# Tagged Questions

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

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### $1000$th decimal digit of $(8+\sqrt{63})^{2012}$

Find the digit at the $1000$th position at the right of the decimal point of the number $(8+\sqrt{63})^{2012}$ I took this problem from a Mexican Math Olympiad called Galois-Noether. It's the last ...
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### Prove there are 3 points on the circle having same colour [closed]

All the points of a circle are randomly coloured red or blue. Prove there are 3 points on the circle having same colour, representing an isosceles triangle.
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### Find all polynomials $P(x)$ such that $P(x^2)=P(x)^2$

Find all polynomials $P:\mathbb{C}\rightarrow\mathbb{C}$ such that $$P(x^2)=P(x)^2 .$$ Here is what I tried: First, it is easy to see the constant solutions, namely $P\equiv 0,P\equiv 1$. Let $r$ ...
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### Combinatorics olympiad problem (Yandex Data Science School)

I've found quite an interesting problem involving combinatorics and some set theory. It was in Yandex Data Science School admission exam. Please check if my solution is correct. Given arbitrary 100 ...
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### Olympiad Inequality $\sum_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$

$x,y,z >0$, prove $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$ Note: Often Stack Exchange asked to show some work before answering the ...
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### If for all $\displaystyle \theta \in [ 0,\frac{\pi}{2} ]$, we have $| \sin \theta - p \cos \theta - q|\leq \frac{\sqrt{2}-1}{2}$. Then find $p+q$.

If for all $\displaystyle \theta \in [ 0,\frac{\pi}{2} ]$, we have $| \sin \theta - p \cos \theta - q|\leq \frac{\sqrt{2}-1}{2}$. Find $p+q$. My Work: When $p=-1,q=\frac{\sqrt{2}+1}{2}$, we have ...
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### How to plot this graph $y^3=x^2$

I was solving a problem related to area under the integral. When I got a question with the curve $y^3=x^2$. Now this might seem trivial with plotting calculator and for some without plotting ...
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Here is the Problem: 1)Suppose $p$ is a prime. prove that for any integer $k$, there exist integers $x$ and $y$ such that $x^2+y^2 \equiv k\ \pmod p$. 2)Are there infinitely many composite ...
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### Algebraic Manipulations [duplicate]

Let a, b and c be such that $a+b+c = 0$ and $l^2 = \frac{a^2}{2a^2+bc} + \frac{b^2}{2b^2+ac} + \frac{c^2}{2c^2+ba}$ The what is the value of l My approach : I could just put in the adequate ...
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### Application of A.M. -G.M. inequality

Let x, y,z be positive numbers. The least value of $\frac{x(1+y)+y(1+z)+z(1+x)}{(xyz)^{.5}}$ is a) $\frac{9}{2^{.5}}$ b) 6 c) $\frac{1}{6^{.5}}$ d.) None of the above I tried applying the A.M. ...
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### IMO Shortlist 1995 G3 by inversion

The incircle of $\triangle ABC$ is tangent to sides $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Point $X$ is chosen inside $\triangle ABC$ so that the incircle of $\triangle XBC$ ...
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### The number of integral solutions $(x,y)$ of $x^3+3x^2y+3xy^2+2y^3=50653$

This was a wonderful question given to me by professor in my last class test. He asked for the solution with the least number of steps. Find the number of integral solutions $(x,y)$ of the ...
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### Find real parametar $a,b,c$ such that function $f$ become convex function $f(x) = \begin{cases}ax^2+bx+c,& x<0\\1 ,& x \ge 0\end{cases}$

Find real parametar $a,b,c$ such that function $f$ become convex function $$f(x) = \begin{cases}ax^2+bx+c,& x<0\\1 ,& x \ge 0\end{cases}$$ My work: If $f(x)$ is convex function that means ...
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### Find the middle number in the $29$th row in the Pascal's Triangle

This question is taken from the Singapore Mathematical Olmpiad training notes for Primary school. Find the middle number in the $29$th row of the Pascal's triangle. For example, the middle number ...
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### $\alpha$ exists so that for any points $x_n$ there is a point at average distance $\alpha$ from the $x_n$.

Let $X$ be a connected and compact metric space. Prove a real number $\alpha$ exists so that for every finite set of points $x_1,x_2,\dots, x_n\in X$ (not necessarily distinct) there exists $x\in X$ ...
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### If three cevians are concurrent at a point and form triangles of equal area, the point is the centroid

Let D,E,F be points on side BC,CA,AB of triangle ABC. The three cevians are concurrent at a point G. The areas of triangles BGD, CGE and AGF are equal. Prove that G is the centroid of ABC I have ...
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### Isogonal Conjugate of point outside of triangle

I was wondering about reflections of lines over the external bisectors instead of external bisectors in a triangle. Here is a problem that brought it up: Let $P$ be a given point inside quadrilateral ...
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### Prove that matrix $a_{ij}=|A_i\cap A_j|$ is positive semi-definite

Let $A_i, i=\overline{1,n}$ be finite sets. Define the elements of $n\times n$-matrix $A$ as $$a_{ij}=\big|A_i\cap A_j\big|.$$ The problem is to prove that this matrix is positive semi-definite. I ...
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### Congruence - Number Theory

Prove that $2005^{2005}$ is not the sum of two perfect cubes. I have looked at some mods but none have given me anything useful as of yet. I looked at the usual mods such as $4, 5, 7, 11, 13$ but ...
### Find the coefficient of $x^{19}$ in the expression $(x+1)(x+2)(x+3)\cdots (x+400)$
Find the coefficient of $x^{19}$ in the expression $(x+1)(x+2)(x+3)\cdots (x+400)$ I have no clue how to start. Any kind of help will be appreciated.