# Tagged Questions

Many people have their first exposure to real math in the form of a mathematics competition. Math contests show the competitors that math is a subject that makes great use of creativity. For those already initiated, math contests are a source of fantastic problems that often lead to extended ...

3answers
485 views

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

How many rationals of the form $\large \frac{2^n+1}{n^2},$ $(n \in \mathbb{N} )$ are integers? The possible values of $n$ that i am able to find is $n=1$ and $n=3$, so there are two solutions ...
2answers
407 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 ...
6answers
1k views

### Contest problem about convergent series

The following is probably a math contest problem. I have been unable to locate the original source. Suppose that $\{a_i\}$ is a set of positive real numbers and the series \sum_{n = 1}^\infty ...
1answer
270 views

### Prove that $\sum_{k=1}^n \frac{2k+1}{a_1+a_2+…+a_k}<4\sum_{k=1}^n\frac1{a_k}.$

Prove that for $a_k>0,k=1,2,\dots,n$, $$\sum_{k=1}^n \frac{2k+1}{a_1+a_2+\ldots+a_k}<4\sum_{k=1}^n\frac1{a_k}\;.$$
4answers
519 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: ...
3answers
2k views

### The easy(?) part of IMO 2011 Problem 3

Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies $$f(x + y) \leq yf(x) + f(f(x))$$ for all real numbers $x$ and $y$. How can I prove that ...
4answers
503 views

### Irrational painting device

Part a) of the following problem appeared in one of the Putnam Exams (sorry, don't know which year exactly). If you want to solve Part a) don't read Part b). You have a painting device, which given ...
1answer
215 views

### Mathematical problem with square numbers in the decimal system

Moderator Note: this is a question from the Federal Mathematics Competition 2013. Good morning, here's another (pretty difficult) mathematical problem... The task may sound a little strange (I'm ...
1answer
276 views

### Is it possible for the number created by ordering $1$ to $n$ where $n > 1$ be a palindrome?

Is it possible for the number created by the consecutive numbers $1$ to $n$ where $n > 1$ be a palindrome eg. $1234567\ldots n$? I believe this is a contest problem, but how would one solve ...
1answer
166 views

### Smallest possible value on Fibonacci Function

Moderator Note: this is an open problem on brilliant.org Suppose $f$ is a polynomial with integer coefficients, such that for all non-negative integers $n$ the $n$-th Fibonacci number $u_n$ ...
1answer
195 views

### Applying a Function to Square Matrices

Moderator Note: This question is from a contest which ended 1 Dec 2012. Consider a polynomial $f$ with complex coefﬁcients. Call such $f$ broken if we can find a square matrix $M$ such that \$M ...
3answers
228 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?
1answer
86 views

### Uniformly distributed probability problem

May you have an idea for the following exercise I found from some olympiad. Each day you have to bring home one full can of water. To do so you go to the next well and make the can full. On the way ...
1answer
480 views

### Pennies on a checkerboard.

Here is a question on pennies on checkerboard. It isnt a homework question. I saw it in a book. ...
1answer
257 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...
1answer
170 views

### Game Theory Matching a Deck of Cards

Moderator Note: This question is from a contest which ended 1 Dec 2012. Suppose we have a deck of cards labeled from $1$ to $52$. Let them be shuffled in a random configuration, then made ...
1answer
215 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 $-39$.
1answer
530 views

### Does there exist a sequence $\{a_n\}_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}$ such that $\sum_{n=1}^{\infty}a_n$ converges?

Does there exist a sequence $\{a_n\}_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}$ such that $\sum_{n=1}^{\infty}a_n$ converges? Does there exist a sequence with the same property but with each term ...
4answers
584 views

### Solving for the implicit function $f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$ and $f(1)=1$

How can I find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(1)=1$ and $$f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$$ for all real numbers $x$ and $y$ with $y\neq0$? PS. This is ...
6answers
689 views

### Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$

Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$
3answers
320 views

### Finding all integer solutions of $5^x+7^y=2^z$

Find all integers $x,y,z$ such that $5^x+7^y=2^z$. This one comes from an online contest that I arranged some years ago, and I can assure that a completely elementary solution exists.
2answers
1k views

### Olympiad calculus problem

This problem is from a qualifying round in a Colombian math Olympiad, I thought some time about it but didn't make any progress. It is as follows. Given a continuous function \$f : [0,1] \to ...
2answers
504 views

### Find a way from 2011 to 2 in four steps using a special movement

USAMTS 6/2/22 states: The roving rational robot rolls along the rational number line. On each turn, if the robot is at $\frac{p}{q}$, he selects a positive integer $n$ and rolls to ...
5answers
766 views

### Probability of random integer's digits summing to 12

What is the probability that a random integer between 1 and 9999 will have digits that sum to 12? As a user suggested, I could make a spreadsheet and count them, but is there a quicker way to do ...
2answers
471 views

### Translations of Kolmogorov Student Olympiads in Probability Theory

I am deeply interested in Kolmogorov's probability contest whose tests could be found in English for the five first years but there is no English translation to its problems from round 6 onward. I ...
1answer
448 views

### Contest problems with connections to deeper mathematics

We all know that problems from for example the IMO and the Putnam competition can sometimes have lovely connections to "deeper parts of mathematics". I would want to see such problems here which you ...
2answers
477 views

### How do people come up with difficult math Olympiad questions?

The problems that appear in difficult math competitions such as the IMO or the Putnam exam are usually very difficult and require some ingenuity to solve. They also usually don't look like they can be ...
3answers
682 views

### Olympiad Inequality Problem

Consider three positive reals $x,y,z$ such that $xyz=1$. How would one go about proving: $$\frac{x^5y^5}{x^2+y^2}+\frac{y^5z^5}{y^2+z^2}+\frac{x^5z^5}{x^2+z^2}\ge \frac{3}{2}$$ I really dont know ...
1answer
220 views

### Maximizing the volume of a rectangular prism

A rectangular prism has a surface area of $300$ square inches. What whole number dimensions give the prism the greatest volume? This is a math olympiad problem. It involves the volume and surface ...
2answers
476 views

### Proving there are an infinite number of pairs of positive integers $(m,n)$ such that $\frac{m+1}{n}+\frac{n+1}{m}$ is a positive integer

The question is: Show that there are an infinite number of pairs $(m,n): m, n \in \mathbb{Z}^{+}$, such that: $$\frac{m+1}{n}+\frac{n+1}{m} \in \mathbb{Z}^{+}$$ I started off approaching this ...
3answers
145 views

### Is there a non-constant function $f:\mathbb{R}^2 \to \mathbb{Z}/2\mathbb{Z}$ that sums to 0 on corners of squares?

A problem in the 2009 Putnam asks about functions $f:\mathbb{R}^2 \to \mathbb{R}$ such that whenever $A,B,C,D$ are corners of some square we have $f(A)+f(B)+f(C)+f(D)=0$. Without spoiling the problem ...
3answers
379 views

### Expected value uniform decreasing function

Moderator Note: This is part of a current contest. We are given a function $f(n,k)$ as for(i=0;i < k;i++) n = rand(n); return n; rand is defined as a ...
3answers
164 views

### Evaluate $\lim_{x\to\infty}\left(1+\frac{\ln x}{f(x)}\right)^{\displaystyle\frac{f(x)}{x}}$

Let's consider the function $f:\mathbb{R}\rightarrow(0,\infty)$, with $f(x)\cdot \ln f(x)=e^x$, $\forall x \in \mathbb{R}$. Then compute \lim_{x\to\infty}\left(1+\frac{\ln ...
1answer
183 views

### (Olympiad) Minimum number of pairs of friends.

I gave up, my approaches didn't work (induction, pigeon-hole, parity; though obviously there's a good chance I didn't use them cleverly): In a group of 12 people, every pair of them has a common ...
2answers
125 views

### Functional Equation: a little tricky

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f[f(x)^2+f(y)]=xf(x)+y$ for all real numbers $x$ and $y$. Clearly $f(x)=x$ is a solution, check by substitution. I'm at a loss as ...
1answer
228 views

### Proving or disproving $f(n)-f(n-1)\le n, \forall n \gt 1$, for a recursive function with floors.

The Olympiad-style question I was given was as follows: A function $f:\mathbb{N}\to\mathbb{N}$ is defined by $f(1)=1$ and for $n>1$, by: ...
2answers
692 views

### Let a; b; c and d be non-negative numbers such that a+b+c+d = 4. Prove that 4/(abcd) ≥ a/b + b/c + c/d + d/a

How would I approach this using only the AM - GM inequality? Are there any other methods that does not involve the AM-GM inequality?
0answers
210 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 ...
1answer
70 views

### Sequence $a_k=1-\frac{\lambda^2}{4a_{k-1}},\ k=2,3,\ldots,n$.

Consider the sequence $a_1, a_2,\ldots,a_n$ with $a_1=1$ and defined recursively by $$a_k=1-\frac{\lambda^2}{4a_{k-1}},\ k=2,3,\ldots,n.$$ Find $\lambda>1$ such that $a_n=0$. The answer is ...
3answers
416 views

### Finding a diagonal in a trapezoid given the other diagonal and three sides

The figure below is a trapezoid, what is the length of the red line? Thank you very much in advance!
2answers
185 views

### A question with the sequence $e_{n}=\left(1+\frac{1}{n}\right)^{n}$

Prove that $a$) the following sequence is increasing $$e_{n}=\left(1+\frac{1}{n}\right)^{n},\quad n\ge1;$$ $b$) the inequality below holds $$e_{n} \leq3,\quad n\ge1.$$
1answer
234 views

### Multiplication Table with a frame and picture of equal sum

Is there an $n \times n$ multiplication table such that if you form a border of width $k$ ("the frame") and sum its elements, the total will equal the sum of the remaining elements ("the picture")? ...
1answer
115 views

### Estimate variance given a sample of size one (7th Kolmogorov Student Olympiad)

This is problem 10 of the seventh Kolmogorov Student Olympiad in Probability Theory as translated by Jonathan Christensen in this thread. Given a sample of size one from the random variable \$\xi ...
1answer
212 views

### Putnam A-6: 1978: Upper bound on number of unit distances

Let n distinct points in the plane be given.prove that fewer than $2n^\frac{3}{2}$ pairs of them are at unit distance apart
2answers
370 views

### Exploring Properties of Pascal's Triangle $\pmod 2$

Moderator Note: This question is from a contest which ended 1 Dec 2012. Consider Pascal's Triangle taken $\pmod 2$: For simplicity, we will call a finite string of 0's and 1's proper if it ...
5answers
485 views

### Least wasteful use of stamps to achieve a given postage

You have sheets of 42-cent stamps and 29-cent stamps, but you need at least \$3.20 to mail a package. What is the least amount you can make with the 42- and 29-cent stamps that is ...
2answers
117 views

### Resource for Vieta root jumping

I can't seem to find a good resource on Vieta's root jumping, which would explain various scenarios that suggest using the technique. Does anyone have a suggestion for a reference? P.S. Not certain ...
3answers
216 views

### prove that $\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$

If $A$ is a $m \times n$ matrix and $B$ a $n \times k$ matrix, prove that $$\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$$ Also show when equality occurs.
1answer
95 views

### A quick question on general mathematics

I have the following question that I am currently unable to satisfactorily answer myself. My question is: Does the inequality \frac{a}{b} + \frac{b}{a} < \frac{f(a)}{f(b)} + ...
2answers
150 views

### Understanding Less Frequent Form of Induction? (Putnam and Beyond)

I won't paste the question here since my problem is not a technical one but a conceptual one. Book is here: (Page 22 of the pdf) I do not understand why it is necessarily to induct $2^{k}$ to show ...
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