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

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1answer
103 views

Convex optimization problem: linear equality and inequality constraints

When linear equality constraints can be converted in an inequality constraints for a strongly convex optimization problem? I mean, I got the same solution for both the following problem: 1) $\min_x ...
0
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2answers
36 views

why the all the coefficient terms of this integral share the least common factor 1/594

why the all the coefficient terms of this integral share the least common factor 1/594? Refer to this: $\int 1/(x^{23}+x^{50}) dx$ There are a lot of weird terms in the answer but they all share the ...
0
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1answer
99 views

Sum of Number of non-decreasing sequences [duplicate]

I know that the number of non-decreasing sequences of length $n$ and numbers in the sequence lying in the range $[l,r]$ is given by $$\binom{n+r-l}{n}$$ What is the formula to find the ...
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1answer
33 views

PIE Problem with divisors

Find the number of positive integers that are divisors of at least one of $10^{10},15^7,18^{11}$. Let $n(A)$ be the number of positive integers that divide $10^{10}$ let $n(B)$ be the number of ...
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1answer
360 views

What is the difference between the largest and smallest possible positive roots?

I am faced with the following question: What is the difference between the largest and the smallest possible positive roots of $4x^5 + 3x^3 -5x^2 + 7x - 12$? Now, my first attempt was to try ...
3
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1answer
68 views

Find $k$ max such as $I_n-A_1A_2…A_k$ is invertible, then so is $I_n-A_{\pi(1)}A_{\pi(2)}…A_{\pi(k)}$ for every permutation. [closed]

Let $ n \ge 2$ be an integer. Find the largest integer $ k \ge 1$ with the following property: for any $k$ matrices: $A_1,A_2,...,A_k \in \mathcal{M}_n(\mathbb{C})$, if $I_n-A_1A_2...A_k$ is ...
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1answer
76 views

BMO1 2008/09 Question 6 Trigonometry Problem

The obtuse-angled triangle $ABC$ has sides of length $a,b$ and $c$ opposite the angles $\angle A, \angle B$ and $\angle C$ respectively. Prove that $$a^3 \cos A + b^3 \cos B + c^3 \cos C \lt abc.$$ ...
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1answer
76 views

How many ordered triples $(a, b, c)$ of positive integers satisfying the given conditions exist?

Find the number of ordered triples $(a,b,c)$ where $a$, $b$, and $c$ are positive integers, $a$ is a factor of $b$, $a$ is a factor of $c$, and $a+b+c=100$ $b = ax, c = ay$ so: $$a + b + c = ...
2
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0answers
52 views

$\lim_n \sum_k^{n-1} \tfrac1{1-\rho^k-\rho^{n-k}}$

If $|\rho|<1$, show that, when $n\to\infty$: $$ \frac1{n-1} \sum_{k=1}^{n-1} \frac1{1-\rho^k-\rho^{n-k}} = 1 + \frac1n \frac{2\big(\psi_{\rho}(1)+\log\big(1-\rho)\big)}{\log \rho} + ...
1
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1answer
169 views

How to calculate sum of combinations with different n and k

Input: $[X,Y]$ and $L$ Output : no of increasing sequence of length L and all elements should be $X\le i \le Y$ e.g: for $[6,7]$ and $2$ sequences are $6,66,67,7,77.$ For the above question my ...
2
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2answers
51 views

Find $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 ,\ n\ge 1$ has $n$ roots $x_1,x_2,\ldots,x_n \le -1$ and such that $a_0^2+a_1a_n=a_n^2+a_0a_{n-1}.$

Let $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 ,\ n\ge 1$ have $n$ roots $x_1,x_2,\ldots,x_n \le -1$ and $a_0^2+a_1a_n=a_n^2+a_0a_{n-1}$. Find all such $P(x)$ (Poland 1990). I used Viete Theorem ...
4
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1answer
80 views

BM01 2008/09 Question 5 Sequences Problem

Determine the sequences $a_0 , a_1 , a_2 ,\dots$ which satisfy all of the following conditions: a) $a_{n+1} = 2a_n^2 − 1$ for every integer $n ≥ 0,$ b) $a_0$ is a rational number and c) $a_i ...
9
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2answers
300 views

diophantine equation $x^3+x^2-16=2^y$

Solve in integers: $x^3+x^2-16=2^y$. my attempt: of course $y\ge 0$, then $2^y\ge 1$, so $x\ge 1$. for $y=0,1,2,3$ there is no good $x$. so $y\ge 4$ and we have equation $x^2(x+1)=16(2^z+1)$, ...
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2answers
819 views

Finding Sum of all Distict number whose LCM is N

The problem was : For a given positive integer N, what is the maximum sum of distinct numbers such that the Least Common Multiple of all these numbers is N. for n=1) Only possible number is 1, so the ...
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2answers
48 views

$x^3-3x^2+(a^2+2)x-a^2$ has 3 roots $x_1,x_2,x_3$ such that $\sin \tfrac{2\pi x_1}{3}+\sin \tfrac{2\pi x_3}{3}=2\sin \tfrac{2\pi x_2}{3}$. Find $a$.

$x^3-3x^2+(a^2+2)x-a^2$ has 3 roots $x_1,x_2,x_3$ such that $\sin \dfrac{2\pi x_1}{3}+\sin \dfrac{2\pi x_3}{3}=2\sin \dfrac{2\pi x_2}{3}$. Find $a$ (Bulgari 1998)
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2answers
35 views

Existence of polynomials $g$, $h$, with non-negative coefficients, such that $f(x)= \frac{g(x)}{h(x)}$ [closed]

Suppose $a$ and $b$ are real numbers such that the quadratic polynomial $f(x) = x^2 + ax + b$has no non-negative real roots. Prove that ther exist two polynomials g,h, whose coefficients are ...
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2answers
39 views

Let $ (x-1)^n\mid P(x)$ Prove that $P(x)$ has $n+1$ coefficients not zero

Let $ (x-1)^n\mid P(x)$ Prove that $P(x)$ has $n+1$ coefficients not zero It's is 1977 Bulgaria contest, i tried but not succeed
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1answer
76 views

Find minimum of $P=\frac{\sqrt{3(2x^2+2x+1)}}{3}+\frac{1}{\sqrt{2x^2+(3-\sqrt{3})x +3}}+\frac{1}{\sqrt{2x^2+(3+\sqrt{3})x +3}}$

For $x\in\mathbb{R}$ find minimum of $P$. $P=\dfrac{\sqrt{3(2x^2+2x+1)}}{3}+\dfrac{1}{\sqrt{2x^2+(3-\sqrt{3})x +3}}+\dfrac{1}{\sqrt{2x^2+(3+\sqrt{3})x +3}}$ Source : Viet Nam national test for high ...
3
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1answer
68 views

$(a_1,\cdots a_n)\rightarrow (|a_1-a|,\cdots ,|a_n-a|)\rightarrow\cdots\rightarrow (0,\cdots ,0)$

NOTE: I only need verification of part (b) of this question. But feel free to comment on anything about this question. Given an initial sequence $a_1,\cdots a_n$ of real numbers, we perform a ...
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4answers
86 views

Solving for $f(2004)$ in a given functional equation

Given that $$f(1)=2005$$ and $$f(1)+f(2)+...f(n) = n^{2}f(n)$$ for all $n>1$. Determine the value of $f(2004)$. My progress: I first substituted $n-1$ into the equation to get ...
4
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2answers
251 views

Denesting a square root: $\sqrt{7 + \sqrt{14}}$

Write: $$\sqrt{7 + \sqrt{14}} = a + b\sqrt{c}$$ Form. $$7 + \sqrt{14} = a^2 + 2ab\sqrt{c} + b^2c$$ $a^2 + b^2c = 7$ and $2ab = 1$, and $c = 14$ But that doesn't seem right as $a, b,$ wont be ...
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1answer
40 views

Show that $29 | N$ Problem

Let $\frac{29}{25} x_1$ and $\frac{39}{50}x_2$ equal $N$ for some $x_1,x_2$. If $x_{1,2}$ are positive integers show that: $$29 | N,\space \text{and} \space 39 | N$$ So, $$29 | N \implies ...
3
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1answer
109 views

British Olympiad; Combinatorics Recursion

Isaac is planning a nine-day holiday. Every day he will go surfing, or water skiing, or he will rest. On any given day he does just one of these three things. He never does different ...
6
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1answer
110 views

Solving $xyt = 1000$

How many nonegative integer solutions (triples), $(x, y, t)$ exist for: $$xyt = 1000$$ I found the prime factorization being, $$1000 = 2^3 \cdot 5^3$$ Let $x = 2^{a} \cdot 3^{b}$, let $y = 2^{c} ...
4
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2answers
85 views

Prove that $n(r) < 2\pi \sqrt[3]{r^{2}}$

Suppose that $n(r)$ denotes the numbers of points with integer coordinates on a circle of radius $r > 1$. Prove that $$ n(r) < 2\pi \sqrt[3]{r^{2}} $$ What process would you use to resolve ...
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2answers
103 views

Find $p,q$ s.t. $2q^2-p^2=\Box$ and $2p^2-q^2=\Box$

Problem. Find all integers $p,q$ such that $2q^2-p^2$ and $2p^2-q^2$ are perfect squares. I think this is only true when $p=\pm q$ but I have not been able to prove it. One approach I tried is ...
4
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3answers
69 views

$x_1 + x_2 + x_3 \le 50$ solutions

The book shows the answer as attached. Their equation, $$x_1 + x_2 + x_3 + y = 50 \implies x_1 + x_2 + x_3 = 50 - y$$ How is that the same as solving, $$x_1 + x_2 + x_3 \le 50$$ ???
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1answer
123 views

Question about the chakravala method on solving Pell's equation

I am currently reading on this old way of Pell's equation: http://en.wikipedia.org/wiki/Chakravala_method Looking at the section where they consider $N = 61$, it is not clear to me if the solution ...
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1answer
69 views

Finding other problems similar to a math contest problems?

*I don't know if I can ask these type of questions here. Tell me and I will delete it right away if it's doesn't belong here. I'm preparing for a math contest, but I'm done answering all the previous ...
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1answer
93 views

BMO1 2009/10 Problem 6

Long John Silverman has captured a treasure map from Adam McBones. Adam has buried the treasure at the point $(x,y)$ with integer co-ordinates (not necessarily positive). He has indicated on the map ...
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2answers
45 views

probability contest problem

The question asks Daniel and Scott are playing a game where a player wins as soon as he has two points more than his opponent. Both players start at par, and points are earned one at a time. If ...
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1answer
37 views

Find the smallest postive integer $n$ such $H(n)<H(n+1)$

Let $$H(x)=\dfrac{\sin{\frac{\pi}{6}x}}{x}$$ Find the smallest postive ineteger $n$ such $$H(n)<H(n+1)$$ My approach is the following: I use wolframalpha found $n=9?$ Now I don't know how to ...
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1answer
30 views

Find all positive integer $n$ such that there exists $m$ with $2^n-1|m^2+17^2$.

Find all positive integer $n$ such that there exists $m$ with $2^n-1|m^2+17^2$. I have tried to mod $2^n-1$ and use the fact that $2^n \equiv 1 \pmod{2^n-1}$. I have also tried to factorize ...
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1answer
66 views

BMO1 2009/10 Question 5 Functional Equations Problem

Find all functions $f$, defined on the real numbers and taking real values, which satisfy the equation $f(x)f(y) = f(x + y) + xy$ for all real numbers $x$ and $y$. Thanks in advance for any ...
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1answer
113 views

Complete Solution (Icosahedron Proof Putnam)

I posted a similar question earlier, but then I noted an issue. Again the problem: A1: Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are ...
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1answer
34 views

How many integers can be made?

The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. How many integers $n$ can be made? Since there is: $$0, 1, 2, 3, 4, 5, 6, 7, ...
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1answer
77 views

BMO1 2009/10 Question 4 Geometry Problem

Two circles, of different radius, with centres at B and C, touch externally at A. A common tangent, not through A, touches the first circle at D and the second at E. The line through A which is ...
1
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2answers
204 views

2013 Putnam A1 Proof understanding (geometry)

Problem A1: Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is ...
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2answers
29 views

Sum $\pmod{1000}$

Let $$N= \sum_{k=1}^{1000}k(\lceil \log_{\sqrt{2}}k\rceil-\lfloor \log_{\sqrt{2}}k \rfloor).$$ Find $N \pmod{1000}$. Let $\lceil x \rceil$ be represented by $(x)$ and $\lfloor x \rfloor$ be ...
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1answer
84 views

How many perfect squares exist (multiples of $24$)

How many positive perfect squares less than $10^6$ are multiples of 24? I quickly realized: $$24 = 2^{3}*3*5^0$$ $$10^6 = 2^6 * 5^6*3^0$$ We are finding numbers in the form $24(k^2)$. But I ...
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2answers
232 views

Probability Question (Colored Socks)

In a drawer Sandy has 5 pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the 10 socks in the drawer. On Tuesday Sandy selects 2 of the ...
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2answers
41 views

Probability using Combinations

I am confused on how this works. Normally, probability is: $$P = \frac{\text{Number of successes}}{\text{Number of total trials}}$$ For a problem like: If you flip a fair coin $8$ times, what is ...
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1answer
75 views

Difficult Probability mixed with combinatorics problem

Melinda has three empty boxes and $12$ textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold ...
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2answers
88 views

INMO Problem with even function proof. [duplicate]

Let $n$ be a natural number. Show that $$\left[ \frac{n}{1} \right ] + \left[ \frac{n}{2} \right ] + \left[ \frac{n}{3} \right ] + \cdots + \left[ \frac{n}{n} \right ] + [\sqrt{n}]$$ is even. ...
1
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2answers
39 views

Interpretation of a Problem involving permutations

[USAMO 1999 submission, Titu Andreescu] Let $n$ be an odd integer greater than $1$. Find the number of permutations $p$ of the set $\{ 1, 2, …, n\}$ for which $$\def\x#1{\lvert p(#1)-#1\rvert} ...
2
votes
1answer
302 views

Sum of GCD and LCM

If $a,b \in \mathbb{N}$ and $ab > 2$ show that: $$\text{lcm}(a, b) + \gcd(a, b) \le ab + 1$$ Let the lcm be $l$ and let the gcd be $g$. We have to show: $$g + l \le ab + 1$$ I know that: ...
2
votes
1answer
152 views

Is this 5th root in the set of natural numbers?

Is $$\sqrt[5]{x(x+1)(x^4 + x^2 + 1)} \in \mathbb{N}$$ for some $x$? I am not asking for all $x$, but just for some natural number $x$? I don't believe so, but I may be wrong? Suppose $x=1$, ...
0
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1answer
52 views

AMC $12A$ Problem (Sequence lengths)

For each positive integer $n$, let $S(n)$ be the number of sequences of length $n$ consisting solely of the letters $A$ and $B$, with no more than three $A$s in a row and no more than three $B$s in ...
0
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1answer
39 views

Combinatorics Chess Spot Problem

Very tough problem, I must say. NOT CONSIDERING the squares both can go in from one of the black square not considering the squares both can go to. The horse can go to is: $$4 + 4 = 8 \space ...
3
votes
2answers
116 views

Product of repeated cosec.

$$P = \prod_{k=1}^{45} \csc^2(2k-1)^\circ=m^n$$ I realize that there must be some sort of trick in this. $$P = \csc^2(1)\csc^2(3).....\csc^2(89) = \frac{1}{\sin^2(1)\sin^2(3)....\sin^2(89)}$$ I ...