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

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0
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1answer
16 views

Inequality with condition $x+y+z=xy+yz+zx$

I'm trying to prove the following inequality: For $x,y,z\in\mathbb{R}$ with $x+y+z=xy+yz+zx$, prove that $$ \frac{x}{x^2+1}+\frac{y}{y^2+1}+\frac{z}{z^2+1}\ge-\frac{1}{2} $$ My approach: After ...
0
votes
0answers
10 views

What is the isotomic conjugate version of the six point circle of isogonal conjugates?

As it is well known, the pedal triangles of a pair of isogonal conjugates in a triangle share a circumcircle. This is a nice theorem, but is there an analogous version of it for a pair of isotomic ...
2
votes
0answers
45 views

incorrect rejection of a true null hypothesis?

We have a contest 1 weeks ago. One question is a bit strange for us as follows: $X\sim B(4,p). $ for test $H_0:p=0.2$ versus $H_1:p>0.2$. if $X=4$, $H_0$ assumption is rejected. calculate ...
9
votes
3answers
134 views

Sum of digits of $11\dots 11^2$ where $11\dots 11$ is a 1992 digit number with all digits $1$ [duplicate]

I read this on a non-math forum where the OP says this is a question for Grade 6 elementary school students. Grade 6 elementary school level is somehow ambiguous but clearly this means no advanced ...
1
vote
3answers
60 views

AIME I 2015 #14:Area under a function

(This isn't the exact wording of the problem on the AIME) Find the number of $n,2\le n \le 1000$ such that $$\int_1^n x \lfloor \sqrt x \rfloor dx\in \Bbb Z$$ During the test, I noticed that for ...
0
votes
0answers
26 views

Congruent Angles with Condition [on hold]

Let A be a point in the interior of triangle BCD such that $AB · CD = AD · BC$. Point P is the reflection of point A with respect to BD. Prove that $\angle PCB = \angle ACD$. I don't know how to ...
2
votes
1answer
20 views

End digit of numbers raised to a certain power

In a math competition I came across the following question: What digit does the result of 2^2006 end with? This competition tested how fast you are at solving math problems. So, I was wondering ...
0
votes
1answer
19 views

if $f(n+1)-f(n)=P(n)$, exist a polynomial $Q(x)$ such that for all $n \in \mathbb{Z}$ : $Q(n)=f(n)$

Let $f:\mathbb{Z} \to \mathbb{Z}$ such that, exist a polynomial $P(x)$: $$f(n+1)-f(n)=P(n)$$ for all $n \in \mathbb{Z}$ Prove that exist a polynomial $Q(x)$ such that for all $n \in ...
1
vote
0answers
31 views

Combination problems

During numerous math contests I have come across questions such as: I have __ shirts, __ shoes and ___ pants... How many combinations of the __ are possible... As well as many other combination ...
3
votes
1answer
51 views

What are some good problem solving techniques for Math Olympiad style tests? [duplicate]

I am taking part in a Math Olympiad style test at my school in a few weeks. This test is mainly problem solving based and tests you on topics such as counting techniques, algebra, geometry as well as ...
0
votes
1answer
33 views

Why do (the ranges of) these sequences intersect?

Let $\{(a_n,b_n)\}$, ($1\le n\le N$) be a finite sequence and $\{(s_n,t_n)\}$ ($n\ge 1$) be an infinite sequence, both in $(\{0\}\cup \mathbb{Z}^{+})^2$. We have $a_1=0$ and $b_N=0$. Also, either ...
0
votes
1answer
26 views

Find a number that is evenly divisible by all numbers between 1 and 20

I'm solving this for a programming challenge, in fact I already solved it but I'd like to know if there's some kind of rule that could improve such thing? For example if I needed the numbers ...
11
votes
0answers
85 views
+50

How prove this geometry inequality $R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2$

Zhautykov Olympiad 2015 problem 6 This links discuss olympiad problem none of student solve it,therefore, meaning this problem is so hard. Question: The area of a convex pentagon $ABCDE$ is $S$, ...
-1
votes
0answers
30 views

Snow White split 3 liters [duplicate]

Snow White split 3 liters of milk into the cup of the Seven Dwarfs. Before the meal, the Dwarfs play a game as follows: Dwarves are first divided all his cup of milk into the cup of the remaining six ...
3
votes
4answers
227 views

Writing numbers as a sum of 2s and 3s

Is there a way to count the number of ways a positive integer N, can be written as a sum of twos and threes? Are there any patterns? Re-arranging the twos and threes are distinct..(makes sense right?? ...
-1
votes
0answers
24 views

What resources are necessary for IMC (International Mathematical Competition among Undergraduate Students)?

I am studying Azerbaijan as a undergraduate student. This year I am going to participate in the IMC, which is organised every year in Bulgaria. But unfortunately there is not a math department in my ...
0
votes
1answer
57 views

How many natural numbers less than $10^{2015}$ have their digits in non-decreasing order?

I am having pretty hard time with combinatorics. Could someone explain me step-by-step how to get to solution? Note: digits are observed from left to right.
0
votes
1answer
26 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 ...
2
votes
1answer
46 views

BMO1 2007/08 Question 3 Geometry Problem [closed]

Let ABC be a triangle, with an obtuse angle at A. Let Q be a point (other than A, B or C ) on the circumcircle of the triangle, on the same side of chord BC as A, and let P be the other end of the ...
0
votes
2answers
34 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
votes
1answer
55 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 ...
0
votes
1answer
20 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 ...
0
votes
1answer
28 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 ...
1
vote
0answers
39 views

Sum of number of rows with max value

Suppose i have an N by N matrix, each element in the matrix my contains 0 or 1, so there are 2^(N*N) different matrix. Let's define the function F that takes a matrix and calculate the sum for each ...
3
votes
1answer
57 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 ...
3
votes
0answers
102 views

If $a_7 = 120$ then find $a_8$. [closed]

Let $a_1, a_2, \dots a_n, a_{n+1}, \dots$ be an increasing sequence of numbers following the recurrence $a_{n+2} = a_{n+1}+ a_n$. If $a_7 = 120$ then find $a_8$. What could be a general process in ...
1
vote
1answer
61 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 $$a^3 \cos A + b^3 \cos B + c^3 \cos C \lt abc.$$ ...
0
votes
1answer
47 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
votes
0answers
48 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
vote
1answer
98 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
votes
2answers
46 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
votes
1answer
64 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 ...
6
votes
2answers
155 views
+100

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 $x^2(x+1)=16(2^z+1)$, ...
-1
votes
2answers
734 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 ...
-1
votes
2answers
57 views

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)
1
vote
2answers
40 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)
2
votes
2answers
30 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
votes
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
vote
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
votes
1answer
57 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
votes
3answers
65 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
votes
1answer
116 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
votes
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
votes
0answers
14 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
votes
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
votes
1answer
92 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} ...
2
votes
1answer
45 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
votes
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
votes
1answer
33 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 ...