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

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1answer
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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
79 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
votes
1answer
59 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
31 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 ...
0
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3answers
66 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
58 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 ...
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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
52 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
54 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 ...
10
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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. ...
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2answers
93 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
117 views

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

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 ...
1
vote
1answer
88 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
45 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
60 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 ...
-1
votes
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
76 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 ...
0
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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 ...
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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
75 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
votes
4answers
159 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
45 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
75 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
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1answer
47 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
54 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 ...
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2answers
73 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
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2answers
359 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 ...
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0answers
51 views

Find all $f:\mathbb{R}\rightarrow\mathbb{R}$ for which $f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy))$ [duplicate]

Problem Find all $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfy $$f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy))$$ for all $x,y\in\mathbb{R}$. This is a contest math problem, and I have very little ...
2
votes
1answer
51 views

Sum of Square-Weights

For positive reals $a,b,c$, prove that $$\frac{a^3+b^3+c^3}{3abc}+\sum_{\text{cyc}} \frac{a(b+c)}{b^2+c^2}\geq 4$$ I've heard of a lemma stating if a polynomial expression $f(a,b,c)$ satisfies both ...
0
votes
1answer
23 views

Diagonals of $2n$-gon bisecting area implies?

For a convex (or no diagonals connecting opposite vertices leaving the hexagon) hexagon $ABCDEF,$ if $AD, BE,$ and $CF$ all bisect the area, then they are concurrent. This can be proven in multiple ...
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0answers
47 views

Fixed Points of Function from Rationals to Reals

Consider a function $f$ from the positive rationals to the reals such that $f(x)f(y)\ge f(xy)$ and $f(x+y)\ge f(x)+f(y)$. Further assume this function has a fixed point greater than $1$. Prove this ...
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1answer
39 views

Cutting the Plane

Into how many parts at most is a plane cut by $n$ lines? Into how many parts is space divided by $n$ planes in general position? My approach: $$p(n+1)=p(n)+n+1$$ $$s(n+1)=s(n)+p(n)$$ This solution ...
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1answer
50 views

Another olympiad question related to External principle (regarding geometry problem)

Into how many parts at most is a plane cut by $n$ lines? (b) Into how many parts is space divided by $n$ planes in general position First i was thinking about the approach (not able to find it). ...
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1answer
23 views

Circular cross-sections characterize spheres

The intersection of a set $A \subset \mathbb{R^3}$ with all planes is always a circle. Prove that $A$ is a sphere. We regard single points and $\emptyset$ as degenerate circles & spheres. I ...
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1answer
39 views

another olympiad problem from Arthur Engel related to invariant

The vertices of an n-gon are labeled by real numbers $x_1,...,x_n$ . Let $a, b, c, d$ be four successive labels. If $(a − d)(b − c) < 0$, then we may switch $b$ with $c$. Decide if this switching ...
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0answers
37 views

problem from olympiad book by Arthur Engel(invariant problem)

There are $a$ white, $b$ black, and $c$ red chips on a table. In one step, you may choose two chips of different colors and replace them by a chip of the third color. If just one chip will remain at ...
2
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1answer
22 views

problem in solving this problem from olympiad(use of invariant)

Start with the set $\{3, 4, 12\}$. In each step you may choose two of the numbers $a$, $b$ and replace them by $0.6a − 0.8b$ and $0.8a + 0.6b$. Can you reach $\{4, 6, 12\}$ in finitely many steps: ...
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3answers
31 views

Finding the maximum of an expression in three variables

I am trying to find the max of $\frac{(a)(a-1)(9-a)}{2} + \frac{(b)(b-1)(9-b)}{2} + \frac{(c)(c-1)(9-c)}{2}$ subject to $a+b+c=9$ over nonnegative integers. I originally tried to see if something ...
3
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1answer
64 views

Evaluating $ 1+\frac{\cos(\theta)}{1!}+\frac{\cos(2\theta)}{2!}+\frac{\cos(3\theta)}{3!}+\cdots $

Problem: We are asked to evaluate the infinite sum $$ 1+\frac{\cos(\theta)}{1!}+\frac{\cos(2\theta)}{2!}+\frac{\cos(3\theta)}{3!}+\cdots $$ Thoughts on the problem: At first the infinite sum looks ...
3
votes
1answer
61 views

Mathematical Reflections J339: Solving $\frac{x-1}{y+1} + \frac{y-1}{z+1} + \frac{z-1}{x+1} = 1$ in positive integers

I have been trying to solve $\frac{x-1}{y+1} + \frac{y-1}{z+1} + \frac{z-1}{x+1} = 1$ in positive integers and I'm shaky on one specific step. I solved the problem through case-by-case analysis and I ...
0
votes
0answers
36 views

BMO1 2005/06 Question 5 Geometry Problem

Let $G$ be a convex quadrilateral. Show that there is a point $X$ in the plane of $G$ with the property that every straight line through $X$ divides $G$ into two regions of equal area if and only if ...
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1answer
39 views

BMO1 2005/06 Question 4 Combinatorics Problem

The equilateral triangle $ABC$ has sides of integer length $N$. The triangle is completely divided (by drawing lines parallel to the sides of the triangle) into equilateral triangular cells of side ...
3
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1answer
64 views

If $a+b+c$, $a^2+b^2+c^2$, $a^3+b^3+c^3$ are real, then so are a,b,c

Let $a,b,c$ be complex numbers with distinct magnitudes such that $a+b+c$, $a^2+b^2+c^2$, $a^3+b^3+c^3$ are real. Prove that $a,b,c$ are real numbers as well. I tried to go for ...
3
votes
1answer
40 views

Prove that for any positive integer $n$ the number $1 - {n \choose 2}a + {n \choose 4}a^2 - {n \choose 6}a^3+\cdots$ is divisible by $2^{n-1}$.

Let $a=4k-1$, where $k \in \mathbb{Z}$. Prove that for any positive integer $n$ the number $$1 - {n \choose 2}a + {n \choose 4}a^2 - {n \choose 6}a^3+\cdots$$ is divisible by $2^{n-1}$. My ...
0
votes
1answer
30 views

Show $f(x)\geq 0$ for all $x$ if and only if it has even degree and is of the form $f(x)=r(x)^2(x^2+a_1x+b_1)(x^2+a_2x+b_2) \cdots (x^2+a_nx+b_n)$

Suppose $f(x)$ is a polynomial with real coefficients. Show that $f(x)$ is nonnegative for all real numbers $x$ if and only if it has even degree and is of the form ...
1
vote
1answer
64 views

Showing that $(a^2-a+1)(b^2-b+1)(c^2-c+1) \leq 7$

How can I show that $(a^2-a+1)(b^2-b+1)(c^2-c+1) \leq 7$ given that $a+b+c = 3?$ Attempt: Setting $x=2a-1, y=2b-1, z=2c-1,$ we obtain that $[(2a-1)^2+3][(2b-1)^2+3][(2c-1)^2+3] \leq 448,$ $s=x+y+z ...
0
votes
1answer
42 views

Midpoint of a set as Mean? [closed]

Given a set of an odd number of terms: $x = \{a, b, c, ..., \}$ Consisting of $n$ elements. How is the midpoint of the set. A proof and explanation would be helpful: $$\frac{a + b + c + ... }{n} ...