# Questions tagged [contest-math]

For questions about mathematics competitions or the questions that typically appear in math competitions. Provide enough information about the source to confirm the question doesn't come from a live contest.

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### Probability Question on Stanford Math Tournament

On Day 1, Aaron draws a smiley face on the board. From then on, on each day he does the same thing as the previous day (draw a smiley face or not) with probability $\dfrac{2}{3}$. What’s the ...
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### Proving Symmedian intersects intersection of tangents

I'm going through Evan Chen's "Euclidean Geometry in Math Olympiads" and I've come to Chapter 4's section on Symmedians. Proposition 4.24 says: Let $X$ be the intersection of the tangents to ...
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### AMC 10 Math Question About Formula For A Sequence... [closed]

Find the value of $\frac{{1^2 + 1 \cdot 2 + 2^2}}{{1^3 \cdot 2^3}} + \frac{{2^2 + 2 \cdot 3 + 3^2}}{{2^3 \cdot 3^3}} + \cdots + \frac{{10^2 + 10 \cdot 11 + 11^2}}{{10^3 \cdot 11^3}}$ This is a ...
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### How to maximize the area of the triangle? [duplicate]

Let $W$ be a focus of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. $P$ and $Q$ are two points on the ellipse such that for any position of $P$ and $Q$ on the ellipse, perimeter of $\Delta PQW$ is ...
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### Prove $f'(r_{+}) + f'(r_{-}) < 0$ for roots of $1 - \frac{M}{r^2} + \frac{Q}{r^4} - r^2 = 0$

Problem. Let $M, Q > 0$ be given. Let $$f(r) := 1 - \frac{M}{r^2} + \frac{Q}{r^4} - r^2, \quad r > 0.$$ If $f$ has three distinct positive real roots $r_c > r_{+} > r_{-} > 0$, prove ...
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### Find $m \in\mathbb{R}$ such that $mX^4 + (1-3m)X^3 + 3(m-1)X^2 + (3-m)X - 1 = 0$ has only one root which is strictly greater than $1$.

Find $m \in\mathbb{R}$ such that $mX^4 + (1-3m)X^3 + 3(m-1)X^2 + (3-m)X - 1 = 0$ has only one root which is strictly greater than $1$. I'm having trouble in solving this type of problems. I tried ...
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### Is this proof correct for Putnam 1986 B4?

I have been practicing on some old Putnam questions, and I attempted to solve this 1986 Putnam B4: For a positive real number $r$ define $G(r)$ to be the minimum value of $|r-\sqrt{m^2+n^2}|$ for all ...
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### Maximize $f(x)=(1-x)^5(1+x)(1+2x)^2$

For which value of $x$ is the product $(1-x)^5(1+x)(1+2x)^2$ a maximum, and what is this value? This is easy with calculus, but how would you do it without calculus? $f(x)=(1-x)^5(1+x)(1+2x)^2 \geq 0$ ...
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### Prove $\sum\limits_{\mathrm{cyc}} \sqrt{a+b} \ge \sum\limits_{\mathrm{cyc}} \sqrt{a + bc}$ for $ab + bc + ca + abc =4$

Problem: Given non-negative real numbers $a,b,c$ satisfying $ab+bc+ca+abc=4.$ Prove that $$\color{blue}{\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\ge \sqrt{a+bc}+\sqrt{b+ca}+\sqrt{c+ab}.}$$ When does equality ...
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### Finding the real roots of an octic polynomial (degree eight). [duplicate]

I have been spending some time on a question which I encountered in the PYQs of a small maths competition from $2$ years ago. It is from the topic quadratic equations and polynomials, it however deals ...
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### Given positive $a,b,c$, show that $\frac{1}{4a^2-ab+4b^2}+\frac{1}{4b^2-bc+4c^2}+ \frac{1}{4c^2-ca+4a^2}\geq \frac{9}{7(a^2+b^2+c^2)}$

This is question 41 of this inequality exercise sheet, and a solution is provided. However, the author's solution is purposefully cumbersome, showing how complicated brute force solution can get. Here ...
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