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

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1answer
107 views

Math contest question - prove unsolvability of equations.

Prove that the system of equations has no real solutions: $$\begin{cases} y=\sqrt{x+\sqrt{1-x}} \\ x=\sqrt{y-\sqrt{1+y}}\end{cases}. $$ This is a former problem from a national math contest which ...
3
votes
5answers
88 views

How many $5$ element sets can be made?

Let $m$ be the number of five-element subsets that can be chosen from the set of the first $14$ natural numbers so that at least two of the five numbers are consecutive. Find the remainder when $m$ ...
3
votes
3answers
56 views

Double Factorial Sum

Let $ n!!$ to be $ n(n-2)(n-4)\ldots3\cdot1$ for odd $ n$ values and let $ n(n-2)(n-4)\ldots4\cdot2$ for even $ n$ values. Also let $ \displaystyle \sum_{n=1}^{2009} \frac{(2n-1)!!}{(2n)!!}$ be ...
5
votes
2answers
53 views

Floor Function Equation

How many positive integers $ N$ less than $ 1000$ are there such that the equation $ x^{\lfloor x\rfloor} = N$ has a solution for $ x$? (The notation $ \lfloor x\rfloor$ denotes the greatest ...
5
votes
2answers
550 views

Olympiad question on Pigeonhole principle

Given a set $M$ of $1985$ distinct positive integers, none of which has a prime divisor greater than $26$, prove that $M$ contains at least one subset of four distinct elements, whose product is ...
0
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2answers
69 views

Find the coefficient of $x^{17}$

Find the coefficient of $x^{17}$ in:$$ (1 + x^5 + x^7)^{20}$$ $x^{17} = x^{5} x^5 x^{7}$ I would say: $$\frac{17!}{5!5!7!} $$ But this isnt the correct answer. I know I need to use ...
4
votes
2answers
48 views

A positive integer is equal to the sum of digits of a multiple of itself.

Let $n$ be a positive integer, prove there is a positive integer $k$ so that $n$ is equal to the sum of digits of $nk$. I'm not really sure how I should approach this problem, I tried to do a ...
1
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1answer
61 views

“At least” type probability question.

Recently, I asked a question: Team A has more Points than team B Though I ultimately got the right answer, it took extreme casework, and long computations. My question is: suppose the question was ...
0
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1answer
37 views

Maximum number of teams of three people such that each team is built in one of two ways

A coach picks team members in two ways:   A. The team of three people should consist of one experienced participant and two newbies. Thus, each experienced participant can share the ...
5
votes
1answer
65 views

Elegant applications of advanced techniques to “olympiad” problems

I am interested in applications of somewhat "advanced machinery" (with respect to the usual machinery involved in these cases, which is usually elementary) to olympiad or (high school-level) contest ...
14
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1answer
203 views

There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x^b_{1}+x^b_{2}+\cdots+x^b_{k}|\ge 1$

This problem is a 2014 Sydney mathematics competition problem (11 grade). It seems difficult to solve. (I previously posted the n=2 case for which André Nicolas and Dan Robertson proposed solutions) ...
9
votes
2answers
141 views

There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x_{1}+x_{2}+\cdots+x_{k}|\ge 1$

Edit: This problem 1 is a 2014 Sydney mathematics competition problem (8th grade). It seems difficult to solve. Show that: There exist complex numbers $x_{1},x_{2},\cdots,x_{k}(k\ge 2)$ such ...
2
votes
4answers
88 views

Probability that team $A$ has more points than team $B$

Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the ...
3
votes
0answers
488 views

Two circumcircles of triangles defined relative to a fixed acute triangle are tangent to each other (IMO 2015)

I'm posting here the question because I want to see a nice synthetic solution (not using complex numbers or inversive geometry) for the 3rd problem from IMO 2015. The problem is as follows: Let ...
5
votes
4answers
354 views

Find the roots of the summed polynomial

Find the roots of: $$x^7 + x^5 + x^4 + x^3 + x^2 + 1 = 0$$ I got that: $$\frac{1 - x^8}{1-x} - x^6 - x = 0$$ But that doesnt make it any easier.
0
votes
2answers
20 views

discount and percentage question, how to solve this

To attract more visitors, Zoo authority announces $20\%$ discount on every ticket which cost $\$25$. For this reason, sales of tickets increases by $28\%$. Find the $\%$ of increase in the number of ...
2
votes
0answers
88 views

Expected Power Product of rolling a dice .

A 15 sided dice is rolled 1000 times. Let k1,k2,k3,k4,..k15 denote the number of times 1,2,3...15 appears. How can I compute the following expected value :$$E( (k_1 k_2 k_3 k_4)^5).$$ My attempts:: ...
3
votes
2answers
154 views

IMO 1995 Shortlist problem C5

IMO 1995 Shortlist problem C5 At a meeting of $12k$ people, each person exchanged greetings with exactly $3k+6$ people. For any two people, the number who exchange greetings with both is ...
0
votes
2answers
35 views

Triangle and Ratio : Find the length of a side.

Let $\theta = \angle CAD, \phi = \angle CDB, \varphi=\angle DBC, \alpha = \angle BCD$ and $\beta=\angle ACD$. Then we have the following system of equations $\theta + \varphi = 90^{\circ},$ ...
6
votes
0answers
106 views

Balanced, center-free set. [closed]

We say that a finite set $\mathcal{S}$ of points in the plane is balanced if, for any two different points $A$ and $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC=BC$. We say ...
3
votes
3answers
75 views

Arrangements of Chairs in a Circle

Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs. Hints only please! This is a confusing worded-problem. We ...
0
votes
3answers
67 views

BMO2 1996 Questio 4 - Algebra Problem

Let $a,b,c$ and $d$ be positive real numbers such that $a + b + c + d = 12$ and $abcd=27+ab+ac+ad+bc+bd+cd$. Find all possible values of $a,b,c,d$ satisfying these equations. I found this problem ...
2
votes
1answer
38 views

Clerks sorting files

A group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are reassigned to ...
3
votes
2answers
53 views

Ratio of $\frac{\sin x_1 }{\sin x_2}$ where $f(x_1)=f(x_2)$ for a trigonometric sum $f(x)$

If $f(x) = \cos(x+a_1)+\frac12\cos(x+a_2)+\frac14\cos(x+a_3)+\cdots+\frac1{2^{n-1}}\cos(x+a_n)$, where $a_1, a_2, ... a_n$ are some constants and $f(x_1)-f(x_2)=0$, where $x_2 \neq m\pi$, find ...
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votes
4answers
64 views

How many arrangements exist (a + b + c = 4) [duplicate]

For example, $a + b + c = 4$ Solving this using stars and bars You have $4$ stars and $2$ bars: $$ x | x | xx$$ For example. Then what does $\binom{6}{2}$ mean? The number of arrangements ...
22
votes
3answers
2k views

Finding $(a, b, c)$ with $ab-c$, $bc-a$, and $ca-b$ being powers of $2$

This is a 2015 IMO problem. It seems difficult to solve. Find all triples of positive integers $(a, b, c)$ such that each of the numbers $ab-c$, $bc-a$, and $ca-b$ is a power of $2$. Four such ...
6
votes
1answer
43 views

Relationship between groups that have the same group of homomorphisms to another group

Say, there are two groups $A$ and $B$. We are given that $\mathrm{Hom}(A,G)$ and $\mathrm{Hom}(B,G)$ are isomorphic, where $G$ is another group that may or may not be trivial. What can we say about ...
2
votes
2answers
54 views

Linear Recurrence In Faster Time

I am trying to solve this linear recurrence using matrix exponentiation:- $$f(n) = 2f(n-1) - f(n-2) + c,$$ where $c$ is a constant. What I have come up with is this - Let the matrix $M$ be $$ ...
1
vote
1answer
66 views

How to find the least number with maximum trailing zeroes when multplying with numbers containing 4 or 7 only!

Example: 15 -> 15*4=60 - minimum number with max trailing zeros 125 -> 125*4*4=2000 400 -> 400 will be the answer as its the minimum number with max trailing zeros. Can you think of any other ...
3
votes
3answers
65 views

Probability that no two consecutive heads occur?

A fair coin is tossed $10$ times. What is the probability that no two consecutive tosses are heads? Possibilities are (dont mind the number of terms): $H TTTTTTH$, $HTHTHTHTHTHTHT$. But ...
1
vote
2answers
27 views

time and work question how to solve this

A,B,C are employed to do a piece of work for Rs 5290. A and B together are supposed to do 19/23 th of the work and B and C together 8/23 th of the work then A should be paid A.Rs 4250 B.Rs 3450 ...
3
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0answers
57 views

Is the set of integers so that $n!+1$ divides $(2012n)!$ finite or infinite?

I am having trouble with this problem. We have to determine whether the set of integers such that $n!+1$ divides $(2012n)!$ is finite or infinite. Basically we have to determine if the prime factors ...
0
votes
1answer
33 views

What is the average or above-average score range for the AMC 1o and AIME?

I'm not sure if this is the right place to ask, but what is the average/above-average score range for the AMC 10 and AIME? For example, what points would you say are below-average, average, ...
2
votes
1answer
43 views

How do you calculate variables as exponents in a polynomial without a calculator?

Good day The problem is as follow: Find all solutions $(x, y)$, where $x, y \in \mathbb {Z^+}$ to the equation: $$1+3^x=2^y$$ Two solutions are $(0,1)$ and $(1,2)$ but how do you go about ...
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3answers
98 views

Why is Binomial Probability used here?

A test consists of 10 multiple choice questions with five choices for each question. As an experiment, you GUESS on each and every answer without even reading the questions. What is the ...
1
vote
1answer
111 views

IMC 2008 first problem first day. Finding continuous functions so $x-y\in \mathbb Q \implies f(x)-f(y)\in \mathbb Q$

I would like an alternate solution and proof verification for the following problem: Find all continuous functions $f:\mathbb R \rightarrow \mathbb R$ so that if $x-y$ is rational then $f(x)-f(y)$ is ...
0
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2answers
54 views

Find integer solution of sysem of quadratic equations [closed]

If: $a,b,c$ positive integers, where $a\geq b\geq c$. such that: $$a^2 - b^2 - c^2 +ab=2011$$ $$a^2 +3b^2 +3c^2 -3ab-2ac-2bc=-1997.$$ Find the value of $a$ I tried, but I got nothing. Source: 2012 ...
0
votes
0answers
28 views

Confused between cyclic sum and symmetric sums.

four variables $a, b, c, d$ are given, what is the symmetric and cyclic sum? I thought: $$\sum_{cyc} ab = ab + ac + ad + bc + bd + cd$$ And $$\sum_{sym} ab = 2(ab + ac + ad + bc + bc + ...
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votes
1answer
242 views

How to find minimum number with max trailing zeros when multiplying with 4 or 7?

For example , 15 - 15*4=60 - minimum number with max trailing zeros when multiplying with 4 or 7 125 - 125*4*4=2000 400 - 400 will be the answer as its the minimum number with max trailing zeros. ...
1
vote
3answers
57 views

Maximum determinant of $3 \times 3$ matrix

Good one guys! I'm studying to the maths olympiads in my college and I ran to the following problem: What is the possible matrix $3 \times 3$, that you can write using digits from $0 $ to $9$, (you ...
1
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1answer
44 views

Maths challenge problem: Why is the number of teams which require 4 substitutions 32?

I came across the following problem on a UKMT senior maths challenege: A hockey team consists of 1 goalkeeper, 4 defenders, 4 midfielders and 2 forwards. There are four substitutes: 1 goalkeeper, 1 ...
6
votes
0answers
59 views

Given $100$ coplanar points, no $3$ collinear, then at most $70$ percent triangles formed using these points are acute-angled

(IMO-$1970$) Given $100$ coplanar points, no $3$ collinear, prove that at most $70$ percent of the triangles formed using these points are acute-angled. I know that one solution proceeds by ...
0
votes
2answers
78 views

2014 IMC first problem first day (eigenvalues of a product of symmetric matrices).

This was the first problem of the IMC 2014. Let $A$ and $B$ be two $n\times n$ symmetric matrices with real entries which have all their eigenvalues strictly larger than $1$. Prove all the ...
0
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1answer
38 views

A strange growth speed equation

This question has had me stumped for months, now... It is as quotes: The population of fish in a bay (measured in thousands of fish) at time $t$ is described by the function $p(t) = t^4 + t^2 + ...
2
votes
0answers
40 views

Differentiable and concave functions with the following properties? [closed]

What are all differentiable and concave function $f: [0, \infty) \to [0, \infty)$ with the following properties: $f'(0) - 1 = 0$. $f(f(x)) = f(x)f'(x)$, whenver $x \in [0, \infty)$.
0
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1answer
47 views

Maximum number of positive integers $x\neq y$ such that $\frac{xy}{100}\leq|x-y|$

I've been trying to solve the next problem but I have no idea of how to find the solution: Find the largest number of positive integers in such a way that any two of them $x$ and $y$ ($x\neq y$) ...
2
votes
4answers
135 views

What is the value of the expression: $(1+\frac 12)(1+\frac 13)(1+\frac 14)…(1+\frac {1}{2004})(1+\frac {1}{2005})$?

What is the value of the expression: $(1+\frac 12)(1+\frac 13)(1+\frac 14)...(1+\frac {1}{2004})(1+\frac {1}{2005})$? This question appeared on the UKMT senior maths challenge 2005, and I can't find ...
5
votes
1answer
65 views

Find all primes $a,b,c$ and integer $k$ satisfying the equation $a^2 + b^2 + 16 c^2 = 9k^2 +1$

This was a problem in this year's Junior Balkan Olympiad. So here's what I did first: If $a,b,c,k$ satisfy the conditions, then they satisfy the congruence: $$a^2 +b^2 + c^2 \equiv 1\pmod 3$$ ...
4
votes
1answer
86 views

Find the remainder when the sum is divided by $1000$

Find $S \pmod{1000}$ given: $$S = \sum_{n=0}^{2015} n! + n^3 - n^2 + n - 1$$ $$S_0 = 0! + 0 - 0 + 0 -1 = 0$$ $$S_1 = 1! + 1 - 1 + 1 - 1 = 1$$ $$S_2 = 2! + 8 - 4 + 2 - 1 = 7$$ This isn't ...
1
vote
1answer
50 views

What is the sum of all $k$ values?

In an urn there are a certain number (at least two) of black marbles and a certain number of white marbles. Steven blindfolds himself and chooses two marbles from the urn at random. Suppose the ...