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

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66 views

Given that $\sum\limits_{i=1}^{n}x_i=m+r$, show that $\sum\limits_{i=1}^{n}x_i^2\leq{m+r^2}$

The summation of real numbers $x_i\in (0,1)\, \text{for}\, i=1,\ldots ,n$ is equal to $m+r$, where $m$ is an integer and $r\in [0,1)$. Show that $$\sum_{i=1} ^n x_i^2\leq m+r^2.$$ I pick up this ...
2
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1answer
91 views

Interesting Olympiad Questions.

Rather than through research, I much prefer discovering new maths or interesting theories through doing problems and I also enjoy contest maths which has led me to this question: Which (high school) ...
4
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2answers
55 views

Math Contest Question with Polynomials

Prove that there does not exist a polynomial f(x) with integer coefficients for which f(2008) = 0 and f(2010) = 1867. This is a question from CMOQR (Qualifier for Canadian Math Olympiad , not the ...
1
vote
1answer
49 views

Graph Theory Contest Maths

I have never covered Graph Theory so I've been put into a bit of a quandary over how to do these two questions (I am assuming the second is graph theory, if not I will edit it out of the question). ...
0
votes
1answer
64 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
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1answer
125 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
40 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 + ...
4
votes
1answer
170 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 ...
3
votes
2answers
81 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
27 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
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0answers
91 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
60 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
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0answers
51 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
140 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
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1answer
32 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 ...
0
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0answers
123 views

One dimensional Kingdom

$N$ one dimensional kingdoms are represented as intervals of the form $[ai , bi]$ on the real line. A kingdom of the form $[L, R]$ can be destroyed completely by placing a bomb at a point $x$ on the ...
0
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0answers
190 views

Show that $\displaystyle \displaystyle 2^{2^{\sqrt3}}>10 $ without a calculator [duplicate]

Show that $\displaystyle \displaystyle 2^{2^{\sqrt3}}>10 $ without a calculator. I've tried many methods of inequalities, and had no success. Source: contest problem collection.
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3answers
184 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
32 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 ...
3
votes
3answers
58 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 $(x, y, z)$ ...
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2answers
76 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 ...
2
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0answers
55 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 ...
3
votes
0answers
47 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 $d(G, I), d(G, H), d(G, J)$, ...
2
votes
1answer
98 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 ...
0
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0answers
441 views

Finding $GCD$ excluding some elements from an $array$

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
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2answers
268 views
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2answers
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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
189 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
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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
78 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
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0answers
38 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
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0answers
43 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 ...
24
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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
145 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
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2answers
100 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, ...
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11answers
1k views

Farewell 2014 welcome 2015 - “Math Golf” [closed]

In programming, Code Golf is a competition in which the participants are trying to implement an algorithm with code which is as short as possible. Also, Stack Exchange has a site dedicated to Code ...
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3answers
53 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
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2answers
84 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
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1answer
164 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
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1answer
70 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
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1answer
70 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
31 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 ...
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vote
4answers
151 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
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1answer
169 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
111 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 $\lim_{n\to\infty}\dfrac{a_1+a_2+\dots+a_n}{n}=0$?
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vote
2answers
58 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
111 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
38 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
82 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. ...