Tagged Questions

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

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Prove that all the roots of $p(x)=F_{n}x^{n}+..+F_{1}x+F_{0}$ can't be real

Last night I have created this problem. Let $p(x)=F_{n}x^{n}+..+F_{1}x+F_{0}$ where $F_{n}$ is $n$th Fibonacci number. Prove that all the roots of $p(x)$ can't be real. Edit 1: $n>1$.
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How many integer pairs (x, y) satisfy $x^2 + 4y^2 − 2xy − 2x − 4y − 8 = 0$?

How many integer pairs (x, y) satisfy $x^2 + 4y^2 − 2xy − 2x − 4y − 8 = 0$? My Attempt Let $f(x,y)=x^2 + 4y^2 − 2xy − 2x − 4y − 8$ . So $f(x,0)=x^2 − 2x − 8$ . $f(x,0)$ has two roots $x=4 , -2$ . ...
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Polynomial game problem: do we have winning strategy for this game?

I'm thinking about some game theory problem. Here it is, Problem: Consider the polynomial equation $x^3+Ax^2+Bx+C=0$. A priori, $A$,$B$ and $C$ are "undecided", yet and two players "Boy" and ...
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if $A,B,C$ are real numbers such that ,${ A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 } = 1$ and $A+B+C = 0$ find the maximum value of $(ABC )^2$ [duplicate]

$$A,B,C$$ are real numbers such that ,$${ A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 } = 1$$ and $$A+B+C = 0$$ find the maximum value of ${ (ABC) }^{ 2 }$ I don't know how can I start to solve this ...
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No eight digit perfect fourth powers with distinct digits and not containing 3.

Problem: Prove that there are no perfect fourth powers that have eight distinct digits in their base 10 representation and also don't contain 3 as a digit. My attempt: Since the problem is about ...
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Modular Arithmetic Related Question

I've been trying to solve this problem but I couldn't and I don't get the solutions either (I don't think I get how to use modular arithmetic to solve problems in general. And though I tried to ...
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$15$ integers $m_1 \ldots m_{15}$such that $\sum _{k=1} ^{15} m_k \arctan {k} = \arctan 16$

Determine whether or not there exist $15$ integers $m_1 \ldots m_{15}$ such that $\sum _{k=1} ^{15} m_k \cdot \arctan (k) = \arctan (16)$. This is a question from IMC 2015 Day 1 Problem. Here ...
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What is the probability that the two-seed makes the finals?

I can solve the following problem using brute-force combinatorics, but I'm looking for an elegant way to think about it, since there is a rather elegant answer. Suppose there is a tournament of ...
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Plane geometry problem, Suppose ABP,BCP,CAP have same area&perimeter…

I'm trying to solve following geometry question, but it is quite challenging.(at least for me!) Thanks for your help in advance. On plane, there is some triangle ABC. Also, there is a point P ...
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Let $S$ be the smallest positive multiple of $15$, that comprises exactly $3k$ digits with $k$ $0$'s, $k$ $3$'s and $k$ $8$'s.

The following is taken from Singapore Mathematical Olympiad $2013$ Junior Round $1$. Let $S$ be the smallest positive multiple of $15$, that comprises exactly $3k$ digits with $k$ $0$'s, $k$ $3$'s ...
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Variance of the random variable $|X \cup Y|$? [closed]

Let $X$ and $Y$ be random subsets of $\{1, 2, \dots, k-1, k\}$ picked uniformly at random from all $2^k$ subsets, independent of each other. What is the variance of the random variable $|X \cup Y|$?
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How many sewings are there on a soccer ball?

A soccer ball is obtained by sewing $20$ hexagonal pieces of leather and $12$ pieces of leather of pentagonal shape. A sewing joins together the sides of two adjacent pieces. How many sewings ...
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Weight of watermelons after percentage of water is evaporated.

A stock of watermelons of the initial weight of $500 \space\text{kg}$ has been put in a store for a week. Initially the percentage of water in the watermelons makes up $99 \%$ of the ...
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Arrangement of points in a circle

From the 2015 Moscow Mathematical Olympiad: The numbers $1$ to $1000$ are arranged on a circle such that each number divides the sum of its two neighbors. Suppose that the number $k$ has two odd ...
How do you find $∠XPC$ + $∠XPB$ such that $PB+PC$ is maximum where $P$ is a point on $f(x) = (x-1)(x-3)(x-5)$?
Problem: $f(x) = (x-1)(x-3)(x-5)$ intersects the x axis at $A(1,0)$, $B(3,0)$ and $C(5,0)$. A point $P(t,f(t))$ is selected on the curve such that $PB+PC$ is maximum and $t \in (3,5).$ Let $PX$ be ...