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

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14
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386 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 ...
11
votes
0answers
265 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 ...
11
votes
0answers
284 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 ...
8
votes
0answers
55 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 $$\dfrac{|A_0A_1|\cdot|A_0A_2|\cdot\ldots\cdot|A_0A_n|}{m^n}?$$ ...
7
votes
0answers
182 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 ...
6
votes
0answers
58 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 ...
6
votes
0answers
64 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
60 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]
5
votes
0answers
335 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
69 views
+50

Cover the grid graph with simple cycles

I have a two dimensional n x m grid graph. And I want to find in how many ways this grid can be covered with simple cycles (it can be a one cycle or it can be many ...
4
votes
0answers
72 views

Long polynomial expansion with 34 roots

This is a very tricky problem, I just need a few hints. I think the $(-x^{17})$ is also there for a specific trick. In the end if it is $ax^{17}$, I see that $a = 17 - 1 + 1 = 17$. Also, another ...
4
votes
0answers
53 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 ...
4
votes
0answers
105 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
63 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
84 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 ...
4
votes
0answers
65 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
49 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
62 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
251 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
72 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
158 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
26 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
35 views

Finding relations of variables

Suppose that \begin{align*} x&=t+t^{-1}+t^2s+t^{-2}s^{-1}+ts^{-1}+t^{-1}s-6\\ y&=t+t^{-2}+ts+s^{-1}-4\\ z&=t^{-1}+t^2+t^{-1}s^{-1}+s-4 \end{align*} Find a polynomial $P(x, y, z)=0$ ...
3
votes
0answers
55 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
45 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
39 views

Picking K counters out of K buckets containing NK counters, N of each different colour, up to N in each

This is a generalisation of a question that recently came up while solving a TopCoder problem. Suppose we have N blue counters, N red counters, N white counters, and so forth, K colours in total. We ...
3
votes
0answers
82 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
56 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
54 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
64 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
163 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
411 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
268 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
546 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
273 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
465 views

Two circumcircles of triangles defined relative to a fixed acute triangle are tangent to each other (IMO 2015)

I'm posting here the question because I want to see a nice synthetic solution (not using complex numbers or inversive geometry) for the 3rd problem from IMO 2015. The problem is as follows: Let ...
2
votes
0answers
87 views

Expected Power Product of rolling a dice .

A 15 sided dice is rolled 1000 times. Let k1,k2,k3,k4,..k15 denote the number of times 1,2,3...15 appears. How can I compute the following expected value :$$E( (k_1 k_2 k_3 k_4)^5).$$ My attempts:: ...
2
votes
0answers
49 views

Fixed Points of Function from Rationals to Reals

Consider a function $f$ from the positive rationals to the reals such that $f(x)f(y)\ge f(xy)$ and $f(x+y)\ge f(x)+f(y)$. Further assume this function has a fixed point greater than $1$. Prove this ...
2
votes
0answers
55 views

Proof Verification for Putnam Problem [Alternate Solution] 1997 A4

I have come across an interesting problem from the Putnam 1997 test, question A4: Problem: Let $G$ be a group with identity $e$ and $\phi: G \to G$ a mapping such that $\phi(g_1)\phi(g_2)\phi(g_3) = ...
2
votes
0answers
23 views

Let $p_1, p_2,…,p_n$ be polynomials of $k$ variables $x_1,…,x_k$ and $p_1^2 + \cdots p_n^2=x_1^2 + \cdots + x_k^2$ Prove that $n \geq k$.

Let $p_1, p_2,...,p_n$ be real polynomials of $k$ variables $x_1,...,x_k$ and assume that $$p_1^2 + \cdots p_n^2=x_1^2 + \cdots + x_k^2$$ Prove that $n \geq k$. Out of so many questions that I ...
2
votes
0answers
49 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
58 views

Biggest number divisible by $99$

John has $1012$ stickers on which the numbers $1000,1001,\cdots,2010,2011$ are written.He wants to put them(not necessarily all) in a row so that he gets the biggest number which is divisible by ...
2
votes
0answers
70 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
61 views

Number theory - equation

I´m preparing for math contests and found the following problem from this pdf: http://www.fmf.uni-lj.si/~lavric/Santos%20-%20Number%20Theory%20for%20Mathematical%20Contests.pdf Find all integers $a, ...
2
votes
0answers
73 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
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
59 views

“Massaging” inequalities to prove them (esp. in contest math like the IMO/Putnam)?

What's the contest inequality solving technique where you do something like representing each side as the function of some sequence and replacing the max/min terms of the sequence with their average, ...
2
votes
0answers
54 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 ...