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

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6
votes
4answers
620 views

Seemingly invalid step in the proof of $\frac{a^2+b^2}{ab+1}$ is a perfect square?

Recall the famous IMO 1988 question 6: Suppose that $\displaystyle\frac{a^2+b^2}{ab+1}=k\in\mathbb{N}$ for some $a,b\in\mathbb{N}$. Show that $k$ is a perfect square. Solutions can be found: ...
9
votes
6answers
525 views

Is 1100 a valid state for this machine?

A room starts out empty. Every hour, either 2 people enter or 4 people leave. In exactly a year, can there be exactly 1100 people in the room? I think there can be because 1100 is even, but how do I ...
4
votes
3answers
706 views

In how many ways we can choose three numbers the set of first $11$ natural numbers $(1,2,\cdots,11)$ so that their sum is a multiple of $3$?

In how many ways we can choose three numbers from first $11$ natural numbers $(1,2,\cdots,11)$ so that their sum is a multiple of $3$? I tried using "stars and bars", but as this is about selection ...
0
votes
5answers
185 views

A contest problem on finding the real root of the equation.

This equation have two real roots, the task is to find them all! $x^8+x^6+x^4+x^2=340$ Please do not use check and guess or brute force
1
vote
0answers
70 views

What is a general way to do a “reversal” on these kind of problems?

Is there general way to do "reverse", is there a subject exist for any function that is possible to "reverse" and for solving these problem for all types of functions or equations? Example:What is a ...
0
votes
1answer
188 views

Could you use Miquel point theorem to continue this solution on a USAMO problem?

My friend give me some hint on this problem, i only able to find some clue but not able to finish the whole problem, he said i could use Miquel point theorem to finish the rest of it, how could i able ...
2
votes
0answers
435 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 ...
9
votes
1answer
204 views

solution to $ 7^{a}+1 =3^{b}+5^{c} $ for natural $a$,$b$ and $c$

How do I solve $ 7^{a}+1 =3^{b}+5^{c} $ for natural $a$,$b$ and $c$?All I got after some modular arithmetic is that the $a$,$b$ and $c$ are all odd.The problem was posted on Art of Problem ...
1
vote
0answers
107 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 ...
0
votes
1answer
178 views

finding final state of numbers after certain operations

There are $N$ children sitting along a circle, numbered $1,2,\dots,n$ clockwise. The ith child has a piece of paper with number $a_i$ written on it. They play the following game: In the first round, ...
4
votes
1answer
258 views

If $f(x) = \frac{4^x}{4^x+2},$ find the value of $\sum \limits_{i=1}^{1999} f\left(\frac{i}{1999}\right) $

If $$f(x) = \frac{4^x}{4^x+2} $$ then find the value of $$f\left(\frac{1}{1999}\right) + f\left(\frac{2}{1999}\right) + f\left(\frac{3}{1999}\right) +\cdots+f\left(\frac{1999}{1999}\right).$$ ...
2
votes
1answer
141 views

Explicit formula for sequence with parity-based recursion

How do we find an explicit formula for the sequence $(a_i)_{i=1}^\infty$ in terms of $a_1=C$ if $$a_{i+1}=\begin{cases} a_i-13 & i \text{ even}, \\ 2a_i & i \text{ odd}.\end{cases}\quad i\ge2 ...
4
votes
1answer
509 views

Card Shuffling [SPOJ]

The original question is posted on SPOJ, and included below: Here is an algorithm for shuffling N cards: 1) The cards are divided into K equal piles, where K is a factor of N. 2) The ...
11
votes
2answers
277 views

Prove that $n^{2003}+n+1$ is composite for every $n\in \mathbb{N} \backslash\{1\}$

Prove that $n^{2003}+n+1$ is composite for every $n\in \mathbb{N} \backslash\{1\}$. I tried with expanding $n^{2003}+1$, but I got nothing pretty not useful. I also couldn't get any improvement, let ...
1
vote
1answer
466 views

Tough Geometry Problem--Regular Polygon inside Circle

$ABCDEFG$ is a regular heptagon inscribed in a unit circle centered at $O$. $\ell$ is the line tangent to the circumcircle of $ABCDEFG$ at $A$, and $P$ is a point on $\ell$ such that triangle $AOP$ is ...
12
votes
2answers
300 views

Find all $n$ such that each number containing $n$ $1$'s and one $3$ is prime

This question is O205 from the Mathematical Reflections. I do not own any copyrights to this question. Find all $n$ such that each number containing $n$ $1$'s and one $3$ is prime. For example, ...
2
votes
1answer
129 views

What is a short way to deal with this cubic polynomial problem?

i tried to denote the roots to be $a,(a+1),b$ in this problem and i set up a bunch of equations but they are too complicated to solve. What is a way to solve this problem and check it within 3-4 ...
2
votes
2answers
557 views

Euclidean Geometry Intersection of Circles

Two circles intersect in the Cartesian Coordinate system at points $A$ and $B$. Point $A$ lies on the line $y=3$. Point $B$ lies on the line $y=12$. These two circles are also tangent to the x-axis at ...
3
votes
0answers
241 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 ...
2
votes
1answer
45 views

Probability/ Counting question about Error Rates

This is probably a simple question to most of you but I wasn't seeing a clear solution. I was programming the other day and in one part there is a batch insert into a database of about 30 items at a ...
4
votes
1answer
358 views

How to find all rational numbers satisfy this equation?

Find all rational number $a,b,c$ satisfy: $$a+b+c=abc$$ I try to change this in different forms like $(ab-1)c = a+b$, $(ac-1)b = a+c$, $(cb-1)a = b+c$ etc but it won't help...
0
votes
2answers
380 views

Solving $\arctan(a) + \arctan(b) + \arctan(c) = \pi$ for $0 < a < b < c < 10$

This is a trigonometry math contest problem. Which ordered triple of numbers $(a,b,c)$ with $0 < a < b < c < 10$ satisfies the equation $$\arctan(a) + \arctan(b) + \arctan(c) = ...
1
vote
4answers
725 views

Arranging Couples in a Row

Three couples are sitting in a row. Compute the number of arrangements in which no person is sitting next to his or her partner. Answer is 240. From wikipedia, This problem is called the menage ...
2
votes
1answer
199 views

Inequality, Vojtěch Jarník Competition 2006

This is the problem from Vojtěch Jarník Competition 2006. Given real numbers $0=x_1,x_2<\dots<x_{2n}<x_{2n+1}=1$ such that $x_{i+1}-x_{i}\leq h$ for $1\leq i \leq 2n$, show that ...
6
votes
0answers
142 views

“Will we ever get a palindrome” 1996 Problem. [duplicate]

Possible Duplicate: Is it possible for the number created by ordering $1$ to $n$ where $n \geq 1$ be a palindrome? I've been thinking about this problem for a while now, and have come up ...
5
votes
3answers
305 views

Find remainder when dividing $9^{{10}^{{11}^{12}}}-5^{9^{10^{11}}}$ by $13$.

Find remainder when dividing $$9^{{10}^{{11}^{12}}}-5^{9^{10^{11}}} \hspace{1cm} \text{by} \hspace{1.2cm} 13.$$ I tried transforming these who numbers separately to form $13k+n$ but failed.
3
votes
3answers
281 views

Prove $a^ab^bc^c\ge (abc)^{\frac{a+b+c}3}$ for positive numbers.

Prove taht the following inequality holds $$a^ab^bc^c\ge (abc)^{\frac{a+b+c}3}$$ if $a,b,c$ are positive. I'm not sure how to handle these kinds of powers. Are there any "famous" but not so ...
10
votes
1answer
264 views

Prove that there can't be 985 divisors of $123456…19841985$

Numbers from 1 to 1985 are written one after another so they form a new number, $n=123456\ldots19841985$. Prove that there can't be 985 divisors of $n$. This should be solved on paper, without ...
5
votes
2answers
300 views

Prove this trigonometric identity in quadrilateral

If $\alpha,\beta,\gamma,\delta$ are angles in quadrilateral different from $90^\circ$, prove the following: $$ ...
23
votes
3answers
551 views

Find the number of pairs of positive integers $(a, b)$ such that $a!+b! = a^b$

How many pairs of positive integers $(a, b)$ such that $a!+b! = a^b$? A straight forward brute-force reveals that $(2,2)$ and $(2,3)$ are solutions and this seems to be the only possible solutions, I ...
3
votes
1answer
266 views

Contest problem on domain and range of square root function

I have no clue how to do this problem: Let $f(x)=\sqrt{ax^2+bx}$. For how many real values $a$ is there at least one positive real value of $b$ for which the domain of $f$ and the range of $f$ are ...
9
votes
1answer
777 views

Olympiad Style Inequality

I don't quite remember where this problem is from. I came across is sometime last summer, when I was in an olympiad-problem mood and I decided to improve my inequality skills. Suppose $a,b,c > 0$. ...
3
votes
1answer
676 views

Two theorems about an inscribed quadrilateral and the radius of the circle containing its vertices

I think those two theorem are two of the most complicated formulas I have ever seen; please prove it because I am not able to find proofs on the internet: It is known that if the sides of an ...
8
votes
3answers
361 views

Motivation for solution to constructing a set of 1983 distinct integers such that no three are consecutive terms of an arithmetic progression

Problem: Is it possible to choose $1983$ distinct positive integers, all less than or equal to $100,000$, no three of which are consecutive terms of an arithmetic progression? (Source: IMO 1983 Q5) ...
8
votes
5answers
771 views

How many rationals of the form $\large \frac{2^n+1}{n^2}$ are integers?

This was Problem 3 (first day) of the 1990 IMO. A full solution can be found here. How many rationals of the form $\large \frac{2^n+1}{n^2},$ $(n \in \mathbb{N} )$ are integers? The possible ...
5
votes
2answers
1k views

Putnam problem of the day, roots of a polynomial

I've been lately having fun with some Putnam problems (http://www.math.harvard.edu/putnam/) and I would like to see how todays problem can be solved and for somebody more experienced to check my ...
13
votes
1answer
303 views

How many $N$ of the form $2^n$ are there such that no digit is a power of $2$?

How many $N$ of the form $2^n,\text{ with } n \in \mathbb{N}$ are there such that no digit is a power of $2$? For this one the answer given is the $2^{16}$, but how could we prove that that this ...
3
votes
3answers
536 views

Graph theory resource for mathematical Olympiads

I would like to learn a bit of Graph theory for mathematical Olympiads.Can anyone please point out a resource from where I can learn it? Here's my background: I have limited knowledge of linear ...
1
vote
1answer
132 views

Five tangent circles inscribed in the same angle

Given: Five circles have been inscribed in an angle (their centers are contained in the angle bisector). Adjacent circles are tangent. Express the radius of the middle circle in terms of the radii of ...
1
vote
1answer
279 views

What is shortcut to this contest algebra problem about polynomial?

The polynomial $P(x)=x^4 + ax^3 + bx^2 +cx + d$ has the property that $p(k)=11k$ for $k=1,2,3,4$. Compute $c$. The answer is $-39$.
1
vote
1answer
230 views

A perpetual calendar cubes spinoff problem

Perpetual calendar cubes keep track of the date all year around. They must be turned (or even transposed) once a day. The following is a spinoff problem I'm having trouble with. Any hints are much ...
14
votes
2answers
1k views

Asking 2011 Putnam B6

I wish to ask today's Putnam problem B6: Suppose $p$ is an odd prime. Prove that for $n\in \{0,1,2...p-1\}$, at least $\frac{p+1}{2}$ number of $\sum^{p-1}_{k=0} k! n^{k}$ is not divisble by $p$. ...
10
votes
1answer
484 views

Four kissing circles

How can one go about solving the following problem? Inscribe a circle in an arbitrary triangle. Call it's radius $r_1$. Inscribe three more circles so that each one is tangent to two sides of ...
0
votes
1answer
227 views

Four points and the distances between all but one pair are given (check my example?)

Given: four points in the Euclidean plane and the distances to all but one pair of them. Find the remaining distance. My solution attempts to be exhaustive--I consider each case individually: four ...
1
vote
2answers
125 views

Is there an easy way to determine when this fractional expression is an integer?

For $x,y\in \mathbb{Z}^+,$ when is the following expression an integer? $$z=\frac{(1-x)-(1+x)y}{(1+x)+(1-x)y}$$ The associated Diophantine equation is symmetric in $x, y, z$, but I couldn't do ...
1
vote
1answer
169 views

Finding positive real numbers $x$,$y$ and $z$ IMO Shortlist 1995 A4

How can we find all of the positive real numbers like $x$,$y$,$z$, such that : 1.) $x + y + z = a + b + c$(here $a$,$b$ and $c>0)$ and 2.) $4xyz = a^2x + b^2y + c^2z + abc$ ?(Both the conditions ...
8
votes
3answers
566 views

Help remembering a Putnam Problem

I recall that there was a Putnam problem which went something like this: Find all real functions satisfying $$f(s^2+f(t)) = t+f(s)^2$$ for all $s,t \in \mathbb{R}$. There was a cool trick to ...
2
votes
1answer
257 views

The number of pairs of disjoint chords in a circle

$n$ points are given on a circle. I'm looking for a formula for the number of pairs of disjoint chords connecting them. What I have so far: I count the number of pairs of interior intersecting chords ...
4
votes
2answers
365 views

Counting ordered triples of non-negative integers not greater than 100

Can we find the number of ordered triples $(x,y,z)$ of non-negative integers satisfying (i) $x \leq y \leq z$ (ii) $x + y + z \leq 100$? Source:Regional Mathematics Olympiad India (2003) Thank you.I ...
2
votes
1answer
156 views

Question about infinite descent method in this proof

How do we conclude that:"We have therefore constructed another pair $(a_1, b_1)$ in $S(c, k)$ with $a_1 + b_1 < a + b$. However, $S(c, k)$ is contain in $Z^+ × Z^+$, so using the argument of ...