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

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-3
votes
0answers
41 views

Mathematics help please

Suppose that the dollar cost of producing q appliances is $c(q)=1000-0.05q+0.3q^2$, and the demand function is given by $p=20-0.025q$ Q1: compute the marginal cost when the quantity is equal to 1 ...
3
votes
1answer
70 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 ...
4
votes
0answers
26 views
+100

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 ...
3
votes
0answers
25 views

Doing a magic trick with limited memory (from a problem solving course)

I got the following question in a problem solving course: There are four different objects lying on places 1, 2, 3, 4. A magician closes his eyes and someone from the audience comes. He switches ...
-2
votes
1answer
33 views

Challanging problems on [Grade-12]Complex Number [on hold]

recently we are introduced to interesting world of complex number but except for 3-5 problems in the my books,all the problems are just plug-and chug,expression manipulation,etc.. which bores me out ...
2
votes
0answers
28 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 ...
14
votes
1answer
405 views

How many $n$-element subsets $A$ of $\{1,2,3,\cdots,2n\}$ with specified sum are there?

Question: Let $ n$ be an postive integer number.and let $A=\{x_{1},x_{2},\cdots,x_{n}\}$, How many $ n$-element subsets $ A$ of $ \{1,2,\dots,2n\}$ are there,such ...
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 ...
26
votes
4answers
829 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. ...
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
68 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 $(ab+bc+ca)$.
37
votes
1answer
1k views

Do $p,q$ exist such $|p-q|+|a_{p}-a_{q}|=2014$

Let $\{a_1,a_2,\ldots,a_{2016}\}=\{1,2,3,\ldots,2016\}=A$ be such $$\dfrac{a_i-a_j}{i-j}\neq 1,\forall i,j\in A\text{ with } i\neq j.$$ Show that there exists $p,q\in A$ such that ...
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?
5
votes
4answers
152 views

How to simplify the nested radical $\sqrt{1 - \frac{\sqrt{3}}{2}}$ by hand?

I was solving a Mock Mathcounts Contest Mock contest (.pdf) written by a user on the Art of Problem Solving Forums. In problem #24 the only thing I couldn't do by hand was simplify the radical ...
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 ...
0
votes
3answers
100 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 ...
9
votes
7answers
360 views

$211!$ or $106^{211}$:Which is greater?

A BdMO question: Let $a=211!$ and $b=106^{211}$. Show which is greater with proper logic. By matching term by term,it is pretty easy to note that $106!<106^{106}$ $106^{105}<107\cdot ...
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 ...
-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?
2
votes
2answers
124 views

Analytical solution to a nonlinear ODE

How might I analytically solve the following differential equation? $$yy'' = y' + y^3$$ I've tried certain substitutions ($y = ux$ etc.) but none of them work.
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.
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
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 ...
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 ...
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) ...
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 ...
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} \\ ...
11
votes
2answers
388 views

IMC 2015 - Problem 10 - Inequality between polynomials and exponential

This is problem 10 from the International Mathematical Competition for University Students of 2015, from day 2, in Bulgaria. I think it is an interesting problem! Let $n$ be a positive integer, and ...
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{$n$}}} = ...
0
votes
0answers
23 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 ...
0
votes
1answer
685 views

Triangle inscribed inside a circle: prove that $abc = 4 \times area \times R$

The solution to Putnam 2000 A5 uses this formula, for which the following proof is given: (source: https://mks.mff.cuni.cz/kalva/putnam/psoln/psol005.html) Let the sides (of triangle $ABC$) have ...
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 ...
7
votes
1answer
158 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, ...
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 ...
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 ...
0
votes
0answers
49 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 ...
30
votes
7answers
36k views

Expected Number of Coin Tosses to Get Five Consecutive Heads

A fair coin is tossed repeatedly until 5 consecutive heads occurs. What is the expected number of coin tosses?
1
vote
0answers
74 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
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 -> ...
0
votes
0answers
19 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 ...
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 ...
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 ...