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

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4
votes
3answers
106 views

$1000$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 ...
2
votes
1answer
41 views

Tournament of Towns Geometry Problem, Proof by Construction?

I was to prove the following proposition from an old Tournament of Towns problems archive: Problem. A circle $\omega_{1}$ with center $O_{1}$ passes through the center $O_{2}$ of another circle $\...
6
votes
1answer
86 views

Putnam 1985 B-1 Polynomial Problem

Problem: Let $k$ be the smallest positive integer for which there exist distinct integers $m_1, m_2, m_3, m_4, m_5$ such that the polynomial $$p(x)=(x-m_1)(x-m_2)(x-m_3)(x-m_4)(x-m_5)$$ has exactly $k$...
0
votes
1answer
40 views

recursive definition of a palindrome help

Recall that a bit string is a string using the alphabet {0, 1}. A palindrome is a string that is equal to the reversal of itself. Consider the following recursive definition of a palindrome: Basis ...
2
votes
2answers
35 views

Calculating cosine of dihedral angle

Let $O,A,B,C$ be points in space such that $\angle AOB=60^{\circ},\angle BOC=90^{\circ},\angle COA=120^{\circ}$ Let $\theta$ be the acute angle between the planes $AOB$ and $AOC$. Find $\cos\theta.$ ...
12
votes
2answers
681 views

Give an example of a real function so that every rational is a strict local minimum

Give an example of $f : \mathbb R → [0, \infty) $ so that every $r \in \mathbb Q$ is a strict local minimum for $f$. Strict local minimum means there is a vicinity $V$ of $r$ such that $f(y) > ...
6
votes
3answers
1k views

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.
0
votes
1answer
29 views

Calculate the limit of recursively defined sequence

Given a sequence $x_{n}$, $x_0=0, x_1=1, x_{n+1}=\frac{x_n + nx_{n-1}}{n+1}$. Prove, that $x_{n}$ converges and find the limit. $$x_{k+1}=\frac{x_k + kx_{k-1}}{k+1} \\ (k+1)x_{k+1}=x_k + kx_{k-1} \\ (...
-1
votes
1answer
73 views

Find the equation of the ellipse

An ellipse with centre at $(4,3)$ touches $x$-axis at $(0,0)$. If the slope of the major axis of ellipse is 1, then find the equation of the ellipse?
0
votes
0answers
24 views

prove de Rham cohomology of S,the “spherical universe,” is 0-dimensional?

How to prove de Rham cohomology of S,the "spherical universe," is 0-dimensional?(Here, S is a rectangle where if you exit the right, the enter from the top and if you exit the left, the enter from the ...
1
vote
1answer
61 views

Which version of this question is right?

Find digits $x,y,z$ such that the equality $$\sqrt{\smash[b]{\underbrace{\overline{xx\cdots x}}_\text{$2n$}}-\smash[b]{\underbrace{\overline{yy\cdots y}}_\text{$n$}}} = \overline{\underbrace{zz\cdots ...
5
votes
3answers
114 views

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 $r$ ...
1
vote
1answer
84 views

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 ...
36
votes
6answers
2k views

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 ...
3
votes
1answer
47 views

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 ...
4
votes
1answer
82 views

A contest math problem

Let $P(x)$ be a polynomial with integer coefficients of degree $d>0$. If $\alpha $ and $\beta $ are two integers such that $P(\alpha)=1$ and $P(\beta)=-1$, then prove that $|\beta -\alpha | ...
0
votes
0answers
62 views

USAMO 2005, Problem3 (Triangle Geometry)- Is my solution correct?

USAMO 2005, Problem 3: Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on its side $BC$. Construct a point $C_{1}$ in such a way that the convex quadrilateral $APBC_{1}$ is ...
1
vote
0answers
85 views

System of Equations which can be solved by inequalities: $(x^3+y^3)(y^3+z^3)(z^3+x^3)=8$, $\frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}=\frac32$.

S367. Solve in positive real numbers the system of equations: \begin{gather*} (x^3+y^3)(y^3+z^3)(z^3+x^3)=8,\\ \frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}=\frac32. \end{gather*} Proposed by ...
2
votes
1answer
43 views

Looking for a simpler solution to a problem about the divisibility of combinatorial numbers

Here is the problem: For every positive integer r, there exists a natural number $n_r$ such that for every integer $n>n_r$, there is at least one $k$, where $1\leq k \leq n-1$,such that $C_n^k$...
0
votes
2answers
129 views

Formula for smallest multiple of given number, whose every digit is 1

Introduction I've been solving a problem, which says which number is the smallest multiple of $x$ which only has digits with value 1. For example: $minOnes(3) = 3 -> 111$; $minOnes(7) = 6 -> ...
1
vote
2answers
67 views

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 ...
2
votes
1answer
47 views

Looking for a simpler solution about quadratic congruence

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 ...
0
votes
1answer
28 views

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 ...
0
votes
2answers
31 views

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. ...
4
votes
1answer
105 views

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 $\triangle XBC$ ...
2
votes
1answer
84 views

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 ...
2
votes
2answers
47 views

Find the minimum $k$

Find the minimum $k$, which $\exists a,b,c>0$, satisfies $$ \frac{kabc}{a+b+c}\geq (a+b)^2+(a+b+4c)^2$$ My Progress With the help of Mathematica, I found that when $k=100$, we can take $a=1,b=...
0
votes
0answers
74 views

What is the value of $k^2$

For all $f(x)$ and $g(x)$ functions that are differentiable in $\mathbb{R}$, and satisfy the following conditions: Condition A: $$f(1)=1,~f(3)=3.$$ Condition B: $$g(x)=4{f\left(\frac{\pi}{2}\...
7
votes
1answer
162 views

Any math competitions dedicated to calculations by hand (on a college level)?

Most of the people consider hand calculations the thing of the past. However, I recently started thinking about it and there are many interesting ways to do basic arithmetics on large numbers, ...
0
votes
0answers
39 views

Combinatorial Nullstellenatz riddle

I've been unable to solve the last problem here: http://www.mit.edu/~evanchen/handouts/BMC_Combo_Null/BMC_Combo_Null.pdf Let $n ≥ 2$ be even and let $v_1, v_2, . . . , v_k ∈ \{±1\}^n$ be vectors of ...
0
votes
3answers
82 views

Number of real root of the equation $8x^3-6x+1$ lying between -1 and 1 is

Number of real root of the equation $8x^3-6x+1$ lying between -1 and 1 is: I am lagging in solving the inequality portion. Let the roots be $m_1,m_2,m_3$ then $m_1m_2m_3=-\frac{1}{8}$ which means ...
0
votes
1answer
89 views

Logarithm in the exponent

$$(2x)^{\log 2} = (3y)^{\log 3} \\ 3^{\log x} = 2^{\log y}$$ Solve for $x$ and $y$. My intuition for solving such problems is taking the logarithm on both sides but it does not work. I also ...
1
vote
2answers
51 views

Power of a point proof

I found the question on page 13 of this link. Let $P$ be a point inside a circle such that there exist three chords through $P$ of equal length. Prove that $P$ is the center of the circle. I ...
10
votes
2answers
152 views

A nice and hard colouring problem

This question is a generalization of a problem recently appeared in a Italian mathematical competition. $A$ and $B$ are two coprime integers, both greater than $2$. A non-constant colouring $$ ...
0
votes
0answers
30 views

mathematical formula to compute sum of all sub sequences of a number N

We have a number say N and we list down all its sub- sequences and sum them up.SAY for n=123 ,the sum is 177(123+12+23+13+1+2+3). I came across this mathematical formula which computes the sum taking ...
2
votes
1answer
140 views

A problem of olympiad. [closed]

This nice functional equation was proposed in the “VIII Olimpíada Iberoamericana de Matemáticas” held in Mexico (1993). Find all the functions $f:\mathbb N^* \to \mathbb N^*$ such that i) ...
2
votes
1answer
86 views

Olympiad Inequality AM-GM (easy)

Prove that $(1 + x + y)^2 + (1 + y + z)^2 + (1 + z + x)^2 ≤ 3(x + y + z)^2$, with equality if and only if $x = y = z = 1$ ($xyz \ge 1$) ($x,y,z$ positive reals) This simplifies to $x^2 + y^2 + z^2 +...
2
votes
5answers
155 views

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 ...
-1
votes
3answers
74 views

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 ...
5
votes
1answer
87 views

$\alpha$ 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 $x\in X$ ...
1
vote
2answers
73 views

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 ...
0
votes
0answers
15 views

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 ...
0
votes
0answers
44 views

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 ...
0
votes
1answer
80 views

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 ...
4
votes
1answer
160 views

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.
0
votes
0answers
22 views

Prove Concurrency using Radical Axis of Circumcircles

Let the incircle of $\triangle ABC$ touch sides $BC,CA,AB$ at $D,E,F$, respectively. Let $\omega,\omega_1,\omega_2,\omega_3$ be the circumcircles of $\triangle ABCm,\triangle AEF,\triangle BDF,\...
-2
votes
2answers
25 views

Find the total number of Chair in the hall [closed]

In a School hall, $\frac{7}{31}$ of the chairs are arranged in rows of 5, and $\frac{11}{31}$ of the chairs are arranged in rows of 13. The rest of the chairs are stacked up. If there are less than ...
3
votes
1answer
64 views

Comparison of $ ( 1^a + 2^a+ … n^a)^n$ and $n^n(n!)^a $

For a given real number $a>0$ , define $ d_n =( 1^a + 2^a+ ... n^a)^n $ and $ b_n = n^n(n!)^a $ for $ n = 1,2,\ldots$ Then a) $ d_n< b_n $ for $ n> 1$, b) There exists an integer $n>1$...
2
votes
2answers
59 views

linear algebra (matrices) - challenging problem (determination of method/algorithm)

I wonder about the following method/algortithm about square matrices $A_{n \times n}$ $\in$ $M_{n\times n}(\mathbb{K})$, where $\mathbb{K} $ $\in$ {$ \mathbb{R}, \mathbb{C}$ }. Given certain value of ...
0
votes
2answers
65 views

Inequality from IMO 2000 problem 4 question $\Pi_{cyc}\left(a-1+\frac{1}{b}\right)\leq 1$ $abc=1$

I know the problem is repeated but my question is somehow different. I want to know whether my proof is correct because I have troubles with the last part. Since $abc=1$ we can homogenize the ...