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

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Let a polyminal : $P(x)$ is a irreducible in $\mathbb{Q}[X]$. If $x_0 \in \mathbb{R} :P(x_0)=0$ prove that $P'(x_0) \not=0$ [closed]

Let a polyminal : $P(x)$ is a irreducible in $\mathbb{Q}[X]$. If $x_0 \in \mathbb{R} :P(x_0)=0$ prove that $P'(x_0) \not=0$ Vietnam 2014 (College)
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$x^3-3x^2+(a^2+2)x-a^2$ 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)
2
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2answers
31 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 ...
3
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2answers
35 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
1
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1answer
71 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
58 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 ...
2
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3answers
66 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 ...
5
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1answer
121 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 doesnt seem right as $a, b,$ wont be ...
0
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1answer
31 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 ...
0
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0answers
17 views

Generic Equation of 4D

Generic Equation of 3D is Ax2+By2+Cz2+Dxy+Eyz+Fxz+Gx+Hy+Iz+J=0 Like this I have to write Generic Equation of 4D This I have to write with my own logic I think as A,B,C are for x2,y2,z2 and then D,E,F ...
3
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1answer
68 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
94 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} ...
3
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1answer
48 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 ...
7
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2answers
93 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
votes
3answers
59 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, $$x_1 + x_2 + x_3 \le 50$$ ???
0
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1answer
34 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 ...
1
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1answer
52 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 ...
3
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1answer
74 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
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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 ...
0
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1answer
33 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 ...
1
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1answer
27 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 ...
1
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1answer
56 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 ...
0
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1answer
59 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 ...
2
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1answer
24 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
57 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 ...
0
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2answers
133 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
28 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 ...
0
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1answer
33 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 ...
0
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2answers
54 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
33 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 ...
0
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1answer
43 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 ...
0
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2answers
75 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. ...
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2answers
31 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
223 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
134 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
41 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 ...
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1answer
35 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
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2answers
101 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 ...
3
votes
2answers
119 views

Ball and urn method (counting problems)

How many ordered triples $(a, b, c)$ of positive integers exist with the property that $abc = 500$? Since, $500 = 2^2 5^3$ I believe this can be solved using Ball and Urn let $a = ...
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2answers
94 views

Putnam 2009 A1 Points in a plane

HINTS PLEASE! Let $f$ be a real-valued function on the plane such that for every square $ABCD$ in the plane, $f(A)+ f(B)+ f(C)+ f(D) = 0$. Does it follow that $f(P) = 0$ for all points $P$ in ...
0
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1answer
25 views

AMC12B Problem, probability

An unfair coin lands on heads with a probability of $\tfrac{1}{4}$. When tossed $n$ times, the probability of exactly two heads is the same as the probability of exactly three heads. What is the ...
2
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2answers
201 views

Putnam 2009 B1 (rational number as factorial)

Show that every positive rational number can be written as a quotient of products of factorials of (not necessarily distinct) primes. For example, $ \frac{10}9=\frac{2!\cdot 5!}{3!\cdot 3!\cdot ...
8
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3answers
118 views

nonzero digits in decimal representation of $\sqrt{2}$

let $1,d_1d_2d_3\dots$ be a decimal representation of $\sqrt{2}$. Prove that at least one $d_i$ with $10^{1999}<i<10^{2000}$ is nonzero. I have no idea how to solve it. I think that the given ...
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2answers
58 views

Fraction of area covered by three circles

Take a square with edges of size $10$. Now take take three circles of radius $5$. Prove that you can't cover the square with these three circles. Find the maximum proportion of the area of the ...
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0answers
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Determine points of tangency (Putnam 2007)

Find all values of $ \alpha$ for which the curves $ y=\alpha x^2+\alpha x+\frac1{24}$ and $ x=\alpha y^2+\alpha y+\frac1{24}$ are tangent to each other. This is an old Putnam Problem (2007 A1) ...
0
votes
3answers
71 views

All means integer

$a$ and $b$ are distinct positive integers such that $\frac{a+b}{2}$, $\sqrt{ab}$, and $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ are integers. Find the smallest possible value of $|a-b|$. My work led me ...
1
vote
1answer
80 views

Find all nonnegative integers

Determine all nonnegative integers $x$ and $y$ so that $$3^x + 7^y$$ is a perfect square and $y$ is even. Without trial-and-error of course. $$3^x + 7^y = a^2$$ For some integer $a$. ...
3
votes
1answer
50 views

Geometry question posed in RMO 1999

Let $ ABCD $ be a square and $ M, N $ points on sides $AB, BC $ respectively, such that $\angle MDN = 45°$ . if $R$ is the midpoint of $MN$ show that $RP=RQ$ where $P,Q$ are the points of ...
1
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0answers
46 views

Putnam 2000 A1 Series square problem

Let $A$ be a positive real number. What are the possible values of $\displaystyle\sum_{j=0}^{\infty} x_j^2, $ given that $x_0, x_1, \cdots$ are positive numbers for which ...
0
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1answer
31 views

Find the solution to the system (not linear)

Find all $(x, y, z) \in \mathbb{R^3}$ satisfying: $$x^2 + 4y^2 = 4xz \tag1$$ $$y^2 + 4z^2 = 4xy \tag2$$ $$z^2 + 4x^2 = 4yz \tag3$$ This is a very difficult problem. I added $-4(1) + (3)$ to ...