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

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20
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867 views
+50

Prove this inequality with $xyz\le 1$

if $x,y,z>0$ and $\color{red}{xyz\le 1}$, show that $$\color{blue}{\dfrac{x^2-x+1}{x^2+y^2+1}+\dfrac{y^2-y+1}{y^2+z^2+1} +\dfrac{z^2-z+1}{z^2+x^2+1}\ge 1}$$ Sorry,This is not 2015 TST ...
15
votes
0answers
350 views

IMO programs of different nations?

We in Albania have a good team in the IMO, and this year I will probably be part of it. Since Albania does not have a public training programme, I have to consult the training programmes of other ...
15
votes
0answers
843 views

Quadratic Recurrence Relation

The following sequence appeared in IMC 2012 (a math competition): $$a_1 = \frac{1}{2}, \qquad a_{n+1} = \frac{n a_n^2}{1+(n+1)a_n}$$ I am trying to find an explicit formula for the sequence. It ...
12
votes
0answers
91 views
+50

How prove this geometry inequality $R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2$

Zhautykov Olympiad 2015 problem 6 This links discuss olympiad problem none of student solve it,therefore, meaning this problem is so hard. Question: The area of a convex pentagon $ABCDE$ is $S$, ...
11
votes
0answers
233 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 ...
9
votes
0answers
220 views

How many $n$-element subsets $A$ of $\{1,2,3,\cdots,2n\}$ with specified sum are thereļ¼Ÿ

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 ...
8
votes
0answers
216 views

How to prove there exists $n_{1}a_{n_{0}}+n_{2}a_{n_{1}}+\cdots+n_{k}a_{n_{k-1}}<3(a_{1}+a_{2}+\cdots+a_{N})$

Let $a_{1},a_{2},\cdots,a_{N}$ be nonnegative reals, not all $0$. Prove that there exists a sequence $$1=n_{0}<n_{1}<\cdots<n_{k}=N+1$$ of integers such that ...
8
votes
0answers
49 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 $$\dfrac{|A_0A_1|\cdot|A_0A_2|\cdot\ldots\cdot|A_0A_n|}{m^n}?$$ ...
6
votes
0answers
53 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 ...
5
votes
0answers
52 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]
4
votes
0answers
88 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
53 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
64 views

Centroids and Harmonic Means

A triangle $ABC$ with centroid $G$ is such that a line $l$ passing through $G$ intersects $AB$, $BC$, and $AC$ at $H, I, J$, respectively. Show that out of the 3 distances $d(G, I), d(G, H), d(G, J)$, ...
4
votes
0answers
94 views

Integer solutions of $a^3+2a+1=2^b$

What are the solutions in integers of $a^3+2a+1=2^b$? [Source: Serbian competition problem]
4
votes
0answers
45 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
48 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
232 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
70 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
89 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). ...
4
votes
0answers
152 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 $$ ...
4
votes
0answers
330 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 ...
3
votes
0answers
72 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
55 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
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0answers
50 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 ...
3
votes
0answers
63 views

A function on sets which is constant for all permutations

Let $U=\{1, 2,\ldots, 2014\}$. For positive integers $a$, $b$, $c$ we denote by $f(a, b, c)$ the number of ordered 6-tuples of sets $(X_1,X_2,X_3,Y_1,Y_2,Y_3)$ satisfying the following conditions: ...
3
votes
0answers
144 views

Number of collinear subsets in a set

Call a set of points $(x,y)$ good if all the points in the set are collinear (i.e. they all lie on a line).Let S be the set of points $(x,y)$ such that $0\leq x,y \leq n$ ( $ x,y $ are restricted to ...
3
votes
0answers
404 views

An easy question on geometry.

The question says that I have to derive a formula to find the maximum number of enclosed regions formed by $n$ lines. This is how I proceeded: Let $f(n)$ be the maximum number of enclosed regions ...
3
votes
0answers
267 views

Equation with divisors II

This is a link to my first question about this problem . Upd$^{*}$: I've followed Matthew Conroy advice and found "amazing" numbers such as $2^6 \cdot p$, $3^6 \cdot p$. Upd$^{**}$: If $n=p^6 ...
3
votes
0answers
528 views

Is it possible to use inversion to solve this USAMO problem in 2007?

I've no previous experience to solve any problems by inversive geometry but I am willing to see how it works. But I think I know some of the basic definition about inversion in geometry. Also I expect ...
3
votes
0answers
269 views

What is a way to do this combinatorics problem that could generalize to do any of problems similar to this but with more path?

A bug travels from $A$ to $B$ along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never ...
3
votes
0answers
109 views

If $0\leq \cdots \lt s'''\lt s''\lt s'\lt s$ and $s''=((s')^2-k)/s$, $s'''=((s'')^2-k)/s',\ldots$, then $k$ is a perfect square

This is an IMO problem from 1988, problem 6. The book does not provide a proof of this part and it is eluding me. Let $$\cdots \lt s''' \lt s'' \lt s' \lt s$$ all be ...
2
votes
0answers
48 views

$\lim_n \sum_k^{n-1} \tfrac1{1-\rho^k-\rho^{n-k}}$

If $|\rho|<1$, show that, when $n\to\infty$: $$ \frac1{n-1} \sum_{k=1}^{n-1} \frac1{1-\rho^k-\rho^{n-k}} = 1 + \frac1n \frac{2\big(\psi_{\rho}(1)+\log\big(1-\rho)\big)}{\log \rho} + ...
2
votes
0answers
57 views

Graph Algorithm and Cycle Detection

In $O(|V|+|E|)$, we can detect whether a Directed Graph has a cycle or not. ---> True In depth-first seach on DAG, there is no Back Edge. ---> True With known Number of Edges, in $O(|V|)$ and not ...
2
votes
0answers
68 views

How to solve this equation $x^5 +4^y=2013^z$ in positive integers?

I think to solve the equation in positive integers. It appears in a contest and I don't remember where. I obtain that $x$ must be an odd number and further $x=1 \, mod\, 4$. Any hint is appreciated.
2
votes
0answers
67 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 ...
2
votes
0answers
30 views

NIMO 16.8 Expected Value + Probability

Let $p=2^{16}+1$ be a prime. A sequence of $2^{16}$ positive integers $\{a_n\}$ is monotonically bounded if $1\leq a_i\leq i$ for all $1\leq i\leq 2^{16}$. We say that a term $a_k$ in the sequence ...
2
votes
0answers
53 views

Prove that $a,b,c$ are the sides of a triangle

$a,b,c\in\mathbb R_{>0}$ are such that $$\begin{cases}a^2x+b^2y+c^2z=1\\xy+yz+zx=1\end{cases}$$ has a unique solution $(x,y,z)\in\mathbb R^{3}$. Prove that $a,b,c$ are the sides of a ...
2
votes
0answers
74 views

Local informatics Olympiad and Algorithm

I see one of recent local informatics Olympiad question. i have a trouble to solve it. any idea? hint? or solutions? thanks to all creative man. We have two function $P_1, P_2$ and input an array $n$ ...
2
votes
0answers
43 views

Diophantine equations which are easier to solve using $\mathbb{Z}[i]$ compared to $\mathbb{Z}$

I wanted to know applications of arithmetic in $\mathbb{Z}[i]$ that helps in some problems of $\mathbb{Z}$. I found a wonderful set of notes by Keith Conrad. Now I want to read more on a similar ...
2
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0answers
94 views

Find all pair(s) of positive integer $(a,b)$ such that $\frac{a^2}{2ab^2 -b^3+1}$ is also positive integer too?

Another number theory problem. I can find the small value of $b$ such that 0,1,2. But, I cannot find the upper limit of $b$, such that the value of $b$ is limited. How can I find the solution ...
2
votes
0answers
137 views

Problem solution by model theory

Sorry if that's not the right place for asking this, but didn't have anywhere else to go. I was cheking out some math problems in the Mathematical Olympiad site, and I found this one: Let $\mathbb ...
2
votes
0answers
103 views

The name of a game from the 2013 Putnam

Does the following game from the 2013 Putnam have an official name? Based on my searches, it seems to have been created just for the exam. Let $n\geq 1$ be an odd integer. Alice and Bob play the ...
2
votes
0answers
113 views

Short list of the IMO 2003

Let $b$ a integer greater than $5$. For each positive integer $n$, consider the number $$x_n=\underbrace{11\ldots1}_{n-1}\ \underbrace{22\ldots 2}_{n}\ 5$$ written in base $b$. Prove that the ...
2
votes
0answers
72 views

An algorithm for solving linear diophantine equations?

I am entering an interesting team based math contest called the purple comet, and quite a lot of questions on this contest involve Diophantine equations. For this contest, you are given a computer, ...
2
votes
0answers
113 views

quadrilateral geometry question

I recently took the AIME, and the following question was one I was not able to answer: On square $ABCD$, points $E,F,G,$ and $H$ lie on sides $\overline{AB}$,$\overline{BC}$,$\overline{CD}$, and ...
2
votes
0answers
179 views

How Find $3x^3+4y^3=7,4x^4+3y^4=16$

if postive real number $x,y$ such $$\begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases}$$ Find $x+y=?$ My try: $$4x^4-3x^3+3y^4-4y^3=9$$ $$x^3(4x-3)+y^3(3y-4)=9$$
2
votes
0answers
209 views

Korean Math Olympiad 2005 (trapezoid & tangent circles)

In a trapezoid $ABCD$ with $AD||BC$, $O_1$, $O_2$, $O_3$, $O_4$ denote the circles with diameters AB, BC, CD, DA, respectively. Show that there exists a circle with center inside the trapezoid which ...
2
votes
0answers
99 views

Rearranging numbered cards to reverse their order

I have been thinking about this question for a long time, but I can't solve it. Here is the question: We have $9$ cards, with numbers one to nine written on them (in the order $1, 2, \ldots , 9$). ...
2
votes
0answers
54 views

Prove $\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$

Let $f\in C^1([0,1])$. Prove the following: $$\sup_{0\le x\le 1}|f(x)|\le\int_0^1(|f(t)|+|f'(t)|)dt$$ and $$|f(1/2)|\le\int_0^1(|f(t)|+\frac12|f'(t)|)dt$$ Note that ...
2
votes
0answers
135 views

How many ways to fill the $N \times N$ board by nonnegative integers, such that sum of the numbers of each row and each column is $R$?

How many ways to fill the $4 \times 4$ board by nonnegative integers, such that sum of the numbers of each row and each column is $3$? I wrote a brute-force and got $2008$ which seems to be the ...