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

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0answers
11 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
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1answer
85 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 ...
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2answers
39 views

Find integer solution of sysem of quadratic equations

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 ...
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0answers
16 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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1answer
46 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. ...
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3answers
45 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 ...
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1answer
41 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 ...
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0answers
37 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 ...
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2answers
63 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 ...
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3answers
35 views

The sum of the multiples of 2 and 17 under 767 [on hold]

What is the sum of the multiples of 2 or 17 under 767
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0answers
29 views

russian math help [on hold]

i have been learning russian from 10 months ago. and its not so good. at least i can read but not understand properly. when people speak slowly then i can understand properly but not conpletely. now i ...
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1answer
36 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
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0answers
40 views

Differentiable and concave functions with the following properties? [on hold]

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 $x \in [0, \infty)$.
-2
votes
1answer
38 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
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4answers
125 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
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1answer
62 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
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1answer
80 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
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1answer
45 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 ...
3
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2answers
63 views

Find the sum$\pmod{1000}$

Find $$1\cdot 2 - 2\cdot 3 + 3\cdot 4 - \cdots + 2015 \cdot 2016 \pmod{1000}$$ I first tried factoring, $$2(1 - 3 + 6 - 10 + \cdots + 2015 \cdot 1008)$$ I know that $\pmod{1000}$ is the last ...
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1answer
20 views

Ratio and Mensuration of Figures

Is there any quick solution to the problem? Thanks.
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1answer
62 views

Proof inequality using Lagrange Multipliers

Is it possible: $a,b,c$ are non-negative real numbers for which holds that $a+b+c=3.$ Prove the following inequality: $$ 4\ge a^2b+b^2c+c^2a+abc $$ Is it possible using Lagrange Multipliers. I ...
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2answers
82 views

Proving inequalities using Calculus

In general how do you prove inequalities using calculus, I believe it is using maxima or minima right? For example $$a^2b+b^2c+c^2a \le 3, \qquad a,b,c \ge 0,\quad a+b+c=3.$$ How would you use ...
5
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1answer
64 views

Function equation, find the function evaluated at the certain point.

Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1,$ $f(2)+f(3)=125,$ and for all $x$, $f(x)f(2x^{2})=f(2x^{3}+x).$ Find $f(5).$ The constant term, $a_0 = f(0) = 1$. Let: ...
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2answers
32 views

Summation of “products”

Ok, I am pretty sure I won't get an answer because the question is somewhat hard, and I have done research to no prevail but how can I find the summation of ab if I know the summation of a and the ...
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3answers
70 views

Polynomial whose one of its roots is $\cos(\pi/7)$

Let $P(x)$ be a 3rd-degree polynomial with integer coefficients, one of whose roots is $\cos(\pi/7)$. Compute $\frac{P(1)}{P(-1)}$ I saw this question in a contest math problem, and I know that it ...
3
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2answers
67 views

Logic problem involving sum of digits

Good one guys! I'm studying to the national maths olympiad (brazil) by myself, and I ran up to the following question: Let $S(n)$ be the sum of the digits of n. For example $S(77) = 14$ and ...
0
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2answers
36 views

Find the sum of the roots of the exponential equation

The equation $$2^{333x - 2} + 2^{111x + 2} = 2^{222x + 1} + 1$$ has three real roots. Find their sum. I'll simplify it first as: $$\frac{1}{4}2^{333x} + (4)2^{111x} = (2)2^{222x } + 1$$ Let ...
2
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3answers
53 views

Find the sum of the roots given no multiple roots.

Find the sum of the roots, real and non-real, of the equation $$ x^{2001} + \left( \frac{1}{2} - x \right)^{2001} = 0 $$ given that there are no multiple roots. I am in a weird situation here. ...
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2answers
58 views

What kind of geometry is useful to study for mathematical competitions?

I'm bad in geometry but I would like to be better. What kind of geometry is useful to learn olympiad level geometry? I mean, can projective geometry solve more problems than geometry with complex ...
11
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4answers
1k views

N gunmen in a field

Let n be an odd integer. In some field, n gunmen are placed such that all pairwise distances between them are different. At a signal, every gunman takes out his gun and shoots the closest gunman. ...
6
votes
2answers
94 views

$ \lim_{n\to+\infty} \frac{1\times 3\times \ldots \times (2n+1)}{2\times 4\times \ldots\times 2n}\times\frac{1}{\sqrt{n}}$

Knowing that : $$I_n=\int_0^{\frac{\pi}{2}}\cos^n(t) \, dt$$ $$I_{2n}=\frac{1\times 3\times \ldots \times (2n-1)}{2\times 4\times \ldots\times 2n}\times\dfrac{\pi}{2}\quad \forall n\geq 1$$ ...
6
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0answers
122 views

why this three complex are equal (2014 Pan African olympaid problem) [closed]

EDIT(This following is 2014 Pan African olympiad problem) Let $$H(p,q)=\dfrac{\omega p}{\omega-1+a(\omega p-q)},a>0$$ where $\omega^3=1,\omega\neq 1$,if ...
2
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3answers
68 views

$I_{2n}=\dfrac{1\times 3\times \ldots \times (2n-1)}{2\times 4\times \ldots\times 2n}\times\dfrac{\pi}{2}\quad \forall n\geq 1$

let $$I_n=\int_0^{\frac{\pi}{2}}\cos^n(t) \, dt$$ show that $$I_{2n}=\frac{1\times 3\times \ldots \times (2n-1)}{2\times 4\times \ldots\times 2n}\times\dfrac{\pi}{2}\quad \forall n\geq ...
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1answer
89 views

how can one show that if $m$ and $n$ are co-prime, then $U_n$ and $U_m$ are also co-prime?

Given that $$U_n=\underbrace{1\cdots1}_{n\text{ times}}$$ and $n >2$, how can one show that if $m$ and $n$ are co-prime, then $U_n$ and $U_m$ are also co-prime? Because $U_m= ...
1
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1answer
48 views

What is the depth of water above the prism?

I have been practising for a math competition and came across the following question: A fishtank with base $100\,\rm cm$by $200\,\rm cm$ and depth $100\,\rm cm$ contains water to a depth of ...
4
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0answers
62 views

Long polynomial expansion with 34 roots

This is a very tricky problem, I just need a few hints. I think the $(-x^{17})$ is also there for a specific trick. In the end if it is $ax^{17}$, I see that $a = 17 - 1 + 1 = 17$. Also, another ...
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1answer
47 views

Proof exist infinitely many $n$ such that $f_{n}(x)$ has two integers roots

The two integer sequence $\{a_{n}\},\{b_{n}\}$ such $$a_{n+1}=a_{n}+1,2b_{n+1}=a_{n}+2b_{n}$$ Define function $f_{n}(x)=x^2+a_{n}x+b_{n}$, if there exisit $k$ such $f_{k}(x)=0$ has two ...
3
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2answers
78 views

If $\frac{(b−c)}{a} + \frac{(a+c)}{b} + \frac{(a−b)}{c}=1$ and $a-b+c \neq 0 $, then prove that $\frac 1a = \frac 1b + \frac 1c$

The question given is If $\dfrac{(b−c)}{a} + \dfrac{(a+c)}{b} + \dfrac{(a−b)}{c}=1$ and $a-b+c \neq 0 $ then prove that $\dfrac 1a = \dfrac 1b + \dfrac 1c$ I tried to take $abc$ on the right ...
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1answer
28 views

How do I formulate a specific formula for a sequence?

I have three arrays, for instance s = [1:2], j = [1:20] and b = [1:8], and I am trying to build a single row. The problem that I actually have is that I need to find a formula f(s,j,b) such that ...
4
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0answers
25 views

Proving that there exist products of $a_k \equiv 1 \pmod {a_i}$ [closed]

Let $n>2$ be an integer. Prove that there exist numbers $a_1, a_2, \ldots ,a_n$ such that $$a_1a_2\cdots \widehat{a_i}\cdots a_n \equiv 1 \pmod{a_i}$$ for $i=1,2,3,\ldots,n$. Here ...
2
votes
4answers
78 views

Find all functions $\mathbb{R}^{+}\rightarrow \mathbb{R}$

Find all functions $f$: $\mathbb{R}^{+}\rightarrow \mathbb{R}$ such that $$f\left ( \frac{x}{y} \right )= f(x)+f(y)-f(x)f(y)$$ for all $x,y\in\mathbb{R}^{+}$. Here, $\mathbb{R}^{+}$, denotes the ...
2
votes
1answer
131 views

If $a,b,c>0, a+b+c=3$, minimize $\frac{2-a^3}{a}+\frac{2-b^3}{b}+\frac{3-c^3}{c}$ [duplicate]

Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Find the minimum value of the expression $A= \frac{2-a^3}{a}+\frac{2-b^3}{b}+\frac{3-c^3}{c}$ I tried solving it, but I got nothing
12
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4answers
161 views

Integrate $\frac{1}{(1+x^2)(1+x^c)}$ from $0$ to $\infty$ for any $c$.

The question is to evaluate $$ \int_0^\infty \frac{dx}{(1+x^2)(1+x^c)} $$ for arbitrary $c\geq0$. Here are my attempts: (The methods behave somewhat differently for $c=0$ but that case is trivial so ...
0
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1answer
47 views

Evaluate this continued trigonometric sum

$$\sin^2(4) + \sin^2(8) + \sin^2(12) + ... + \sin^2(176)$$ Where the number is in degrees not radians. $$\cos(x) = \sin(90 - x) \implies \cos(x) = \sin(90 + x)$$ $$\implies \sin(x) = \cos(x - ...
1
vote
3answers
76 views

Maximize the following sum

Let $a, b, c, d, e$ be nonnegative integers such that $625a + 250b + 100c + 40d + 16e = 15^3$ . What is the maximum possible value of $a + b + c + d + e$? Quick arithmetic gives: ...
3
votes
2answers
76 views

Partitioning $\{1,2,\cdots ,n\}$ into $2$ sets guarantees $3$ numbers $a,b,c$ in the same set with $ab=c$ for some $n$

(ISL-20-$1988$) Find the least natural number $ n$ such that, if the set $ \{1,2, \ldots, n\}$ is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct ...
0
votes
1answer
49 views

3 variable symmetric inequality

Show that for positive reals $a,b,c$, $\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\geq \frac{3a^3+3b^3+3c^3}{2a^2+2b^2+2c^2}$ What I did was WLOG $a+b+c=1$ (since the inequality is homogenous) ...
1
vote
2answers
55 views

Tough Polynomial Root Problem

Let $S$ be the set of all polynomials of the form $z^3 + az^2 + bz + c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either ...
0
votes
2answers
75 views

BMO1 2004/05 Question 2 Geometry Problem

$2$. Let $ABC$ be an acute-angled triangle, and let $D$,$E$ be the feet of the perpendiculars from $A$, $B$ to $BC$, $CA$ respectively. Let $P$ be the point where the line $AD$ meets the semicircle ...
5
votes
2answers
363 views

Prove the inequality using AM-GM inequality

Given that $a,b,u,v \geq 0$ and $$a^5+b^5 \leq 1$$ $$u^5+v^5 \leq 1$$ Prove that $$a^2u^3+b^2v^3 \leq 1$$ This looks like Holder's inequality, but I found this problem in a book just after the AM-GM ...