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

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2
votes
1answer
58 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
593 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
121 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
92 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
159 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
99 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
230 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
557 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
526 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
204 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
368 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
186 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
54 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
276 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
143 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
201 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
150 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 ...
24
votes
2answers
520 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
411 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
160 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
198 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
422 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
263 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
243 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
552 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
338 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
366 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
236 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 ...
2
votes
2answers
487 views

how prove GL(n,R) is not connected subset and open subset of$M_n (\mathbb{R})$with this distance

let n>1 be natural and fix number, $S:=${A : $M_n (\mathbb{R})$ be all real matrix,define this meter for all $A=[a_{ij}]$ $B=[b_{ij}]$ d(A,B):=max{|$a_{ij}-b_{ij}$|:i,j=1,2,2...,n} and GL(n,R) is ...
1
vote
2answers
154 views

How to prove that this linear operator is nilpotent?

Let $A\in M_n(\mathbb{C})$ be an arbitrary matrix , $\mathbb{C}$ is complex fields, and $L$ a mapping that is defined by $L:M_n(\mathbb{C})\to M_n(\mathbb{C})$, $L(X):=AX+XA$. How can we show that ...
0
votes
2answers
229 views

Prove that there is a $\delta$ such that $\int_{0}^{1} (f(x))^2dx\leq \delta$$\int_{0}^{1} (f'(x))^2dx$ for all $f$ with these conditions

Let $S=\{f:\mathbb{R} \to \mathbb{R}\}$ that satisfies: $\forall f\in S$, $f'$ exists and $f'$ is continuous and $f(0)=f(1)=0$. Please prove that $\exists \delta :\forall f\in S$ s.t. $\int_{0}^{1} ...
4
votes
1answer
89 views

* if $\sum_{n=1}^\infty u_{n}$ be divergent then $\sum_{n=1}^\infty n u_{n}$ is convergent or divergent *

let $\sum_{n=1}^\infty u_{n}$ be divergent $\sum_{n=1}^\infty n u_{n}$ this series is divergent or convergent? thanks in advance
3
votes
3answers
767 views

Cyclic sums — How do you use them?

Can someone give me an example of how cyclic sums are used? I don't really understand how they're used in problem-solving. For example, $$\sum_{a,b,c}a^2$$ Any help would be appreciated, and I'm not ...
1
vote
2answers
120 views

A question around concept of derivative

Suppose that $f$ be a real valued function that both $f',f''$ exist and satisfy in the following conditions: $f(0)=0$ and $f'(0)>0$, For all $x≥0$ , $f''(x) ≥f(x)$, We want to prove ...
17
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 $b-a>\pi$. ...
15
votes
5answers
936 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 ...
1
vote
2answers
121 views

Table clock and wall clock

I have two clocks - table clock and wall clock A table clock gains 2 mins every 12 hours and a wall clock loses 1 min every 12 hours both are set at 12 noon on tuesday(date is not known ) we need to ...
6
votes
1answer
141 views

$x^4 + y^4 = z^2$

$x, y, z \in \mathbb{N}$, $\gcd(x, y) = 1$ prove that $x^4 + y^4 = z^2$ has no solutions. It is true even without $\gcd(x, y) = 1$, but it is easy to see that $\gcd(x, y)$ must be $1$
4
votes
2answers
139 views

having trouble with this well known fact $ab | a^2 + b^2 +1 =>a^2 + b^2 +1 = 3ab $

I have seen people use this without proof as a well known fact. Can someone give a proof or a reference?
42
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 ...
2
votes
2answers
231 views

The rooks problem

3 rooks are arranged on a 27 times 27 chess board so that no rook is attacking another. How many places can a 4th rook be placed so that it is attacking exactly one other rook on the board