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$,For how many points $(x,y)\in P$ with integer coordinates is it true that $|4x+3y|\le 1000$ $A:38 , B:40 C:42 ...
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2answers
80 views

Contest geometry problem

$|AM|=|CM|$ $\angle BCA = 15^{\circ}$ $\angle CBM = \angle ABH$ $\angle BHC = 90^\circ$ Find $|AC|$ The solution states that $\overline{BM}$ is the isogonal conjugate of $\overline{BH}$ but I ...
4
votes
1answer
121 views

AMC 12 2010B Problem Help #18

Can someone explain this solution? A frog makes 3 jumps, each exactly 1 meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final ...
0
votes
2answers
62 views

Choosing people around a circular table

There are 20 people around a circular table.We have to choose $3$ of them such that at least $2$ of them are sitting together.In how many ways can this be done? Number of ways of choosing 3 people ...
5
votes
1answer
166 views

Sequence where the sum of digits of all numbers is 7

BdMO 2014 We define a sequence starting with $a_1=7,a_2=16,\ldots,\,$ such that the sum of digits of all numbers of the sequence is $7$ and if $m>n$,then $a_m>a_n$ i.e. all such numbers are ...
3
votes
1answer
192 views

The library with 999 books.

In the town of Capibara there is a library with books in 999 themes. Since Capibara is an international town they have books in various languages. We know that for every language we can find exactly ...
1
vote
1answer
267 views

putnam mathematics

The tail of a giant kangaroo is attached by a giant rubber band to a stake in the ground. A flea is sitting on top of the stake eyeing the kangaroo (hungrily). The kangaroo sees the flea leaps into ...
1
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1answer
31 views

Compute number of points having same property

I have been given a cuboid which has either green or red color for each point (integer coordinates) in it. I am also given another cuboid whose lower left corner is (x1, y1, z1) and upper right corner ...
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2answers
105 views

$7$ points inside a circle at equal distances

BdMO 2014 There are $7$ points on a circle.Any 2 consecutive points are at equal distance from one another.How many acute angled triangles can you form taking any 3 of these points? I believe ...
1
vote
3answers
92 views

Show that the number of 5-tuples (a, b, c, d, e) such that abcde = 5(bcde + acde + abde + abce + abcd) is odd

Show that the number of 5-tuples $(a,b,c,d,e)$ such that$$abcde=5(bcde+acde+abde+abce+abcd)$$ is odd.
1
vote
1answer
65 views

Eliminating numbers from the sequence $1,2,3,4,5,6,7…400$

BdMO 2014 Let us take the sequence $1,2,3,4,5,6,7....400$ .We are going to remove numbers from the sequence such that the sum of any 2 numbers of the remaining sequence is not divisible by 7.What ...
0
votes
1answer
60 views

Brazil 2002 first problem neater result?

Brazil's 2002 first problem basically asks to prove that for any positive integer n, there are n integers $m_1,m_2\dots m_n$ where $1\leq m_i\leq9$ such that $m_1^2+m_2^2+\ldots+m_n^2=a^2$ for some ...
1
vote
0answers
44 views

Separating points on a plane

BdMO 2011 There are $25$ points on a plane, no three of which lie on a line. Find the minimum number of lines needed to separate them from one another. Can we assume that the points lie on a ...
1
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1answer
63 views

Arranging red and blue tiles in a line with at least 1 blue tile between any 2 red tiles

BdMO 2010 Nationals: Tom and Jerry have $8$ blue tiles and $6$ red tiles.They want to arrange them in a straight line so that between any $2$ red tiles there is always at least $1$ blue tile.In ...
2
votes
4answers
131 views

Sum $\sum_{k=0}^{2013}2^ka_{k}$

let real sequence $a_{0},a_{1},a_{2},\cdots,a_{n}$,such $$a_{0}=2013,a_{n}=-\dfrac{2013}{n}\sum_{k=0}^{n-1}a_{k},n\ge 1$$ How find this sum $$\sum_{k=0}^{2013}2^ka_{k}$$ My idea: since ...
7
votes
7answers
285 views

$211!$ or $106^{211}$:Which is greater?

A BdMO question: Let $a=211!$ and $b=106^{211}$. Show which is greater with proper logic. By matching term by term,it is pretty easy to note that $106!<106^{106}$ $106^{105}<107\cdot ...