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

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11
votes
1answer
257 views

Show that $4mn-m-n$ can never be a square

Let $m$ and $n$ be positive integers. Show that $$4mn-m-n$$ can never be a square. In my attempt I started by assuming for the sake of contradiction that $$4mn-m-n=k^2$$ for some $k \in ...
2
votes
3answers
82 views

Summation of a finite series

Let $$f(n) = \frac{1}{1} + \frac{1}{2} + \frac{1}{3}+ \frac{1}{4}+...+ \frac{1}{2^n-1} \forall \ n \ \epsilon \ \mathbb{N} $$ If it cannot be summed , are there any approximations to the series ?
19
votes
1answer
534 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 ...
6
votes
2answers
189 views

High school contest question

Some work on it reveals the possibility of using gamma function. Is there any easy way to compute it? $$\lim_{n\to\infty}\left(\frac{1}{n!} \int_0^e \log^n x \ dx\right)^n$$
0
votes
2answers
46 views

Prove that nearly all positive integers are equal to $a + b + c$ where $a | b$ and $b | c$, $a \lt b \lt c$

If a positive integer $n$ is equal to $a + b + c$ where $a | b$, $b | c$ and $a \lt b \lt c$, let it be called "faithful". Prove that nearly all numbers are faithful and list the non-faithful ...
8
votes
1answer
138 views

How prove this $ab+bc+cd\le\dfrac{5}{4}$

let $a,b,c,d\in \Bbb R$ and $a,b,c,d>-1,a+b+c+d=0$ prove that $$ab+bc+cd\le\dfrac{5}{4}$$ I have this solution if $b\le c$, then $$ab+bc+cd=a(b-c)-c^2\le ...
19
votes
1answer
326 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 - ...
7
votes
2answers
558 views

Cross section is a regular hexagon.Is it a cube?

One of the cross sections in a rectangular box is a regular hexagon.Prove that the box is a cube I tried to prove that certain lengths were equal by showing that certain triangles are congruent but ...
4
votes
1answer
109 views

Lowest degree polynomial vanishing on the integers mod $n$?

This problem comes from D.J. Newman's A Problem Seminar. Problem: What is the lowest degree monic polynomial $p(x)$ such that the value of $p(x)$ is divisible by $100$ whenever $x$ is an integer? ...
3
votes
1answer
104 views

Let M, K, and L be points on line (AB), (BC), and (CA), respectively. Find the maximum area of smallest the three triangles MAL, KBM, and LCK?

Let M, K, and L be points on line (AB), (BC), and (CA), respectively. Find the maximum area of smallest the three triangles MAL, KBM, and LCK in respect to ABC? I try to guess the answer and then ...
12
votes
5answers
472 views

Prove that there are infinitely many natural numbers $n$, such that $n(n+1)$ can be expressed as sum of two positive squares in two distinct ways.

Prove that there are infinitely many natural numbers $n$, such that $n(n+1)$ can be expressed as sum of two positive squares in two distinct ways.($a^2+b^2$, is same $b^2+a^2$), $n \in \mathbb{N}.$ ...
5
votes
1answer
191 views

Tips on this olympiad problem

My brother recently brought this problem to me, and while I found it quite interesting I cannot figure out how to solve it: For any positive integer $n$, let $f(n)$ be defined by $$ f(n) = ...
13
votes
1answer
178 views

Polynomial $P(a)=b,P(b)=c,P(c)=a$

Let $a,b,c$ be $3$ distinct integers, and let $P$ be a polynomial with integer coefficients.Show that in this case the conditions $$P(a)=b,P(b)=c,P(c)=a$$ cannot be satisfied simultaneously. Any hint ...
5
votes
3answers
157 views

Pigeonhole Principle Problem combo inequality

Prove that for any subset of $\{1,2,3,...,300\}$ with $102$ elements, there exists elements $M$ and $x$ in that subset such that $100<M-x<200$. I think this is a pigeonhole problem, I wanna ...
1
vote
1answer
251 views

No of labeled trees with n nodes such that certain pairs of labels are not adjacent.

What is the number of trees possible with $n$ nodes where the $i$th and $(i+1)$th node are not adjacent to each other for $i \in \left[0,n-1\right)$ and $$i/2 = (i+1)/2.$$ (integer division) (nodes ...
5
votes
1answer
86 views

Functional Equation help

Came across this problem a little while ago but can't seem to get beyond a certain point. Let $f:\mathbb{N} \rightarrow \mathbb{N}$ such that $f(n+1)>f(n)$ and $$f(f(n))=3n$$ for all $n$. ...
3
votes
1answer
143 views

contest problem in geometry

Suppose the inscribed circle of $\triangle A_1A_2A_3$ touches the sides $A_2A_3, A_3A_1, A_1A_2$ at $T_1,T_2,T_3$. From the midpoints $M_1,M_2,M_3$ of $A_2A_3,A_3A_1,A_1A_2$, draw lines perpendicular ...
7
votes
2answers
134 views

For which integers x, y is $2^x + 3^y$ a square of a rational number?

For which integers x, y is $2^x + 3^y$ a square of a rational number? (Of course $(x,y)=(0,1),(3,0)$ work)
4
votes
2answers
709 views

What's a good book for a beginner in high school math competitions?

Also, I want to make it clear: Beginner. I'm getting really frustrated trying to study for math competitions: On the one hand, there are books teaching the high school curriculum, but that's it. I ...
33
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 ...
5
votes
1answer
1k 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 ...
2
votes
1answer
78 views

assume $f:[a,b] \to \mathbb R$ such that $f '(a)=f '(b)$ how prove $\exists t\in(a,b) : f(t)-f(a)=f '(t)(t-a)$?

let $f:[a,b] \to \mathbb R$ such that $f '(a)=f '(b)$ how prove $$\exists t\in(a,b) : f(t)-f(a)=f '(t)(t-a)$$ Thanks in advance
-1
votes
1answer
70 views

Pascal's theorem in geometry

We denote $P= WX \cap YZ$ to mean point $P$ is the intersection of lines $WX$ and $YZ$. The problem is about pascal's theorem: Let $ABCD$ be a cyclic quadrilateral. Let the tangent lines at A and at ...
1
vote
1answer
91 views

A recurrence relation with words, contest type problem

For a positive integer $n$, a $n$-word is a string of $n$ letters, where each letter is an $A$ or $B$. Let $p_n$ be the number of $n$-words not containing four consecutive $A$ and not containing three ...
1
vote
1answer
78 views

Graph theory problem with friends

There are 9 people and for every 3 people, 2 of them are mutual friends. Please show that there exist 4 people out of the 9 who are all mutual friends.
6
votes
2answers
240 views

Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$.

Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$. Some solutions I found are $f\equiv0,f\equiv1$, $f(x)=0$ if $x\neq0$ and $f(x)=1$ if $x=0$.
4
votes
1answer
355 views

Find all integer solutions to $x^2+4=y^3$. [duplicate]

Find all integer solutions to $x^2+4=y^3$. Some obvious solutions are $(x,y)=(\pm2,2)$. Are these the only ones?
6
votes
1answer
179 views

Probability puzzler involving roots of unity

Problem: Let $v$ and $w$ be roots of $z^{1997} = 1$ chosen at random (uniformly and independently). What is the probability that $|v + w| \ge \sqrt{2 + \sqrt 3}$? This problem comes from the 1997 ...
6
votes
1answer
339 views

Inscribing equilateral triangle in rectangle

Problem: What is the area of the largest equilateral triangle that can be inscribed in a rectangle with sides $10$ and $11$? (The problem comes from an old high school math contest. I believe it's ...
6
votes
2answers
121 views

Find all functions $f$ that assign a real number $f(x)$ to every real number $x$ . . .

Find all functions $f$ that assign a real number $f(x)$ to every real number $x$ such that $$(x+y)f(x)+f(y^2)=(x+y)f(y)+f(x^2)$$ I've tried subbing in heaps of values but I keep getting things like ...
1
vote
2answers
66 views

Is $u_n\le(1-a)^n\forall n\in\mathbb{N}$?

Consider the sequence $\{u_n\}$ where $u_0=1,u_1=1-a$ for some $0< a < 1/4$, and $u_{n+2} = u_{n+1}-au_n$. Is $u_n\le(1-a)^n\forall n\in\mathbb{N}$?
5
votes
2answers
184 views

Does there always exist an odd number of elements?

Given a nonzero integer $k$, does there always exist a positive integer $n$ such that there are exactly an odd number of elements $i\in\{0,1,...,n-1\}$ with $\frac{2^n-1}4 < 2^ik \mod{2^n-1} < ...
11
votes
3answers
242 views

How to find the limit of these sequences?

Let $\{a_n\}$ be a real-valued sequence such that $a_1 \geq 0$ and $$a_{n+1}=\ln(a_{n}+1)$$ for all $n\ge1$. How can we find the following limits? $$\lim_{n\to \infty}na_n=?,$$ $$\lim_{n\to ...
7
votes
1answer
198 views

(USAJMO)Find the integer solutions:$ab^5+3=x^3,a^5b+3=y^3$

Find the integer solutions: $$a·b^5+3=x^3,a^5·b+3=y^3$$ This is the first problem of today's USAJMO (has finished),I only find a trival result that $x\equiv y \pmod6$ and $abxy≠0 \pmod 3$. Thanks in ...
6
votes
2answers
106 views

let$ G=\{M_1,M_2,…,M_k\}$ be a finite group if $\sum _{i=0}^k \operatorname{tr} (M_i)=0$ then how prove $\sum _{i=0}^k M_i=0$

Let $G=\{M_1,M_2,...,M_k\}$ be a finite set such that $ M_i\in M_n(\mathbb R)$ and $(G,\;\cdot\:)$ is group with operations of matrix multiplication If $\sum _{i=1}^k \operatorname{tr} (M_i)=0$ ...
5
votes
3answers
272 views

Verifying a proof that if $x,y,z \geq 0$ and $x+y+z = 1$, then $0 \le xy + yz + zx - 2xyz \le \frac{7}{27}$

I was working some recreational problems from a book (The Art and Craft Of Problem Solving, Zeitz) and came across one from the '84 IMO: Suppose that $x, y, z$ are non-negative reals, with $x + y ...
1
vote
1answer
523 views

Finding the possible lengths and widths, given a surface area.

Short Version of Question: Each of $l$, $w$ and $k$ is a positive integer. Determine all possible values for $l$ and $w$ such that $l \ge w$, and $(k + 1)(l + w - 2k) = 133$. Long Version of ...
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 ...
4
votes
0answers
138 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 $$ ...
8
votes
1answer
186 views

Online math contests

I know from my friends who major in CS that there are many reputable online CS constests. Can you give me examples of reputable online math contests ? It would be better if they are for ...
2
votes
3answers
116 views

To prove $\sum_{i=0}^k\binom{n}{3i}\leq \frac{1}{3}(2^n+2)$

If $n\in \mathbb{Z^+}$ and $k$ is the largest integer for which $3k\leq n$, then is it true that $\sum_{i=0}^k\binom{n}{3i}\leq \frac{1}{3}(2^n+2)$? My work: We can break this into two cases: ...
2
votes
1answer
146 views

Let $G$ be a simple group such that $|G|=p^2q r$ ($p$ and $q$ are distinct prime numbers) then how prove $|G|=60$?

Let $G$ be a simple group such that $|G|=p^2q r$ ($p$ and $q$ are distinct prime numbers, $r$ some positive integer) then how to prove $G$ and $A_5$ are isomorphic or ($|G|=60$)? Thanks in advance
11
votes
1answer
154 views

Proving existence of a square-free sequence

I found this problem and a solution sketch in a MathOverflow answer, and I thought it was nice enough to deserve more attention and a properly written solution. Problem: Prove that for each ...
5
votes
1answer
57 views

An infinite fraction

Here's a problem I saw on the AoPS twitter. I thought I might as well post it so that it could be discussed and a solution recorded. What is the value of the following? $$\cfrac{4}{{1 + ...
10
votes
1answer
319 views

Prove that all prime divisors of $7a^2(a+1)-1$ are of the form $7k\pm1$

Question: Let a be a positive integer. Prove that all prime divisors of $7a^2(a+1)-1$ are of the form $7k\pm1$ $a$ and $k \in \mathbb{N}$ .
4
votes
2answers
707 views

how to prove following matrix is invertible? [duplicate]

how to prove A is invertible or $\ detA\neq 0$ $$A=\begin{pmatrix} \frac11 & \frac12 & \frac13 & \cdots & \frac1n \\ \frac12 & \frac13 & \frac14 & \cdots & ...
11
votes
5answers
10k 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?
0
votes
1answer
401 views

What is the most number of regions that 9 lines can cut the plane into [duplicate]

0 lines cuts the plane into at most 1 region. A line cuts the plane into at most 2 regions. 2 lines cut the plane into at most 4 regions. What is the most number of regions that 9 lines can cut ...
0
votes
3answers
253 views

A circle has diameter $AD$ of length $400$

A circle has diameter $AD$ of length $400$. $B$ and $C$ are points on the same arc of AD such that $|AB|=|BC|=60$. What is the length $|CD|$?
5
votes
2answers
91 views

Follow on from previous question: 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 $y$. The answer to this has already been posted, but it doesn't explain why ...