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

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86
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4answers
3k views

A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$

A friend of mine sent me a math contest problem that I am not able to solve (he does not know a solution either). So, I thought I might ask you for help. Prove: ...
70
votes
6answers
2k 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 ...
59
votes
6answers
8k views

Studying for the Putnam Exam

This is a question about studying for the Putnam examination (and, secondarily, other high-difficulty proof-based math competitions like the IMO). It is not about the history of the competition, the ...
43
votes
7answers
2k views

Computing $999,999\cdot 222,222 + 333,333\cdot 333,334$ by hand.

I got this question from a last year's olympiad paper. Compute $999,999\cdot 222,222 + 333,333\cdot 333,334$. Is there an approach to this by using pen-and-paper? EDIT Working through on paper ...
41
votes
7answers
9k views

There exists a power of 2 such that the last five digits are all 3's or 6's. Find the last 5 digits of this number

I just took an olympiad and I'm wondering how this problem is solved. Problem: There exists a power of 2 such that the last five digits are all 3's or 6's. Find the last 5 digits of this number. ...
37
votes
1answer
736 views

Integral $\int_0^1\frac{x^9\left(x^4+x^2-x-1-5\ln x\right)}{\left(x^{10}-1\right)\ln x}\mathrm dx$

A friend of mine sent me an integral that she had not been able to crack, and me neither. It comes with a result, but without a proof (I suppose it originated in some math contest). Could you please ...
36
votes
4answers
1k views

How to prove $k!+(2k)!+\cdots+(nk)!$ has a prime divisor greater than $k!$

Question: Let $k$ be a positive integer. Show that there exist $n$ such that $$I=k!+(2k)!+(3k)!+\cdots+(nk)!$$ has a prime divisor $P$ such that $P>k!$. My idea: Let us denote by ...
35
votes
3answers
1k views

A generalization of IMO 1983 problem 6

Note: This question has a bounty that will expire in just a few days. Let $a,b,c$ and $d$ be the lengths of the sides of a quadrilateral. Show that $$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$$ ...
34
votes
2answers
1k views

Find all real numbers $x$ for which $\frac{8^x+27^x}{12^x+18^x}=\frac76$

Find all real numbers $x$ for which $$\frac{8^x+27^x}{12^x+18^x}=\frac76$$ I have tried to fiddle with it as follows: $$2^{3x} \cdot 6 +3^{3x} \cdot 6=12^x \cdot 7+18^x \cdot 7$$ $$ 3 \cdot ...
32
votes
2answers
830 views

Triangles packed into a unit circle

2014 triangles have non-overlapping interiors contained in a unit circle.What is the largest possible value of the sum of their areas? What are some ideas that might help me start this? Note that ...
28
votes
4answers
2k views

Integral Contest

Before you answer this OP, please read all the terms and conditions below. Thank you... Today I hold an unofficial little contest on brilliant.org. Now, I will hold it here on Math S.E. It's just for ...
27
votes
3answers
576 views

Square matrices satisfying certain relations must have dimension divisible by $3$

I saw this tucked away in a MathOverflow comment and am asking this question to preserve (and advertise?) it. It's a nice problem! Problem: Suppose $A$ and $B$ are real $n\times n$ matrices with ...
26
votes
7answers
4k views

Find $f(x)$ such that $f(f(x)) = x^2 - 2$

Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$. Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but ...
26
votes
1answer
763 views

A question about series with a strange property.

Does there exist a sequence $\left(a_n\right)_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}, \forall n=1,2,3,\ldots$ such that the series $\displaystyle{\sum_{n=1}^{\infty}a_n}$ converges? This is the ...
25
votes
8answers
3k views

Reasoning that $ \sin2x=2 \sin x \cos x$

In mathcounts teacher told us to use the formula $ \sin2x=2 \sin x \cos x$. What's the math behind this formula that made it true? Can someone explain?
25
votes
2answers
656 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 ...
25
votes
2answers
548 views

New twist on a Putnam problem

A recent Putnam problem: Let $f$ be a real-valued function on the plane such that for every square $ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that $f$ is identically zero? The ...
23
votes
6answers
2k views

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

Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$
23
votes
4answers
665 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 ...
21
votes
4answers
2k views

Sum of four squares not a prime

Let $ a, b, c, d $ be natural numbers such that $ ab=cd $. Prove that $ a^2+b^2+c^2+d^2 $ is not a prime. I am clueless on this one. I tried contradiction, but didn't get anywhere. Can you help? ...
21
votes
3answers
544 views

Math contest: Find number of roots of $F(x)=\frac{n}{2}$ involving a strange integral.

Edit summary: A good answer appeared. CW full answer added, based on given answers. Removing my ugly-looking attempts, as they still remain in the rev. history. Here's a final-round calculus ...
21
votes
2answers
1k 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 ...
20
votes
3answers
1k 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.
20
votes
1answer
844 views

2013 USAMO problem 5

This is currently unsolved in the AoPS site, the problem says: Given postive integers $m$ and $n$, prove that there is a positive integer $c$ such that the numbers $cm$ and $cn$ have the same ...
20
votes
1answer
540 views

Sum involving binomial coefficient satisfies congruence (A contest question)

Let $p$ be an odd prime, and denote $$f(x)=\sum_{k=0}^{p-1}\binom{2k}{k}^2x^k.$$ Prove that for every $x\in \mathbf Z$,$$(-1)^\frac{p-1}2f(x)\equiv f\left(\frac{1}{16}-x\right)\pmod{p^2}.$$ This is a ...
19
votes
3answers
1k views

Find the sum of all real solutions for $x$ to the equation $(x^2 + 2x + 3)^{(x^2+2x+3)^{(x^2+2x+3)}} = 2012.$

Find the sum of all real solutions for $x$ to the equation $(x^2 + 2x + 3)^{(x^2+2x+3)^{(x^2+2x+3)}} = 2012.$ I just know $x^{x^x}$ is increasing in $x$ and hence the equation has a unique solution, ...
19
votes
2answers
729 views

Which functions satisfy the equation $\,\,f(x)\,f(y)-f(x+y)=\sin x\,\sin y\,$?

Find all continuous functions $f$ which satisfy the functional equation $$ f(x)\,f(y)-f(x+y)=\sin x\,\sin y, $$ for all $x,y\in\mathbb R$. I can prove that $f(n\pi)=\cos\left(n\pi\right)$ for all ...
19
votes
2answers
2k 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 ...
19
votes
1answer
279 views

How to prove $\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$?

How can I prove the following identity? $$\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$$
19
votes
1answer
333 views

To prove that $2^{3n}+2^n +1$ is not a perfect square.

Question: Prove that $2^{3n} + 2^n + 1$ cannot be a perfect square for any natural $n$. I attempted this question and failed in two different ways. 1) I considered a polynomial $p(x) = x^3+ x + 1 - ...
18
votes
2answers
925 views

Tough contest problem

I found this problem in a collection of contest problems of a Russian competition in 1995 and wasn't able to solve it. Solve for real $x$: $$ \cos (\cos (\cos (\cos(x))))=\sin (\sin (\sin (\sin ...
18
votes
2answers
524 views

An upper bound on certain finite trigonometric series given a lower bound

Let $f$ be the function $f(x)=1+a\sin{x}+b\cos x+c\sin{(2x)}+d\cos{(2x)}$, where $a,b,c,d$ are arbitrary real numbers. Prove that if $f(x)>0$ for all $x\in \mathbb R$, then $f(x)<3$ for all ...
18
votes
5answers
1k views

If $f''(x)+f(x)>0$ and $f(x)>0$ $\forall x\in(a,b)$; $f(a)=f(b)=0$; prove that $b-a>\pi$.

Please help me to solve this question: Suppose $f:[a,b] \to \Bbb R$ satisfies: $f''(x)+f(x)>0$ and $f(x)>0$ for all $x\in(a ,b)$; $f(a)=f(b)=0$. Prove that $b-a>\pi$. ...
17
votes
10answers
1k views

Find the integer closest to $\ln(2013)$

I encounter such a problem, in a Maths contest, to find out the closest integer to $\ln(2013)$, without using a calculator. I really get stuck. I tried to turn $\ln(2013)$ into ...
17
votes
1answer
286 views

How to compute the series $\sum\limits_{x=0}^\infty\sum\limits_{y=0}^\infty\sum\limits_{z=0}^\infty\frac{1}{2^x(2^{x+y}+2^{x+z}+2^{z+y})}$

How to compute the series $\displaystyle\sum_{x=0}^\infty\sum_{y=0}^\infty\sum_{z=0}^\infty\frac{1}{2^x(2^{x+y}+2^{x+z}+2^{z+y})}$ ? Thanks in advance.
17
votes
2answers
595 views

Inequality on the side lengths of a triangle: $\left| \frac{a}{b} + \frac{b}{c} + \frac{c}{a} - \frac{a}{c} - \frac{b}{a} - \frac{c}{b} \right| < 1$.

This problem is taken from the Kosovo Mathematical Olympiad for Grade-$ 10 $ students. Let $ a $, $ b $ and $ c $ be the lengths of the edges of a given triangle. How can one prove the following ...
17
votes
3answers
872 views

Examples of math contest problems that can be solved in a 'cheap' way?

What are some examples of math contest problems that can be solved by using a nonrigorous, 'cheap' shortcut? For instance, a problem on the 2011 AMC went: A raft and a motorboat left dock A and ...
17
votes
2answers
884 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 ...
17
votes
2answers
257 views

A game on a graph

Alice and Bob play a game on a complete graph ${G}$ with $2014$ vertices. They take moves in turn with Alice beginning. At each move Alice directs one undirected edge of $G$. At each move Bob chooses ...
16
votes
4answers
1k views

The number $2^{29}$ has exactly $9$ distinct digits. Which digit is missing?

The number $2^{29}$ has exactly $9$ distinct digits. Which digit is missing? I came across this question in a math competition and I am looking for how to solve this question without working it ...
16
votes
5answers
2k 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 ...
16
votes
2answers
2k views

Math Olympiad Prime Number Question

If $p$, $q$ and $r$ are prime numbers such that their product is $19$ times their sum, find $p^2$ + $q^2$ + $r^2$. I came across this question in a Math Olympiad Competition and had no idea how ...
16
votes
1answer
210 views

Closed form for $\sum_{n=0}^\infty\frac{\operatorname{Li}_{1/2}\left(-2^{-2^{-n}}\right)}{\sqrt{2^n}}$

Let $$S=\sum_{n=0}^\infty\frac{\operatorname{Li}_{1/2}\left(-2^{-2^{-n}}\right)}{\sqrt{2^n}},\tag1$$ where $\operatorname{Li}_a(z)$ is the polylogarithm. For $a=1/2$ it can be represented as ...
16
votes
1answer
183 views

$|3^a-2^b|\neq p$, from a contest

I recently came across an old contest problem: (I did not find the solution anywhere) Find the least prime number which cannot be written in the form $|3^a-2^b|$ where $a$ and $b$ are ...
16
votes
1answer
507 views

How find all positive $a^3=b^2+2000000$

Find all positive integer $a$ and $b$,such $$a^3=b^2+2000000$$ This problem is from china Math competition(2013,7.10) So I think this problm have nice methods,because is from competition. Thank you ...
15
votes
5answers
1k views

Is high school contest math useful after high school?

I've been prepping for a lot of high school math competitions this year, and I was just wondering if all the math I learn would actually mean something in college. There is a chance that all of it ...
15
votes
6answers
13k 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?
15
votes
4answers
516 views

Find all bijections $\,\,f:[0,1]\rightarrow[0,1],\,$ which satisfy $\,\,f\big(2x-f(x)\big)=x$.

A friend of mine gave me the following problem: Find all functions $f:[0,1]\to[0,1]$, which are one-to-one and onto and satisfy the following functional relation: $$ f\big(2x-f(x)\big)=x, \tag{1} $$ ...
15
votes
2answers
789 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 ...
15
votes
2answers
751 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 ...