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

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0
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1answer
65 views

Show the integral $\lim_{B\rightarrow\infty}\int_0^B \sin(x)\sin(x^2)\,dx$ converges

Show the integral $$\lim_{B\rightarrow\infty}\int_0^B \sin(x)\sin(x^2)\,dx$$ converges. I guess we should use the equality $$\sin(x)\sin(x^2)=\dfrac{1}{2}[-\cos(x+x^2)+\cos(x-x^2)],$$ so we have ...
8
votes
1answer
134 views

Rational matrix having roots of every degree

As the result of another question, now deleted, I am interested in the following problem. Problem. Let $A\in M_n(\mathbb Q)$ be an invertible matrix with the property that the equation $X^k=A$ has ...
1
vote
1answer
64 views

How prove this idenity this $mv-3nu=m-3u$ with unit circle

Assmue the $m,n,u,v$ be real numbers,and such $$m^2+n^2=1,u^2+v^2=1,nv>0,m>0,u>0$$ and $$5mu=3(1-nv)$$ show that $$mv-3nu=m-3u$$ Following is My methods: let ...
3
votes
2answers
41 views

BMO preparatory question

Q) Let $3\leq n$ be an odd integer and let $a_1,a_2,...a_n$ be fixed positive integers. For each of the $n!$ permutations $\pi=(\pi_1,\pi_2,...,\pi_n)$ of $(1,2,...,n)$, define $$f(\pi) = a_1\pi_1 + ...
5
votes
1answer
184 views

Problem from Iran Olympiad?

Does there exist a positive integer that is a power of $2$ and we get another power of $2$ by swapping its digits? Justify your answer. I gussed the answer is no. Let $\overline{a_n ,...,a_1 ,a_0}$ be ...
4
votes
2answers
95 views

How prove find this value $|AD|+|DF|+|FA|=2$

Question: if $ADB$ and $ACE$ are straight lines with $D,E$ and $B,C$ intersecting at $F$. if $$|AB|=|AC|=1,|AD|+|DE|+|EA|=4$$ show that: $$|AD|+|DF|+|FA|=2$$ I have read this ...
1
vote
1answer
29 views

Exponential GF application [closed]

I have $15$ different books I have $5$ child. I want to give it all to all my child where every my child get at least $1$ book How many way I can distribute it????
8
votes
0answers
96 views

How are contest problems designed? [duplicate]

How are competition questions designed? What techniques do designers employ to design math competition questions? How they know a problem can be solved by introductory methods?Some contest math ...
3
votes
1answer
147 views

How many positive integers less than 1000 are multiples of 5 and are equal to 3 times an even number?

Question: How many positive integers less than 1000 are multiples of 5 and are equal to 3 times an even number? So Multiples of $5$ and $6$ If a number is a multiple of $5$ and $6$ then it is a ...
6
votes
0answers
53 views

Finding a separating family of subsets of $[n]$ of size $n+1$.

I have this friend who always tells me problems I can't solve. Here is the latest one. We are given a family $\mathcal F$ of at least $2^{n-1}+1 $ subsets $[n]$. We must prove that we can ...
12
votes
2answers
156 views

Functions satisfying $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$

Find all functions $f$ such that $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$ Let us plug in $n=1$ $f(f(1))+f(2)=3$ Since the function is from $\mathbb{N}$ to $\mathbb{N}$, ...
0
votes
1answer
34 views

How prove this number of the “Fixed subset” is odd

Let mapping $f:I\to I$ where $I=\{1,2,3,\cdots,n\}$,and the nonempty set $A\subset I$ such $$f(A)=\{b|\exists a\in A,f(a)=b\}$$ we called “Fixed subset”,if such $f(A)=A$ Question: show ...
9
votes
3answers
207 views

$\lim_{x\to +\infty}\frac{x^x}{(\lfloor x \rfloor)^{\lfloor x \rfloor }}$

Determine if the following limits exist $$\lim_{x\to +\infty}\dfrac{x^x}{(\lfloor x \rfloor)^{\lfloor x \rfloor }}$$ note that $\lfloor x \rfloor \leq x < \lfloor x \rfloor + 1 \implies ...
1
vote
1answer
45 views

Determinant of sum of squares of commuting matrices

I have the following question from a math competition, can anyone help me solve this: Let $A,B\in M_n(\mathbb{R})$ be two commuting matrices ($AB=BA$). Prove that $\det(A^2+B^2)\ge0$. Thanks in ...
4
votes
3answers
68 views

Given positive numbers $a, b, c, x, y, z$, such that $a + x = b + y = c + z = S$, prove that $ay + bz +cx < S^2$

Given positive numbers $a, b, c, x, y, z$, such that $a + x = b + y = c + z = S$, prove that $ay + bz +cx < S^2$ One solution is: Denote $T = S/2$. One of the triples $(a, b, c)$ and $(x, y, z)$ ...
1
vote
2answers
83 views

How many ways to tie $2$ ropes so that we do not have a loop

BdMO 2014 Higher Secondary: Avik is holding six identical ropes in his hand where the mid portion of the rope is in his fist. The first end of the ropes is lying in one side, and the other ends ...
3
votes
0answers
68 views

Prove that: $ \left( \sum_{i\neq j}a_{i}b_{j} \right)^2 \geq \left( \sum_{i\neq j}a_{i}a_{j} \right) \left( \sum_{i\neq j}b_{i}b_{j} \right)$

Let $a_{1}, \cdots, a_{n}, b_{1}, \cdots, b_{n}$ be positive real numbers. Prove that: $$ \left( \sum_{i\neq j}a_{i}b_{j} \right)^2 \geq \left( \sum_{i\neq j}a_{i}a_{j} \right) \left( \sum_{i\neq ...
4
votes
0answers
64 views

Centroids and Harmonic Means

A triangle $ABC$ with centroid $G$ is such that a line $l$ passing through $G$ intersects $AB$, $BC$, and $AC$ at $H, I, J$, respectively. Show that out of the 3 distances $d(G, I), d(G, H), d(G, J)$, ...
4
votes
2answers
172 views

Finding a value from 5 systems of equations of 5 variables(CHMMC 2014)

$$\text{For } a_1\cdots a_5\in \mathbb{R},$$ $$\frac{a_1}{k^2+1}+\cdots+\frac{a_5}{k^2+5}=\frac{1}{k^2}$$ $$\forall k=\{2,3,4,5,6\}$$ $$\text{Find }\frac{a_1}2+\cdots+\frac{a_5}6$$ The Provided ...
-1
votes
1answer
471 views

Finding $GCD$ excluding some elements from an $array$ [closed]

I have an array of numbers. I want to calculate $GCD$ of all numbers but excluding numbers from particular index $a$ to index $b$. I need to repeat the same operation multiple times with different ...
5
votes
2answers
283 views
10
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2answers
130 views

Show $1+x+(x^2/2!)+ \cdots + (x^n/n!)=0$ has no rational solutions for all $n>1$.

Prove that the equation $$1+x+\frac{x^2}{2!}+ \cdots + \frac{x^n}{n!}=0$$ has no rational solutions for all $n>1$. Assume there is a rational solution $\frac{p}{q} \in \mathbb{Q}$ with ...
3
votes
2answers
200 views

Solving an equation for two primes

This is from contest preparation: Find all pairs of primes $(p, q)$ that satisfy $$p^q - q^p = p q^2 - 19$$. It looks simple, but I spent hours trying to solve it... and no luck so far. ...
2
votes
0answers
84 views

Problem regarding the speed of two points $A$ and $B$ moving with constant speed in the plane [duplicate]

Consider a Point A that moves linearly on the positive x-axis with the speed 1 m/s and another Point B at a distance L from A with position (L,0). With each forward motion of point A the Point B moves ...
3
votes
1answer
89 views

Putnam Problem A-1 2008 3 variable function

I looked at a Putnam problem from 2008, here it is: Putnam Link " Let $f : R^2 → R$ be a function such that $f(x, y)+ f(y,z)+ f(z, x) = 0$ for all real numbers $x, y, z$. Prove that there exists a ...
0
votes
0answers
43 views

Are there way of proving that polynomials are relatively prime using number theory or abstract algebra?

This question is inspired by question A5 from the Putnam Mathematical Competition: Let $$P_n(x) = 1 + 2x + 3x^2 + \cdots + nx^{n-1}.$$ Prove that polynomials $P_j(x)$ and $P_k(x)$ are ...
0
votes
0answers
46 views

Why doesn't this approach work for integral of $\log(\sin(x))$?

Evaluation of: $$I = \int_{0}^{\pi} \log(\sin(x)) dx$$ Over closed rectangular contour $ABCD$ complex analysis. Kind of like the contour here: Contour Answer complex analysis. BUT INSTEAD the ...
23
votes
4answers
1k views

Prove that $2^{2^{\sqrt3}}>10$

With a computer or calculator, it is easy to show that $$ 2^{2^\sqrt{3}} = 10.000478 \ldots > 10. $$ How can we prove that $2^{2^{\sqrt3}}>10$ without a calculator?
4
votes
1answer
150 views

Summation identity involving the floor function

(Kömal November B. 4666) Prove that $\sum_{k=1}^n (2k-1) [\frac{n}{k}]=\sum_{k=1}^n [\frac{n}{k}]^2$ for every positive integer $n$, where $[n]$ is the largest integer greater than or equal to $n$.
0
votes
2answers
132 views

Is Spivak good preparatory for Putnam Exam?

I am currently a junior in high school, interested in the Putnam exam. I have access to number theory complex analysis, real analysis, any textbooks. Spivak Calculus for one is a rigorous textbook, ...
1
vote
3answers
60 views

On finding the $n$-th term of an arithmetic progression

Given the common difference $d$, and first term $a$ (say). It is very easy to find the $n$th term of an arithmetic progression. My question is if we are given two common differences say $d_1$ and ...
4
votes
2answers
120 views

Three circles having centres on the three sides of a triangle

NOTE: I would appreciate it if you provided a hint and not the whole solution. BdMO 2014 Nationals: In acute angled triangle ABC, considering a portion of side BC as diameter a circle is drawn ...
6
votes
1answer
191 views

IMO 2015 warm up problem

I get this problem from IMO 2015 facebook page. Let $x_i$ be positive integers for $i=1,2,...,11$. If $x_i+x_{i+1}\geq 100$, $|x_i-x_{i+1}|\geq 20$ for $i=1,2,...,10$. And $x_{11}+x_{1}\geq 100$, ...
5
votes
1answer
73 views

Inequality on length of intervals

Let $n\ge 1$ and $\{I_j\}_{j=1}^{n}$ is a set of non-degenerate subintervals of $[0,1]$. Then show that : $$ \overline\sum \dfrac{1}{|I_j\cup I_k|}\geq n^2$$ Here $\overline\sum$ denotes ...
11
votes
1answer
71 views

Coloring $\mathbb R^n$ with $n$ colors always gives us a color with all distances.

I wanted to share a really cool but simple problem. Consider a coloring of the points of $\mathbb R^n$ with $n$ colors. Prove that there is a color $c$ such that for any $r>0$ there are two points ...
2
votes
1answer
51 views

Shuffling cards and laying them out in order

The numbers from 1 to 50 are printed on cards. The cards are shuffled and then laid out face up in 5 rows of 10 cards each. The cards in each row are rearranged to make them increase from left ...
1
vote
4answers
162 views

Absolute convergence in a metric space

Let $(X,d)$ be a metric space, $(a_n)$ and $(b_n)$ are sequences in $(X,d)$. If $\sum_{n=1}^\infty d(a_n,b_n)$ is absolutely convergent, what do I say about the convergence of $(a_n)$ and $(b_n)$?
10
votes
1answer
188 views

Group theoretic solution to an IMO problem

Is there a (strictly) group theoretic interpretation (and possibly a solution) to this problem (taken from the 27th IMO)? "To each vertex of a regular pentagon an integer is assigned in such a way ...
3
votes
3answers
112 views

Convergence of summable sequences

If $(a_n)$ is a sequence such that $$\lim_{n\to\infty}\frac{a_1^4+a_2^4+\dots+a_n^4}{n}=0.$$ How do I show that $\lim_{n\to\infty}\dfrac{a_1+a_2+\dots+a_n}{n}=0$?
1
vote
2answers
66 views

Maximising the Area of a Cyclic Quadrilateral

In cyclic quadrilateral $ABCD$, $AB = AD$. If $AC = 6$ and $AB/BD = 3/5$, find the maximum possible value of $[ABCD]$. (Source: SMT 2014) If we let $AB=AD = 3x$ and $BD=5x$, from Ptolemy, we have ...
4
votes
3answers
122 views

Find $ \int \frac {1-x^2}{1+3x^2+x^4} \, \mathrm{d}x $

Today, the CalcBee sample problems got released. The following problem was my creation and I wanted to see how many solutions people can come up with. The result is very beautiful and I thought it ...
0
votes
1answer
41 views

Limit sup and inf hint

I have problem in finding the Limsup and liminf for the following sequences. Any hint pls? $(s_n) = [1-r^n]\sin \frac{n\pi}{2}$ and $(s_n) = [(-1)^n + 1]n^2$.
3
votes
1answer
88 views

Area of a circumcenter triangle equals area of medial triangle

Let $X$, $Y$, $Z$ be the midpoints of sides $BC$, $AC$, $AB$ respectively in triangle $ABC$. Let $O_{A}$, $O_{B}$, and $O_{C}$ be the circumcenters of triangles $AZX$, $BXY$, and $CYZ$ respectively. ...
6
votes
2answers
202 views

How prove this complex inequality with same as (2014 china CMO) Cauchy-Schwarz inequality

Let $r$ $z_{1},z_{2},\cdots,z_{n}$ be given such that $$ |z_{i}-1|\le r,i=1,2,\cdots,n,r\in(0,1). $$ Show that ...
1
vote
1answer
35 views

Find the volume $z \geq 3x^2+2y^2, \ \ 3x^2+2y^2+5z^2 \le 1$

Find the volume of solid defined by the following inequalities : $$z \geq 3x^2+2y^2, \ \ 3x^2+2y^2+5z^2 \le 1$$ We have an ellipse, which the semi-axis are $\sqrt{\frac{z}{2}}$ and ...
2
votes
8answers
139 views

How to show that $f(x) = 0$ if $\int_a^bf(x)\,\text{d}x=0$ for all $a,b\in\mathbb{R}$?

I found this problem on the web: Let $f(x)$ be a real-valued, continuous function with the property that $$\int_a^bf(x)\,\text{d}x=0$$for all real numbers $a,b$. Prove that $f$ is identically $0$. ...
1
vote
1answer
72 views

2014 Fall OMO #28

Here is a problem from this year’s OMO: Let $S$ be the set of all pairs $(a,b)$ of real numbers satisfying $1+a+a^2+a^3 = b^2(1+3a)$ and $1+2a+3a^2 = b^2 - \frac{5}{b}$. Find $A+B+C$, where $$ A = ...
4
votes
4answers
555 views

Is it possible to permute an unknown binary sequence so that two particular bits are equal? [closed]

A blind mathematician is give a $2015$ bit sequence. The mathematician can take any two bits and switch them (so the bit in position $A$ goes to position $B$ and vice-versa). He knows at what position ...
1
vote
3answers
441 views

How to find Bitwise AND of all numbers for a given range?

How can I find Bitwise AND of all numbers for a given range say from A to B, including both? I found a beautiful answer for finding XOR for such range. http://stackoverflow.com/a/10670524/2046703How ...
1
vote
0answers
60 views

Where can Gaussian Elimination be used?

I have searched for this and came to know about it that it is traditionally used to solve linear equations, finding determinant, rank of matrix, inverse of matrix. There was a problem on codechef: ...