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

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2
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2answers
73 views

How do I solve this Olympiad question with floor functions?

Emmy is playing with a calculator. She enters an integer, and takes its square root. Then she repeats the process with the integer part of the answer. After the third repetition, the integer part ...
1
vote
1answer
56 views

Four different positive integers a, b, c, and d are such that $a^2 + b^2 = c^2 + d^2$

Four different positive integers $a, b, c$, and $d$ are such that $a^2 + b^2 = c^2 + d^2$ What is the smallest possible value of $abcd$? I just need a few hints, nothing else. How should I begin? ...
4
votes
1answer
57 views

$A\subseteq \{1,2,3, \ldots 2000\}, $ and for any $a,b\in A,\; |a-b|$ is not equal to 4 or 7,

$A\subseteq \{1,2,3,\ldots2000\}$, and for any $a,b\in A,$ $|a-b|$ is not equal to 4 or 7. Then, at most, how many element does $A$ contain? For general condition,$|a-b|$ is not equal to $i$ or $j, ...
0
votes
3answers
21 views

Solving for mod indirectly

How many positive integers $n$ exist such that $\frac{680}{n}$ is an integer? So, this is quite obvious, $680 \equiv 0 \pmod{n}$ How should I solve for $n$? There will be multiple $n$?
2
votes
2answers
52 views

A game where starting with 3 boxes, with 10 balls in each, the goal is to remove as many balls as possible following the rules

This is a Norwegian olympiad problem: Peter has three boxes, with ten balls in each. He plays a game where the goal is to end up with as few balls as possible in the boxes. The boxes are each ...
3
votes
1answer
42 views

Show that $x^2 + y^2 + 1 \le \sqrt{(x^3 + y + 1)(y^3 + x + 1)}$

For $x, y \ge 0$ prove that: $$x^2 + y^2 + 1 \le \sqrt{(x^3 + y + 1)(y^3 + x + 1)}$$ What I think would apply is the AM-GM Inequality, so first, $$(x^2 + y^2 + 1)^2 \le (x^3 + y + 1)(y^3 + x + ...
0
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1answer
28 views

number of solutions of these equations.

Find the number of solution for this equation without drawing graph?! Total number of solutions for $2^{\cos x}=|\sin x|$ in $[-2\pi,5\pi]$ a) $14$ b) $15$ c) $16$ d) $17$ [ans given : ...
3
votes
0answers
71 views

Solve an inequality using Cauchy-Schwarz Inequality

Le $a,b,c,d \in \mathbb{R^{+}}$. Using Cauchy-Schwarz Inequality prove that the following inequality holds: $$\frac{1}{\frac{1}{a+c} + \frac{1}{b+d}} \ge \frac{1}{\frac 1a + \frac 1b} + ...
4
votes
2answers
381 views

Where can the knight be?

The answer is 33. I get $24$. Because of $8 \cdot 3 = 24$? How can I do this using combinatorics?
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2answers
43 views

Combinatorics using a geometric diagram

How can I do this without trial-and-error? It has something to do with a triangle and summing the next row?
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1answer
31 views

How many possible paths?

The answer is $32$. Its supposed to be $2^5$ but I do not see how you get that? The way I see it, there are $5$ ways to go up and $5$ ways to go right, total ways = $5x5= 25$
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vote
2answers
66 views

We write all the positive integers run together as follows: $123456789101112131415 . . .$

We write all the positive integers run together as follows: $123456789101112131415 . . .$ What three digit number begins at the $2014th$ digit? I was thinking number theory here. Modulus. Can ...
0
votes
2answers
32 views

Sum of the coefficients of the expansion

Find the sum of the coefficients of the expansion: $$\frac{(1+x)\cdot(2+x^2)\cdot(3+x^3)...(103 + x^{103})}{103!}$$ The answer says let $x=1$, is this the way to go? Why not let $x=0$ ??
-10
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0answers
28 views

uva 12236 - In a Crazy City immediately solution please [on hold]

I live in a crazy city full of crossings and bidirectional roads connecting them. On most of the days, there will be a celebration in one of the crossings, that's why I call this city crazy. Everyday, ...
5
votes
0answers
43 views

A set of integers whose elements all divide $2015^{200}$ but do not divide each other

Let $S$ be a set of natural numbers,such that each element divides $2015^{200}$ but for no two elements $a$ and $b$, $a|b$. Find the maximum number of elements in $S$ . $2015^{200}=(5\cdot ...
2
votes
2answers
51 views

Find the least number b for divisibility

What is the smallest positive integer $b$ so that 2014 divides $5991b + 289$? I just need hints--I am thinking modular arithmetic? This question was supposed to be solvable in 10 minutes...
2
votes
1answer
72 views

How many ordered triples $(a, b, c)$ exist?

How many ordered triples $(a, b, c)$ of positive integers exist with the property that $abc = 500$? Breaking it up, $500 = 2^2\cdot5^3$ $abc = 2^2 \cdot 5^3 = 2\cdot 2 \cdot 5 \cdot 5 \cdot ...
1
vote
1answer
25 views

2013th powered sequence

Let $a_1$, $a_2$, ... be a sequence of integers defined recursively by $a_1=2013$ and for $n \ge 1$, $a_{n+1}$ is the sum of the 2013th power of the digits of $a_n$. Do there exist distinct positive ...
0
votes
1answer
13 views

Cyclic quadrilateral problem

In convex quadrilateral $ABCD$, $AB=2$, $AD=4$, and $2BC+CD=10$. If angle $DAC$ equals angle $DBC$, and the diagonals of $ABCD$ are perpindicular to each other, what is the area of $ABCD$? I have a ...
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0answers
19 views

What quadratic function is created? [on hold]

Directrix of $y = 2$ and a focus of $(3, -4)$ What quadratic equation is created? $f(x) = \frac{1}{12}(x - 3)^2 - 1$ $f(x) = - \frac{1}{6} (x + 3)^2 + 1$ $f(x) = \frac{1}{6}(x - 3)^2 + 1$ ...
3
votes
1answer
90 views

Prove that for any positve real

Prove that for any positive real numbers $x,y,z$ such that $xyz \geq 1$ $$\frac{x^5-x^2}{x^5+y^2+z^2} + \frac{y^5-y^2}{y^5+z^2+x^2} +\frac{z^5-z^2}{z^5+x^2+y^2} \geq 0.$$ This problem is from the ...
2
votes
1answer
58 views

Biggest number of creatures in forest

In crazy forest there are 6 werewolf's,17 unicorns and 55 spiders. Werewolf can eat unicorn and spider,but can't eat another werewolf. Spider can eat unicorn,but can't eat werewolf or another spider ...
2
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0answers
57 views

How to solve this equation $x^5 +4^y=2013^z$ in positive integers?

I think to solve the equation in positive integers. It appears in a contest and I don't remember where. I obtain that $x$ must be an odd number and further $x=1 \, mod\, 4$. Any hint is appreciated.
0
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1answer
33 views

Permutations and Combinations Olympiad

Suppose that all positive integers which are relatively prime to 105 are arranged in an increasing sequence - a1 , a2 ,a3 ,.... Evaluate a1000.
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1answer
31 views

Is there a relation between product of digits of a number and perfect square?

I want to find all numbers less than N whose product of digits is a perfect square. for example if N is equal to 100 then some of possible numbers are 22 (2*2), 49 (4*9=36), 2*8, 8*2 etc. I was ...
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0answers
24 views

Let $P$ with real coefficients satisfies $|P(i)|<1$. Prove that there is a root $z=a+bi$ of $P$ such that $(a^2 + b^2 + 1)^2 < 4b^2 + 1$

A monic polynomial $P$ with real coefficients satisfies $|P(i)|<1$. Prove that there is a root $z=a+bi$ of $P$ such that $(a^2 + b^2 + 1)^2 < 4b^2 + 1$ One solution is: Let us write $P(x) = ...
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0answers
26 views

Prove that the maximum in absolute value of any monic real polynomial of n-th degree on [-1, 1] is not less than $\frac{1}{2^{n-1}}$

One solution is: Note that equality holds for a multiple of the n-th Chebyshev polynomial $T_{n}(X)$ The leading coefficient of $T_{n}$ equals $2^{n-1}$, so $C_{n}(X) = \frac{1}{2^{n-1}}T_{n}(X)$ is ...
0
votes
1answer
65 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
89 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
53 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
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1answer
47 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
123 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
39 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
162 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
79 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 ...
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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????
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27 views

walking in 3Dimension [closed]

Starting from $(0,0,0)$ of a moving object in the coordinate space through a series of steps, each step of length one. Each step to the left, right, up, down, forward or backward with equal ...
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0answers
90 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
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1answer
59 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 ...
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0answers
48 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 ...
11
votes
1answer
124 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
122 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
189 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.
9
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3answers
181 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
31 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
57 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
73 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 ...