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Questions tagged [contest-math]

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

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1answer
55 views

Moscow Seven Sisters

Fix $n$ points in the plane in generic position, i.e. no three of them on the same line, etc. The number of lines joining two of them is ${n \choose 2}$. The number of regions in which $\ell$ lines ...
-1
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0answers
44 views

DO NOT DISCUSS USAMTS YEAR 30 Round 2 [on hold]

USAMTS Year 30 (this year) Round 2 problems are released. http://usamts.org/Tests/Problems_30_2.pdf Please refrain from discussing the problems until the solutions are posted on their website. This ...
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0answers
37 views

Mathematical Olympiad Topics. [on hold]

I really love mathematics. I find it very interesting and I would like to learn the following topics at the level of the International Mathematical Olympiad: Number theory, combinatorics, and ...
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0answers
13 views

4 variables inequality implying $e$ and $W(z)$

It's related to my answer here I make a generalization of the inequality of Michael Rozenberg : Let $x_i$ be $4$ real positives numbers such that $\prod_{i=1}^{4}x_i=1$ then we have : $$\sum_{i=...
0
votes
0answers
30 views

Is there a trick to compute this multinomial-looking sum?

The series I want to sum has this form $\displaystyle \sum_{} 1^{l_1}(1+c)^{l_2} (1+2c)^{l_3} \cdot \ldots \cdot (1+(N-1)c)^{l_{N}}$ for some constant $c$ and positive integers $N$ and $L$. Here ...
4
votes
3answers
124 views

Find all pairs of positive integers $(a,b)$ such that $2^a+5^b$ is a perfect square.

How do you solve such questions when they appear? I know that this problem involves quadratic residues. Moreover, I also know that a=2,b=1 is possible. It may also be the only solution I tried to ...
0
votes
6answers
67 views

Inequality for contests

Prove that for real numbers $x,y,z\in[0;1/2]$ with $x+y+z=1 :$ $$ \sqrt{1-x^2} + \sqrt{1-y^2} + \sqrt{1-z^2} \geq 2\sqrt{2}$$
12
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4answers
161 views

$x=\sqrt[3]{\sqrt{5}+2}+\sqrt[3]{\sqrt{5}-2}$ is rational or irrational?

The number $x$ defined below is rational or irrational? $$x=\sqrt[3]{\sqrt{5}+2}+\sqrt[3]{\sqrt{5}-2}$$ From: IMO 1973 - Longlist My attempt (my real question is at the end): the identity $a^3+...
0
votes
1answer
28 views

Iterations of operator at points remain in the unit cube

Recently, I came across the following problem (Problem 5, Interuniversity Iberoamerican Math. Competition (CIIM)): Let $T:\mathbb{R}^3 \to \mathbb{R}^3$ be the operator defined by $$T(x,y,z) = (...
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0answers
29 views

Solve the problem [on hold]

Its a 6 digit number where the sumation of 6 digits is 43. If the following 2 statements of 3 are true then what is the number? A. The number is a Perfect Square B. The number is a Cubic Number C. ...
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2answers
63 views

INMO 1998 question

This was question 5 of 1998 INMO. Suppose $a,b,c$ are three real numbers such that the quadratic equation $$ x^2 - (a +b +c )x + (ab +bc +ca) = 0 $$has roots of the form $\alpha + i \beta$ where $\...
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0answers
35 views

Making a proof using AM-GM.

Theorem: Let $a_1,a_2,a_3,\cdots a_n$ be a sequence of positive numbers and let $b_1,b_2,\cdots, b_n$ be any permutation of the first sequence. Then $$\frac{a_1}{b_1}+\frac{a_2}{b_2}+\cdots +\frac{...
0
votes
1answer
78 views

Inequality with choose function: $1\sqrt{\binom n1} + 2\sqrt{\binom n2}+3\sqrt{\binom n3}+\cdots+n\sqrt{\binom nn} < \sqrt{{2^{n-1}}{n^3}}$ [duplicate]

From the 1987 Spanish Mathematical Olympiad: Prove, for all natural numbers $n$ with $n > 1$, that $1\sqrt{n\choose1} + 2\sqrt{n\choose2}+3\sqrt{n\choose3}+\cdots+n\sqrt{n\choose{n}} < \sqrt{{2^{...
0
votes
1answer
22 views

Regarding algebric manipulation in order to find $a_0^2 - a_1^2 + a_2^2 + … a_{2n}^2$

In the problem below I am unsure about the validity of the manipulation in Step 3 of the solution. Tldr : How is $(1/x^2+1+x^2)^n = 1/x^{2n}(1 + x^2 + x^4)^n$? (I plugged in small numbers and it ...
12
votes
1answer
296 views

İMO 2011: Prove that, for all integers $m$ and $n$ with $f(m)<f(n)$, the number $f(n)$ is divisible by $f(m)$

Problem: Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m-n)$. Prove ...
1
vote
2answers
111 views

Using binomial coefficients to find sum of roots of a polynomial.

Find the sum of the roots, real and non-real, of the equation $x^{2001}+\left(\frac 12-x\right)^{2001}=0$, given that there are no multiple roots. While trying to solve the above problem (AIME 2001, ...
2
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2answers
63 views

Find an angle in a geometric figure (given) considering triangles

Question: In the figure below, AC=AB and AD=BC. Find angle $x$. My attempt: using a geometric approach, consider the following figure (proportions are not exact). Using the notation $AC=AB=b$ and $...
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2answers
67 views

Randomly permute $\{1,\cdots,100\}$. What is the probability that none of the $S_k$'s defined by $\sigma(1)+\cdots+\sigma(k)$ is divisible by $3$?

After randomly permuting the numbers from $1$ to $100$, what is the probability that none of the $S_k$'s defined by $S_k =\sigma(1)+\cdots+\sigma(k)$ is divisible by $3$? I think I have a ...
2
votes
2answers
85 views

Being successful in mathematics depends on hard work or intelligence? [closed]

I really need to ask this question. Perhaps my question is against the rules of the MSE. I am an IMO participant. I only joined once and I only managed to solve $2$ questions. ($14+1$ points). I've ...
10
votes
0answers
98 views

Prove a strong inequality $\sum_{k=1}^n\frac{k}{a_1+a_2+\cdots+a_k}\le\left(2-\frac{7\ln 2}{8\ln n}\right)\sum_{k=1}^n\frac 1{a_k}$

For $a_i>0$ ($i=1,2,\dots,n$), $n\ge 3$, prove that $$\sum_{k=1}^n\frac{k}{a_1+a_2+\cdots+a_k}\le\left(2\color{red}{-\frac{7\ln 2}{8\ln n}}\right)\sum_{k=1}^n\frac 1{a_k}.$$ The case without $\...
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0answers
37 views

Number of ways to seat people in a rectangular seat arrangment [on hold]

Their is an (2 X n) size seat arrangement is given and we've to make m people sit in this seat arrangement , such that no two people share a side . example n=3 m=2 their are 8 ways of seat ...
0
votes
2answers
18 views

Mathcounts Cutting A Larger Cube into Smaller Cubes

What is the smallest number of cuts required to create 64 unit cubes from a 4 by 4 by 4 unit block of wood? I thought that maybe we could make 3 cuts in the x, y, and z direction, but that would be ...
6
votes
1answer
47 views

Parity of sum of powers of odd numbers

Recently, I came across this exercise: Suppose that $a$ and $b$ are odd numbers. Prove that only for finitely many positive integers $j$ does $2^j$ divide $a^j+b^j$. I tried to solve it using ...
3
votes
1answer
113 views

On minimum distance from one row vector to the linear span of the others

I am dealing with the test of the OBM (Brasilian Math Olympiad), University level, 2017, phase 2. As I've said at other topics (questions 1, 2, 3, 4 and 6 - this last still open), I hope someone can ...
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1answer
31 views

Statistics Interesting Question [closed]

A set of seven integers has a median of 73, a mode of 79, and a mean of 75. What is the least possible difference between the maximum and minimum values in the set? Does anyone know how to do this? ...
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0answers
11 views

Number of paths from the left to the right end of a board with few length restrictions [duplicate]

Suppose we have an $n x m$ board of squares, ie n rows and m columns. We want to count the number of paths starting from a square in the first column and ending in the last column. To continue a path,...
4
votes
2answers
83 views

Completing the proof for a combinatorics question from OIM 1994

The question states: In every square of an $n × n$ board there is a lamp. Initially all the lamps are turned off. Touching a lamp changes the state of all the lamps in its row and its column (...
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2answers
45 views

problem regarding application of Jensen's inequality

question: For $a,b,c,d \in \mathbb{R^+}$ with $a+b+c+d = 4$, Prove $\displaystyle \sum\dfrac{a}{b(b+1)}\geq \dfrac{8}{(a+c)(b+d)}$ my attempt: $f(x)= \dfrac{1}{x^2+x}=\dfrac{1}{x(x+1)}$ is convex ...
0
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1answer
23 views

Help with Rotational Vector Matrix problem

Let $\mathbf{A}$ be two by two matrix [sqrt(3)/2, -1/2; 1/2, sqrt(3)/2]. Then what is $\mathbf{A}^{2018} \begin{pmatrix} 2 \\ 2 \end{pmatrix}?$ I am stuck and cannot think of a method of simplifying....
2
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1answer
82 views

A coordinate geometry problem from Hungarian olympics

Here is the 2nd problem from Hungarian Kurshak maths competition I can’t solve. Please help! Let $v_1,v_2,\dots,v_n$ be different vectors in the 3D space in the Cartesian coordinate system, such that ...
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0answers
63 views

Interesting combinatorics problem from Hungarian Olympiad

This is the third problem from Kurshak Competition (Hungarian) I can’t solve. Please help! In a village (where only dwarfs live) there are $k$ streets, and there are $k(n-1)+1$ clubs, each containing $...
0
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1answer
52 views

limits involving scary roots of binomial coefficients products

Calculate the following limit : $$ \lim_{n\to \infty}\left[\frac{\sqrt[n+1]{^{n+1}C_{1}\cdot^{n+1}C_{2}\cdots^{n+1}C_{n+1}}}{e^{\frac{n+1}{2}}(n+1)^{-\frac{3}{2}}}-\frac{\sqrt[n]{^{n}C_{1}.^{n}C_{2}\...
0
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0answers
146 views

Previous year Olympiad problem (polynomials) [duplicate]

Let $P(x)$ be a nonconstant polynomial whose coefficients are positive integers. If $P(n)$ divides $P(P(n) - 2015 )$ for every natural number n, then prove that $$P (-2015) = 0$$ I've observed that ...
4
votes
3answers
93 views

Find all functions $f\colon \Bbb R\to \Bbb R$ such that $f(1-f(x)) = x$ for all $x \in \Bbb R$ [duplicate]

Find all functions $f\colon \Bbb R\to \Bbb R$ such that $f(1-f(x)) = x$ for all $x \in \Bbb R$. This is a question from the national olympiad in Germany 2018. All i could do is to try with some ...
0
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0answers
31 views

Verification of proof of an olympiad question

I have the following question with me from RMO(India) 2017: "Let $\Omega$ be a circle with chord $AB$ which is not a diameter. Let $\Gamma_1$ be a circle on one side of $AB$ such that it is tangent ...
4
votes
1answer
94 views

Polynomial $f(x)$ such that $\dfrac{f(k)-f(m)}{k-m}\in\mathbb{Z}$

I'm dealing with the test of the International Mathematics Competition for University Students, 2011, and I've had a lot of difficulties, so I hope someone could help me to discuss the questions. I'...
1
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2answers
45 views

Interesting numbers $n$ such that $x^n-1=(x^p-x+1)f(x)+pg(x)$

I'm dealing with the test of the International Mathematics Competition for University Students, 2011, and I've had a lot of difficulties, so I hope someone could help me to discuss the questions. I'...
1
vote
3answers
44 views

Finding an elementary function

can someone please help me find any elementary function that satisfies $f(0) = 6$ $f(1) = f(-1) = 4$ $f(2) = f(-2) = 1$? I have been trying for nearly an hour. I don't know why this is so difficult. ...
3
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2answers
97 views

Simplify $\frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}$

Simplify $$\frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}$$ Found in a book with tag "Moscow 1982", the stated answer is $1+\sqrt[4]{5}$. Used all tricks that I know but without success. ...
0
votes
1answer
37 views

Proving the existence of a particular number

I have the following problem with me: "Let $p$ be a prime, and let $a_1, \dots, a_p$ be integers. Show that there exists an integer $k$ such that the numbers $ [a_1 + k, a_2 + 2k, \dots, a_p + pk] $ ...
2
votes
1answer
35 views

Help in understanding a statement in the solution

I have the following problem from USAMO 2006: "A mathematical frog jumps along the number line. The frog starts at 1, and jumps according to the following rule: if the frog is at integer $n$, then it ...
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2answers
61 views

Show by counting in two ways that $\sum_{k=0}^{n} \binom{n+k}{k} = \sum_{k=0}^{n} 2^{k} \binom{n}{k}$?

How does one approach a question like this? Show by counting in two ways that $\sum_{k=0}^{n} \binom{n+k}{k} = \sum_{k=0}^{n} 2^{k} \binom{n}{k}$ In general, the fact that $k$ can vary from $0$ to $...
3
votes
0answers
57 views

Null sum of vectors over the field of two elements

I'm dealing with the test of the International Mathematics Competition for University Students, 2011, and I've had a lot of difficulties, so I hope someone could help me to discuss the questions. I've ...
1
vote
1answer
63 views

Prove that $n$ is divisible by $6$. [duplicate]

If the quadratic equations $x^2-mx+n=0$ and $x^2+mx-n=0$ both have integral roots, prove that $6|n$. I've proved that $3|n$, and that $2|n$ for odd $m$, but I can't seem to prove it for even $m$. ...
2
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1answer
28 views

Reference to a theorem of T. Nagell

I have the following problem from USAMO 2006: "For an integer $m$, let $p(m)$ be the greatest prime divisor of $m$. By convention, we set $p(\pm 1)=1$ and $p(0)=\infty$. Find all polynomials $f$ with ...
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votes
1answer
78 views

I need to proof something [closed]

The checker with dimensions $2018 \times 2018$ was covered with one a square tile with dimensions of $2 \times 2$ and $\frac{2018^2-4}5$ rectangular tiles with dimensions $1 \times 5$ in such a way ...
3
votes
2answers
98 views

Find $\sqrt[m]{\frac{\sqrt[m]{\frac{\sqrt[m]{\frac{\sqrt[m]{a}}{a}}}{a}}}{\begin{array}{c} a\\\vdots \end{array}}}$

Assuming $m\in \Bbb N\setminus\{0,1\}$ and $a\in \Bbb R_+\setminus\{0\}$, find $$\sqrt[m]{\frac{\sqrt[m]{\frac{\sqrt[m]{\frac{\sqrt[m]{a}}{a}}}{a}}}{\begin{array}{c} a\\\vdots \end{array}}}$$ In a ...
3
votes
1answer
57 views

Show that $\sqrt{a-b\sqrt{a+b\sqrt{a-b\sqrt{a+…}}}}=\sqrt{a-\frac{3b^2}{4}}-\frac{b}{2}$

Assuming that $a>b^2$ show that $$\sqrt{a-b\sqrt{a+b\sqrt{a-b\sqrt{a+...}}}}=\sqrt{a-\frac{3b^2}{4}}-\frac{b}{2}$$ (corrected) This problem listed in a contest-math preparation book with the ...
1
vote
2answers
37 views

Let $p(x)$ be a real polynomial that is bounded below. Prove that there is a real number $x_0$ such that $p(x) ≥ p(x_0)$ for all $x$.

Let $p(x)$ be a real polynomial that is bounded below. Prove that there is a real number $x_0$ such that $p(x) ≥ p(x_0)$ for all $x$. This is listed on some practice problems for a contest math group ...
46
votes
11answers
10k views

$7$ fishermen caught exactly 100 fish and no two had caught the same number of fish. Then there are three who have together captured at least 50 fish.

$7$ fishermen caught exactly $100$ fish and no two had caught the same number of fish. Prove that there are three who have captured together at least $50$ fish. Try: Suppose $k$th fisher caught $r_k$ ...