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

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46
votes
6answers
5k 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 ...
7
votes
1answer
174 views

$f:[a,b]\to(a,b)$ be continuous how prove $f(c)+f(c+d)+\cdots+f(c+nd)=(n+1)(c+\frac{nd}{2})$

let $f:[a,b]\to(a,b)$ be continuous how prove $\forall n\in\mathbb N$ $\exists d\gt0$ ,$\exists c\in(a,b) $ such that $$f(c)+f(c+d)+\cdots+f(c+nd)=(n+1)\left(c+\frac{nd}{2}\right)$$thanks in advance ...
11
votes
1answer
271 views

Prove that $\frac{\pi}{4}\le\sum_{n=1}^{\infty} \arcsin\left(\frac{\sqrt{n+1}-\sqrt{n}}{n+1}\right)$

Prove that $$\frac{\pi}{4}\le\sum_{n=1}^{\infty} \arcsin\left(\frac{\sqrt{n+1}-\sqrt{n}}{n+1}\right)$$ EDIT: inspired by Michael Hardy's suggestion I got that $$\arcsin ...
3
votes
1answer
43 views

$ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right).$

Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: $ \log_{\frac ...
2
votes
1answer
117 views

$\exists a,b\in \mathbb R^+ $such that $|f(x)|\le a|x|+b$

Assuming $f:\mathbb R\to\mathbb R $ be an uniform continuous function, how to prove $$\exists a,b\in \mathbb R^+~~~~\text{such that}~~~~|f(x)|\le a|x|+b.$$ Thanks in advance!
4
votes
2answers
238 views

Can the distance from the vertices of a square of integer width to an inscribed circle all be integer?

I'm looking for solutions to the following British Mathematical Olympiad question: Suppose that $ABCD$ is a square and that $P$ is a point which is on the circle inscribed in the square. Determine ...
5
votes
1answer
74 views

Given $A$ and $B$, how many positive integers $N$ such that $N\times B$ has at least one divisior $D$ that lies in $N \lt D \le A$?

For two integers $A$ and $B$, how can we find the number of positive integers $N$ such that $N\times B$ has at least one divisior $D$ that lies in $N \lt D \le A$? For example, if $A = 100$ and $B = ...
5
votes
3answers
1k views

A Math Olympiad question regarding Geometry

A little bit of a backstory (you may skip this if you want): My high school math teacher knows that I love math, but he also knows that I usually drift off during my classes, perhaps because it's too ...
4
votes
1answer
202 views

$x^2+y^2=z^2(1+xy)$ prove $z=\min \{x;y;z\}$ (with $x,y,z \in \mathbb{Z^+}$)

$x,y,z \in \mathbb{Z^+}$ such that $x^2+y^2=z^2(1+xy)$. Prove $z=\min \{x;y;z\}$ $$x^2+y^2=z^2(1+xy) \iff xy = \frac{x^2+y^2} {z^2} - 1$$. Assum $z>y \implies xy < x^2/z^2$, we have $xy \in Z ...
1
vote
1answer
118 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)} + ...
3
votes
2answers
283 views

LCM($1, 2, 3, …, n$)?

I want to find LCM($1, 2, 3, \ldots, n$) where $2 \le n \le 10^8$ . LCM = Least Common Multiple I'm trying to find a formula . Please Help .
2
votes
1answer
79 views

Assume $A_1,A_2,…,A_n\in M_{m×m}(F)$ that satisfy the following conditions, how to prove that $A_1A_2…A_n=0$?

Assume $A_1,A_2,...,A_n\in M_{m×m}(F)$ (where $F$ is a field) such that $A_jA_i=A_iA_j$ $A_i^2=0, \;\;\forall 1\leq i \leq n.$ If $m\lt2^n$ then how to prove that $A_1A_2...A_n=0.$ Thanks in ...
2
votes
1answer
318 views

Math Competition Practice

I am studying for the Berkeley Math Tournament (BMT) and I was wondering if anyone had any ideas on what I should study to prepare. This is my first math competition and would like to have as much ...
2
votes
1answer
57 views

All diagonals of the regular 30-gons are drawn interior. How many distinct point in the interior of the 30-gon do two or more diagonal interesect?

All diagonals of the regular 30-gons are drawn interior. How many distinct point in the interior of the 30-gon do two or more diagonal interesect? So how do we generalize a formula for any gons? ...
0
votes
5answers
570 views

how many ways to make n by adding k non-negative integers (proof)?

Problem : How many ways are there to make $n$ by adding $k$ non-negative integers, where order matters. Suppose $n=4$ and $k=3$. There are 15 solutions using $0, 1, 2, 3, 4$: $(0,0,4), (0,1,3), ...
2
votes
1answer
118 views

How to find how many number has even value of sigma function?

I have to find how many integers from $1$ to $n$ $(n\leq10^{12})$ have even value of $\sigma$. $\sigma(n)$ = sum of all divisors of $n$ .
0
votes
2answers
91 views

Suppose that $f : U \mapsto \mathbb{R}$ has continuous first partial derivatives.

Let U be an open subset of $\mathbb{R}^n$ and C a compact subset of U. Suppose that $f : U \mapsto \mathbb{R}$ has continuous first partial derivatives. Prove that f is Lipschitz on C. Thoughts: Let ...
2
votes
5answers
158 views

Another limit from a math contest $\lim_{n\to\infty}\frac{x_n^2y_n}{3x_n^2-2x_ny_n+y_n^2}$

Let $(x_{n})_{n\ge1}$, $(y_{n})_{n\ge1}$ be real number sequences and both converge to $0$. Evaluate $$\lim_{n\to\infty}\frac{x_n^2y_n}{3x_n^2-2x_ny_n+y_n^2}$$
7
votes
1answer
98 views

How to prove that exists distinct $x_1,x_2 \in(a,b)$ such that $f '(x_1)f '(x_2)=1$?

Assume $f:[a,b]\to[a,b]$ be continuous and differentiable on $(a,b)$ and $f(a)=a$, $f(b)=b$. How to prove that exists distinct $x_1,x_2 \in(a,b)$ such that $f '(x_1)f '(x_2)=1$? Thanks in advance.
4
votes
3answers
1k 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 ...
10
votes
2answers
228 views

Let the matrix $A=[a_{ij}]_{n×n}$ be defined by $a_{ij}=\gcd(i,j )$. How prove that $A$ is invertible, and compute $\det(A)$?

Let $A=[a_{ij}]_{n×n}$ be the matrix defined by letting $a_{ij}$ be the rational number such that $$a_{ij}=\gcd(i,j ).$$ How prove that $A$ is invertible, and compute $\det(A)$? thanks in advance
26
votes
3answers
551 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 ...
1
vote
0answers
520 views

IMO-2012 Problem 6 (Dušan Djukić, Serbia)

IMO-2012 Problem 6 (Dušan Djukić, Serbia)   Find all positive integers ( n ) for which there exist non-negative integers ${a_1}$, $a_2 $, $ \dots $, $ a_n $ such that \[ ...
2
votes
2answers
203 views

How to factor 30 digit number

I need to find the prime factorization of a number having 30 digits. I used the Pollard rho method but unfortunately it is not sufficient enough. It needs a more advanced prime factorization ...
2
votes
1answer
344 views

Finding the binary representation of the $n$th Fibonacci term

Objective: To find the binary representation ( or no. of 1's in binary representation) of nth term in Fibonacci sequence where n is of the order 10^6. My current approach: Find nth term (in decimal) ...
3
votes
4answers
185 views

how prove $\sum_{n=1}^\infty\frac{a_n}{b_n+a_n} $is convergent?

Let$a_n,b_n\in\mathbb R$ and $(a_n+b_n)b_n\neq 0\quad \forall n\in \mathbb{N}$. The series $\sum_{n=1}^\infty\frac{a_n}{b_n} $ and $\sum_{n=1}^\infty(\frac{a_n}{b_n})^2 $ are convergent. How to prove ...
0
votes
1answer
52 views

how prove $\exists a,b$ that satisfied in following conditions $0 <a\leq b\leq 1, b-a=\frac12, \text{ and }f(a)=f(b)$ [duplicate]

Possible Duplicate: Universal Chord Theorem let $f:[0,1]\mapsto\mathbb R$ be continuous and $f(0)=f(1)$how prove $\exists a,b$ that satisfied in following conditions $$1)0<a\leq b\leq ...
-1
votes
2answers
273 views

Algebra and Geometry book [closed]

Hello can you find two free books: 1) Book for Algebra with Theorems, Techniques and Select Problems 2) Book for Geometry with Theorems, Techniques and Select Problems I need urgent Thanks in ...
0
votes
2answers
131 views

Help understanding train problem

A train $150$ $m$ long passes a km stone in $15$ seconds and another train of the same length traveling in opposite direction in $8$ seconds. The speed of the ...
2
votes
1answer
197 views

Contest Math Geometry

I'm currently prepping for some high school math competitions soon, and I was wondering if anyone knows any resources that are out there with an abundance of contest-math-related geometry problems. ...
2
votes
2answers
164 views

let A,B be complex matrics and $2A(B-A)=A+B$ how prove $AB=BA$

let $A,B\in M_n(\mathbb C)$ $\mathbb C$ is complex field such that $$2A(B-A)=A+B$$ how prove $AB=BA$ thanks in advance
1
vote
2answers
147 views

$\limsup\left(\frac{a_1+a_{n+1}}{a_n}\right)^n\ge c$

Let $a_n>0,n\in\mathbb{N}$ be a sequence of positive real numbers. There exists a positive real number $c$ such that $\limsup\left(\frac{a_1+a_{n+1}}{a_n}\right)^n\ge c$ as $n\to\infty$ for all ...
0
votes
1answer
115 views

Convergence of series with modified denominator

Suppose the series with positive terms $\sum_{n=1}^{\infty} a_n$ converges. Let $r_n=\sum_{k=n}^{\infty}a_k$. Prove or disprove that $\sum_{n=1}^{\infty}\frac{a_n}{r_n}$ diverges, and prove or ...
23
votes
2answers
502 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 ...
7
votes
2answers
400 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?
10
votes
2answers
158 views

Does there exist a sequence of real numbers $\{a_n\}$ such that $\sum_na_n^k$ converges for $k=1$ but diverges for every other odd positive integer?

Does there exist a sequence of real numbers $\{a_n\}$ such that $\sum_na_n^k$ converges for $k=1$ but diverges for every other odd positive integer?
0
votes
2answers
196 views

Find square root of non-rational fraction

If we have to compute this without using calculator, is there a quick way to find the answer approximately of the following problem: which one is smaller ? $$ A = ...
11
votes
3answers
409 views

Reference for combinatorial game theory.

What is a good reference material for elementary combinatorial game theory? By combinatorial game theory I mean chiefly the study of zero-sum, deterministic two-player games (perhaps even more ...
0
votes
1answer
125 views

Proving that $ f(1)=\frac{1-\sqrt{5}}{2}$ for this function

Let $f:(0,+\infty)\mapsto R$ be a strictly increasing function such that $\forall x\ge0,$ $$f(x)+\frac{1}{x}\ge0, \qquad f(x)f\left(f(x)+\frac{1}{x}\right)=1.$$ Show that ...
1
vote
1answer
83 views

Show that a given number has two identical digits(Kosovo TST 2011)

Starting with the number $7^{1996}$ we remove its first digit, and then add that digit to the rest of the number. This process continues until the result has ten digits. Show that the resulting number ...
16
votes
1answer
260 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.
2
votes
1answer
236 views

how prove this integral inequality?

How prove that for all continuous and decreasing function $f:[0 ,1]\mapsto(0,+\infty)$ $$\frac{\int_{0}^1x(f(x))^2dx}{\int_{0}^1xf(x)dx}\leq \frac{\int_{0}^1(f(x))^2dx}{\int_{0}^1f(x)dx}$$ thanks in ...
16
votes
2answers
546 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 ...
6
votes
2answers
109 views

let $A,B\in M_{n}(C)$ such that c is complex field and $AB^2-B^2A=B$ how prove $B^n=0$

Let $A,B\in M_{n}(C)$ such that $C$ is complex field and $AB^2-B^2A=B$. How prove $B^n=0$. thanks in advance
0
votes
5answers
103 views

Proof that a certain number is disivible by 6

Let be number $2^n+n^2$ prime and $n\geq 2$. Proof that number $(n-3)$ is disivible by 6.
7
votes
2answers
334 views

Olympiad Mathematical Kosovo 2012 (Problem grade 9)

Let be $ a_{1},a_{2},a_{3},...,a_{2011},a_{2012} $ integers.Exatly 29 of them divisible by number 3.Show that $ a_{1}^2+a_{2}^2+a_{3}^2+...+a_{2011}^2+a_{2012}^2 $ is divisible by number 3.
3
votes
2answers
356 views

Olympiad Mathematical of Kosovo 2011 (Problem grade 9)

A little boy wrote the numbers $1,2,3,...,2011$ on a blackboard. He picks any two numbers $x,y$ , erases them with a sponge and writes the number $ |x-y |$. This process continues until only one ...
4
votes
2answers
186 views

how to prove this question about derivative and differentiation

Let $$ f:\mathbb{R}\to \mathbb{R} $$ such that $f ',f'',f'''$ exist and $\lim_{x\to+\infty} f(x)=t$ exists if $ \lim_{x\to+\infty} f'''(x)=0$. Then prove that $$ \lim_{x\to+\infty} f'(x) = ...
1
vote
3answers
234 views

Which branches of maths study the ways solving polynomial equations?

I mean , for example , let $$0 = 1 + 2x + 3x^{2} +x^{7} + 19 x^{9 }$$ and we want to solve this equation, what branch of maths doing this? I know that there is no algebraic solution of the problem ...
2
votes
1answer
92 views

how prove $A_1$+$(-1)^nA_n$ is scalar matrix with following condition

let $A_i\in M_n (\mathbb{R})$ ,$i=1,2,...,n$ $$A_1\cdot A_2 \cdot...\cdot A_n=I\hspace{5pt}\&\hspace{5pt}\det A_1=...=\det A_n=1$$ Assume that $A_1-A_k$ for $k=1,2,..,n-1$ are none zero and ...