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

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14
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342 views

Determining information in minimum trials (combinatorics problem)

A student has to pass a exam, with $k2^{k-1}$ questions to be answered by yes or no, on a subject he knows nothing about. The student is allowed to pass mock exams who have the same questions as the ...
10
votes
0answers
122 views

Ratio of product from one point and minimum distance

Given points $A_0,A_1,\ldots,A_n$ in the plane, let $m$ denote the minimum distance among any two points. What is the minimum value of $$\dfrac{|A_0A_1|\cdot|A_0A_2|\cdot\ldots\cdot|A_0A_n|}{m^n}?$$ ...
9
votes
0answers
171 views

solve in positive integers sum of squares of sines equation

Find all positive integer triples $(l,m,n)$ such that $\sin^2\frac{\pi}{n}+\sin^2\frac{\pi}{m}=\sin^2\frac{\pi}{l}$. I have found the solutions $(m,m,1)$ for any $m\in\mathbb{Z}^+$, and also ...
9
votes
0answers
233 views

when $F_n^2+F_m^2$ is a square for fibonacci numbers

This is a curiosity question I'm trying to solve a Diophantine equation and I need some results about fibonnacci numbers, I encountered this problem: For which couple of integers $(n,m)$ the ...
7
votes
0answers
81 views

Finding a separating family of subsets of $[n]$ of size $n+1$.

I have this friend who always tells me problems I can't solve. Here is the latest one. We are given a family $\mathcal F$ of at least $2^{n-1}+1 $ subsets $[n]$. We must prove that we can ...
7
votes
0answers
75 views

Integers neither as sum nor difference of perfect powers

Are there infinitely many positive integers $n$ for which there do not exist integers $a,b\geq 1$ and $c,d\geq 2$ such that $n=a^c+b^d$ or $n=a^c-b^d$? [Source: Hungarian competition problem]
6
votes
0answers
67 views

Game, stealing edges in a graph.

I was inventing a problem for a math contest, I was really pleased with it, but then I found a mistake in my solution and have not been able to solve it. It is as follows: Alice and Bob play a game. ...
6
votes
0answers
56 views

What is the probability of a pen touching a bar given that the length of the pen is $10$ cm and the bars are regularly spaced at $15$ cm?

Problem: If a pen of length $10$ cm is thrown out of infinitely large window having vertical bars regularly spaced at $15$ cm, then find the probability that it will touch any of the bars. (Assume ...
6
votes
0answers
100 views

Find the least possible value of $n$ such that there exist $P(x), Q(x) \in \mathbb{Z}[x]$

Find the least possible value of $n, n \geq 2015$ such that there exists polynomial $P(x)$ with degree $n$, integer coefficients, the coefficient of the term $x^n$ is positive and polynomial $Q(x)$ ...
6
votes
0answers
81 views

Given $100$ coplanar points, no $3$ collinear, then at most $70$ percent triangles formed using these points are acute-angled

(IMO-$1970$) Given $100$ coplanar points, no $3$ collinear, prove that at most $70$ percent of the triangles formed using these points are acute-angled. I know that one solution proceeds by ...
5
votes
0answers
60 views

Smallest $n$-digit number $x$ with cyclic permutations multiples of $1989$

Suppose $x=a_1...a_n$, where $a_1...a_n$ are the digits in decimal of $x$ and $x$ is a positive integer. We define $x_1=x$, $x_2=a_na_1...a_{n-1}$, and so on until $x_n=a_2...a_na_1$. Find the ...
5
votes
0answers
106 views

Pairwise sums are equal

The distinct positive integers $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ with $n\ge2$ have the property that the $\binom{n}2$ sums $a_i+a_j$ are the same as the $\binom{n}2$ sums $b_i+b_j$ (in some order). ...
5
votes
0answers
356 views

A product puzzle

This is from a math contest. I have solved it, but I'm posting it on here because I think that it would be a good challange problem for precalculus courses. Also, it's kind of fun. Write the ...
4
votes
0answers
72 views

Sally and I EACH flip a fair coin.

We then each guess what the other person got: I guess what side Sally's coin landed on, and Sally guesses what side my coin landed on. We win as long as at least one of us is correct. I understand ...
4
votes
0answers
37 views

Functional division $\max(f(x+y),f(x-y))\mid \min(xf(y)-yf(x), xy)$

As the title suggests, the problem here is: Find all functions $f:\mathbb{Z}\to\mathbb{N}$ such that, for every $x,y\in\mathbb{Z}$, we have $$\max(f(x+y),f(x-y))\mid \min(xf(y)-yf(x), xy)$$ I ...
4
votes
0answers
64 views

Roots of a polynomial that is composed n times with itself

Let $f(x)=x(4x^2-3)(64x^6-96x^4+36x^2-3)$ and $f^{(n)}=f(f(f(\cdots f(x))\cdots)$ (composed with itself $n$ times). Prove that for all positive integers $n$, $f^{(n)}(x)=x$ has $9^n$ distinct ...
4
votes
0answers
33 views

Iterated circumcenters - proving collinearity and establishing distance ratios

Let $P_0, P_1, P_2$ be three points on the circumference of a circle with radius $1$, where $P_1P_2 = t < 2$. For each $i \ge 3$, define $P_i$ to be the centre of the circumcircle of $\triangle ...
4
votes
0answers
83 views

Mathematical Olympiad Problem

Let $\Bbb{R}$ be the set of real numbers. Determine all functions $f:\Bbb{R}\longrightarrow \Bbb{R}$ satisfying the equation $$f(x+f(x+y))+f(xy) = x + f(x+y)+yf(x)$$ for all real numbers $x$ and $y$.
4
votes
0answers
120 views

An identity satisfying the divisors of a positive integer

I saw a hard competition problem with long and ugly proof in http://solmu.math.helsinki.fi/olympia/valmennus/2013/vt2013_12var.pdf ? The question is from Australian mathematical olympiad 1985. Is ...
4
votes
0answers
86 views

Sequence of non-collinear integer points.

This is a question from a British Olympiad, I've completed the first 3 but this one had me rather stumped. Given two points $P$ and $Q$ with integer coordinates, we say that $P$ sees $Q$ if the ...
4
votes
0answers
62 views

placing chess knights in a numbered chessboard.

Suppose you have a square board where the number on the square in column $i$ and row $j$ is $(j-1)8+i$ you have to place knights on the board so no two knights threaten each other and the sum of the ...
4
votes
0answers
75 views

Set of Metapolynomials is closed under multiplication

We say that a function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integer $m$ and $n$, it can be represented in the form $$f(x_1,\cdots , x_k ...
4
votes
0answers
274 views

The Monster PolyLog Integral $\int_0^\infty \frac{Li_n(-\sigma x)Li_m(-\omega x^2)}{x^3}dx$

I am trying to solve this integral $$ \int_{0}^{\infty} {{\rm Li}_{n}\left(-\sigma x\right){\rm Li}_m\left(-\omega x^{2}\right) \over x^{3}}\,{\rm d}x $$ which is from some high school training ...
4
votes
0answers
87 views

Smallest value that a certain variable can take in a system of equations.

Consider the solutions $(x,y,z,u)$ of the system of equations: $$\begin{cases} x+y=3(z+u)\\ x+z=4(y+u)\\ x+u=5(y+z)\\ \end{cases}$$ where $x,y,z \text{ and } u$ are positive integers. What ...
4
votes
0answers
177 views

Inequality problem with factorials

I am not sure if this kind of "question" is welcome on MSE. Here is an olympiad-like problem that I would like to share with you: Let $a,b,c$ be nonnegative integers. Prove that $$ ...
3
votes
0answers
14 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 ...
3
votes
0answers
38 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 $x\in X$ ...
3
votes
0answers
132 views

(2016 China team selection Test) with a complex inequality

Let $z_{1},z_{2},z_{3}$ be complex numbers, such that: $z_{1}+z_{2}+z_{3}=0,|z_{i}|<1,i=1,2,3$. Find the minimum of the positive $A$ such that: ...
3
votes
0answers
66 views

Putnam 2015 and Ravi Substitution

Let $T$ be the set of all triples $(a,b,c)$ of positive integers for which there exist triangles with side lengths $a,b,c$. Express $$\sum_{(a,b,c)\in T}\frac{2^a}{3^b5^c}$$ as a rational number in ...
3
votes
0answers
51 views

No eight digit perfect fourth powers with distinct digits and not containing 3.

Problem: Prove that there are no perfect fourth powers that have eight distinct digits in their base 10 representation and also don't contain 3 as a digit. My attempt: Since the problem is about ...
3
votes
0answers
71 views

Math competition for school

I am trying to find a math competition where a 10 year old kid can participate. Can someone suggest a competition in USA?
3
votes
0answers
37 views

Three-gap problem, easy version.

Let $N$ be a positive integer and $\theta$ an angle in $(0, 2\pi)$. Consider the map$$f: \{0, 1, 2, \dots, N-1, N\} \to \text{unit circle}, \text{ }f(k) = k\theta \text{ }(\text{mod } 2\pi).$$Show ...
3
votes
0answers
42 views

Proving that the circumcenters are concyclic.

I was completely lost when handed this at a math competition a couple of weeks ago. I drew the diagram and was able to make sense of the question. My diagram also seemed to show that the ...
3
votes
0answers
39 views

Finding the maximum cardinality of a set

Let $B$ be a subset of $A$ such that for any two elements $b_1$ and $b_2$ in $B$, we always have $2b_1\not \equiv{0}\pmod{b_2}$ if $2b_1\ge b_2$. If $A=\{1,2,...,n\}$ then find the maximum possible ...
3
votes
0answers
89 views

Square of hockey stick identity: $\sum_{i=r}^n{i \choose r}^2$

Evaluate $\sum_{i=r}^n{i \choose r}^2$ where $n,r\in \mathbb{N},n>r$. This looks like the hockey stick identity but I can't find a way to evaluate it without a computer. Can someone help me out?
3
votes
0answers
77 views

Prove that $\int^1_0 \frac{dx}{x^x} = 1+ \frac{1}{2^2} + \frac{1}{3^3}…$.

Prove that $\int^1_0 \frac{dx}{x^x} = 1+ \frac{1}{2^2} + \frac{1}{3^3}...$ Darboux theorem (integral) : Whatever the number $x(k,n) \in [a + \frac{k-1}{n}(b-a),a + \frac{k}{n}(b-a),]$, we have ...
3
votes
0answers
90 views

High School Problem on Differential Geometry (finding new curve's equation)

This is a question in a Differential Geometry test in the last year of high school (which I couldn't solve it!): Suppose there are two pieces of curves in the $x-y$ plane: one is $y=ax^2$ cut by ...
3
votes
0answers
37 views

Integrability of $f(x)=\left(1+\frac{1}{x} \right)^{1+\frac{1}{x}}-a-\frac{b}{x}$

This is from an MCQ contest. For all $x\geq 1$ let $$f(x)=\left(1+\dfrac{1}{x} \right)^{1+\dfrac{1}{x}}-a-\dfrac{b}{x}$$ note that ...
3
votes
0answers
63 views

Find all pairs of prime numbers $p , q$ for which: $p^2 | q^3 + 1$ and $q^2 | p^6 − 1$.

Find all pairs of prime numbers $p , q$ for which: $$p^2 \mid q^3 + 1 \tag{A}$$ and $$q^2 \mid p^6 − 1 \tag{B}$$ The question is from the Bulgaria National Olympiad 2014. I'm looking for ...
3
votes
0answers
55 views

Inequality problem involving log function

Given $|f(x+y)-f(x)-f(y)| \leq x+y$ for all $x > y > 0$, prove that real valued function $f$ satisfies the inequality $|\frac{f(x)}{x} - \frac{f(y)}{y}| \leq M(1+\log_2\frac{x}{y})$ where M is ...
3
votes
0answers
31 views

Show that ordered pairs are solutions to an equation if and only if they are consecutive elements of a recursive sequence (contest question)

The following question appeared on the 1998 Canada National Olympiad. I need help proving that the only solutions to the equation are consecutive elements of the recursively-defined sequence. I ...
3
votes
0answers
95 views

Minimize Value of Function with Constrain

Let $x$ and $y$ be real number with $xy\neq-1$ and $$\frac{x^7y+xy^7}{1+x^5y^5}=4$$ What is the minimum value of $x^2+y^2?$ I've been trying to solve it by Lagrange Multiplier but it's getting ...
3
votes
0answers
62 views

Is the set of integers so that $n!+1$ divides $(2012n)!$ finite or infinite?

I am having trouble with this problem. We have to determine whether the set of integers such that $n!+1$ divides $(2012n)!$ is finite or infinite. Basically we have to determine if the prime factors ...
3
votes
0answers
62 views

Sum of zeros of $P(x)$

I asked this question here before too, but vaguely, hopefully, this time will be a better attempt: There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero ...
3
votes
0answers
61 views

Assume that for any pair of vertices $P_i$ and $P_j$ , there exists a vertex $P_k$ of the polygon such that $∠P_i P_k P_j = \pi/3.$

Let $P_1 P_2 \dots P_n$ be a convex polygon in the plane. Assume that for any pair of vertices $P_i$ and $P_j$ , there exists a vertex $P_k$ of the polygon such that $∠P_i P_k P_j = \pi/3.$ Show ...
3
votes
0answers
115 views

A Tricky Quetiones on Creative Algorithm in Graph

an agent is works between n producer and m consumers. i'th producer, generate $s_i$ candy and j'th consumer, consumes $b_j$ candy, in this year. for each candy that sales, agent get 1 dollar payoff. ...
3
votes
0answers
82 views

Number theory - equation

I´m preparing for math contests and found the following problem from this pdf. Find all integers $a, b, c >1$ and all prime numbers $p, q, r$ which satisfy the equation $p^a=q^b+r^c$ ($a, b, c$ ...
3
votes
0answers
115 views

Prove that: $ \left( \sum_{i\neq j}a_{i}b_{j} \right)^2 \geq \left( \sum_{i\neq j}a_{i}a_{j} \right) \left( \sum_{i\neq j}b_{i}b_{j} \right)$

Let $a_{1}, \cdots, a_{n}, b_{1}, \cdots, b_{n}$ be positive real numbers. Prove that: $$ \left( \sum_{i\neq j}a_{i}b_{j} \right)^2 \geq \left( \sum_{i\neq j}a_{i}a_{j} \right) \left( \sum_{i\neq ...
3
votes
0answers
94 views

Maximum number of acute triangles

Given $n$ points on the plane, no three of which are collinear, what is the maximum number of acute triangles formed by them? [Source: Based on Hungarian competition problem]
3
votes
0answers
64 views

Speed dating/networking challenge

I am trying to organise an event with 54 participants. I want them to participate in 9 different activities at stations around a hall. Obviously this will require 9 sessions to allow the participants ...