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

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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 ...
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47 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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3answers
43 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 ...
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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 ...
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98 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 ...
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26 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 ...
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59 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 ...
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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} ...
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1answer
37 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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BM01 2008/09 Question 2 Algebra Problem [on hold]

Find all real values of $x, y$ and $z$ such that $$(x + 1)yz = 12,\ (y + 1)zx = 4\quad\text{and}\quad(z + 1)xy = 4.$$ Thanks in advance for any contributions.
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81 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 ...
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54 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
20 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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2answers
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Proof by Contradiction Minimum Value Proof $f(x)$

Focusing on $x=a$ first. My Proof: Assume $f'(a) < 0$ $f(x) \le f(x_1)$ for all $x$, this follows from the extreme value theorem. $$f'(x_1) = 0$$ Because it is a maximum. $$\exists x_4 ...
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1answer
46 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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Arc length contest! Minimize the arc length of $f(x)$ when given 3 conditions. [Contest results finally posted!]

Contest: Give an example of a continuous function $f$ that satisfies three conditions: $f(x) \geq 0$ on the interval $0\leq x\leq 1$; $f(0)=0$ and $f(1)=0$; the area bounded by the graph of $f$ and ...
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131 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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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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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 ...
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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 ...
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29 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
26 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
54 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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3answers
595 views

A truth teller and liar puzzle of Ramanujan mathematical olympiad 2013

On an island each person always tells the truth or each person always tells a lie. Three people say $A$ , $B$ and $C$ have a conversation. $A$ says that $B$ is lying , $B$ says that $C$ is lying and ...
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1answer
56 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
50 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 ...
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1answer
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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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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 ...
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1answer
32 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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3answers
2k views

Maths brain teaser. Fifty minutes ago it was four times as many minutes past three o'clock

Fifty minutes ago it was four times as many minutes past three o'clock. How many minutes is it to six o'clock..? I have got the solution online but have doubts in it : ...
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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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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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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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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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1answer
36 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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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} ...
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1answer
131 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 $x=1$, ...
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1answer
221 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: ...
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Integer-sided isosceles triangle with area equal to $120$

BdMO National 2013 Junior Q. 2: Two isosceles triangles are possible with 120 square unit area of each and length of edges are integers. Such one is with 17, 17 and 16 unit edges. Determine the ...
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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 ...
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104 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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138 views

Some challenging Series, maximum value and polynomial factor questions

So I realize that the questions I am gonna ask are going to be a minute's work for some of you but I couldn't do them even after hours of searching for methods or something. They are from a ...
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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 ...
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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 ...
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69 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 ...
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2answers
190 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 ...
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Prove this inequality with $xyz\le 1$

if $x,y,z>0$ and $\color{red}{xyz\le 1}$, show that $$\color{blue}{\dfrac{x^2-x+1}{x^2+y^2+1}+\dfrac{y^2-y+1}{y^2+z^2+1} +\dfrac{z^2-z+1}{z^2+x^2+1}\ge 1}$$ Sorry,This is not 2015 TST ...
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1answer
688 views

Finding the binary representation of the $n$th Fibonacci term

Objective: To find the binary representation ( or no. of 1's in binary representation) of nth term in Fibonacci sequence where n is of the order 10^6. My current approach: Find nth term (in decimal) ...
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2answers
53 views

Volume and surface area of a drilled out cube (BM01 2010/11 Contest Question 2)

Let $s$ be an integer greater than $6$. A solid cube of side $s$ has a square hole of side $x < 6$ drilled directly through from one face to the opposite face (so the drill removes a cuboid). The ...
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2answers
53 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 ...