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

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2
votes
0answers
27 views

A congruence of sum of kth powers of first p-1 numbers [duplicate]

Problem: For $k < p-1$ where $p$ is an odd prime and $k$ is a natural number, prove that $$1^k+2^k+\cdots+(p-1)^k \equiv 0 \mod p.$$ My attempt: It's obvious for odd $k$, as we can pair the ...
3
votes
0answers
13 views

Maximum value of the smallest number of operations to obtain configuration from original configuration

Let $n$ be a positive integer. There are $n(n+1)/2$ marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing $n$ marks. Initially, each ...
2
votes
2answers
47 views

To show that the variables in the system are same in magnitude

I am stuck with this interesting problem, If for non-negative integers $a, b, \text{and} c$, $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ are both integers then ...
1
vote
1answer
33 views

Prove Two Functions are Simultaneously Continuous

Let $f,g,h: \mathbb{R} \rightarrow \mathbb{R}$ so that $f$ is differentiable, $g,h$ monotone and $f'=f+g+h$. Prove that $g$ is continuous in $x_0$ iff $h$ continuous in $x_0$. My ...
5
votes
1answer
52 views

Polynomials bounded by $[-1, 1]$ iff argument is in $[-1, 1]$

Problem: $f(x)$ is a polynomial with complex coefficients, such that $-1 \leq f(x) \leq 1$ iff $-1 \leq x \leq 1$. Find all such $f(x)$. My observations: Now, its easy to see that coefficients are ...
4
votes
3answers
67 views

Right Triangle and Circle Theorem

Let $ABC$ be a triagnle such that $\angle BAC$ is a right angle. Suppose $D$ is a point lying on $BC$ such that $BD=1$, $DC =3$ and $\angle ADB=60^{\circ}$, find the length of $AC$. I was told that ...
0
votes
2answers
95 views

How to solve this equation? $P(x)^2+P(\frac1x)^2= P(x^2)P(\frac1{x^2})$ [closed]

How to solve this equation? Find all polynomials $P$ such that $P(x)^2+P(\frac1x)^2= P(x^2)P(\frac1{x^2})$ Please step by step
-1
votes
0answers
56 views

Find the maximum value of $(ab+bc+ca)$ [closed]

Let $a,b,c$ be three real numbers such that $a+2b+c=4$ then find the maximum value of $(ab+bc+ca)$.
1
vote
1answer
79 views

Sum of two consecutive squares equal square

$N^2 + (N+1)^2 = K^2$, find all solutions for $N<200$ I have done some factoring and also realized that $ K=[n\sqrt{2}]+1$ in eventual solutions, where $[x]$ denotes the greatest integer less ...
1
vote
1answer
59 views

For which $a,b\in \mathbb{N},$ is $\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$ is a rational number. [closed]

I found the following problem on a Olympiad question paper: For which $a,b\in \mathbb{N},$ is $$\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}$$ a rational number. I am unable to solve it. Any help ...
-1
votes
1answer
18 views

Function of product of two uniform random variables [closed]

If X and Y are uniform(0,1) then what is the distribution of $X^kY^m$ for some integers k and m?
0
votes
3answers
99 views

max of $e$ with $a+b+c+d+e=8$ and $a^2+b^2+c^2+d^2+e^2=16$ [closed]

Given that a,b,c,d,e are real number such that: $\begin{cases} a+b+c+d+e=8\\ a^2+b^2+c^2+d^2+e^2=16 \end{cases}$ determine the maximun value of $e$. I started like that : ...
-2
votes
1answer
22 views

Real polynomials from repunits to repunits ( Putnam 2007 A4) [closed]

Find all polynomials $ f$ with real coefficients such that if $ n$ is a repunit, then so is $ f(n).$
-1
votes
1answer
20 views

Arc Length and Area of a Sector

A cake has a circumference of $30 \mathrm{cm}$ and a uniform height of $7\mathrm{cm}$. A slice is to be cut from the cake with two straight cuts meeting at the centre. If the slice is to contain ...
3
votes
1answer
57 views

Singularity at $z=0$ for $1-\cos(z)\sin(\frac{1}{z})$

Any ideas for solving this problem, mentioned in our last exam, is highly appreciated. What is the residue of $f(z)=(1-\cos z)\sin \frac{1}{z}$ at the isolated point $z=0$ ? Our notes say the answer ...
4
votes
3answers
99 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 ...
2
votes
1answer
34 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 ...
2
votes
0answers
61 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 ...
0
votes
1answer
29 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
33 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 ...
12
votes
2answers
659 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) ...
5
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
69 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
22 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
59 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{$n$}}} = ...
4
votes
3answers
109 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 ...
1
vote
1answer
71 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 ...
26
votes
4answers
812 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 question. ...
3
votes
1answer
45 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 ...
4
votes
1answer
75 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 ...
0
votes
0answers
48 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
73 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 ...
0
votes
0answers
18 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 ...
0
votes
2answers
128 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
63 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
45 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
27 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
25 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
99 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
79 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
46 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 ...
0
votes
0answers
67 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: ...
7
votes
1answer
157 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
35 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
78 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
87 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
47 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 ...
9
votes
2answers
139 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 ...