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

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22
votes
1answer
285 views

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 discusses the olympiad problem which none of students could solve , meaning it is very hard. Question: The area of a convex pentagon $ABCDE$ is $S$, ...
1
vote
1answer
654 views

Given n girls and boys how many ways are there to arrange them such that any two boys have atleast 'k' girls between them.

Professor X wants to position $1 \leq N \leq 100,000$ girls and boys in a single row to present at the annual fair. Professor has observed that the boys have been quite pugnacious lately; if two ...
5
votes
0answers
103 views
+50

How prove this systems-equation has least two postive integers solution

Show that: for any $k\ge 100,(k\in N^{+})$, there exsit $p\in N^{+}$, such $$\begin{cases} a+b+c=k\\ abc=p\\ a>b>c \end{cases}$$ has at least two postive integers solution $(a,b,c)$ ...
1
vote
1answer
84 views

A math contest question related to Ramsey numbers

In a group of 17 nations, any two nations are either mutual friends, mutual enemies, or neutral to each other. Show that there is a subgroup of 3 or more nations such that any two nations in the ...
5
votes
3answers
428 views

Prove $a^ab^bc^c\ge (abc)^{\frac{a+b+c}3}$ for positive numbers.

Prove that the following inequality holds $$a^a b^b c^c\ge (abc)^{\frac{a+b+c}{3}}$$ if $a,b,c$ are positive. I'm not sure how to handle these kinds of powers. Are there any "famous" but not ...
3
votes
1answer
53 views

Find all odd positive integers $n$ greater than $1$ such that for any coprime divisors …

Find all odd positive integers $n$ greater than $1$ such that for any coprime divisors $a$ and $b$ of $n$, the number $a + b − 1$ is also a divisor of $n$. This was taken from the Russian ...
1
vote
3answers
45 views

How to apply Chinese Remainder Theorem for $x$

If: $$x \equiv 0 \pmod{17}$$ and $$x \equiv -1 \pmod{9}$$ Then how is: $$x \equiv 17 \pmod{153}$$ I get that since $\gcd(9, 17) = 153 $ the solution will be $\pmod{153}$ but how do you get the $17 ...
2
votes
0answers
39 views

Assume that for any pair of vertices $P_i$ and $P_j$ , there exists a vertex $P_k$ of the polygon such that $∠P_i P_k P_j = \pi/3.$

Let $P_1 P_2 \dots P_n$ be a convex polygon in the plane. Assume that for any pair of vertices $P_i$ and $P_j$ , there exists a vertex $P_k$ of the polygon such that $∠P_i P_k P_j = \pi/3.$ Show ...
0
votes
3answers
41 views

Number of times $2^k$ appears in factorial

For what $n$ does: $2^n | 19!18!...1!$? I checked how many times $2^1$ appears: It appears in, $2!, 3!, 4!... 19!$ meaning, $2^{18}$ I checked how many times $2^2 = 4$ appears: It appears in, ...
1
vote
1answer
36 views

Complex Number - root

The complex numbers $z$ and $w$ satisfy $z^{13} = w$, $w^{11} = z$, and the imaginary part of $z$ is $\sin\left(\frac{m\pi}n\right)$ for relatively prime positive integers $m$ and $n$ with $m < ...
0
votes
2answers
35 views

Strategy to find the most money to use.

As a reward for a week of good behavior, Tommy was given 7 dollars to spend at the canteen. By the time Tommy got to the canteen, there were only chocolate bars, meat pies and pizza pieces left. The ...
2
votes
1answer
19 views

Invariance Principle Question

A circle is divided into six sectors. Then the numbers $1, 0, 1, 0, 0, 0$ are written into the sectors (counter-clockwise say). You may increase two neighboring numbers by $1$. Is it possible to ...
4
votes
2answers
67 views

If $\prod\limits_{k=0}^5(5^{2^k}+6^{2^k})=6^x-5^y$, what is the value of $x-y$?

I think this might be a contest math question, so I'm tagging it as such. I don't know how to do something like this by hand (or if it's even possible, though I would presume it is if it's from a ...
8
votes
5answers
2k views

A Math Olympiad question regarding Geometry

A little bit of a backstory (you may skip this if you want): My high school math teacher knows that I love math, but he also knows that I usually drift off during my classes, perhaps because it's too ...
37
votes
2answers
1k views

Prove this inequality with $xyz\le 1$

if $x,y,z>0$ and $\color{red}{xyz\le 1}$, show that $$\color{blue}{\dfrac{x^2-x+1}{x^2+y^2+1}+\dfrac{y^2-y+1}{y^2+z^2+1} +\dfrac{z^2-z+1}{z^2+x^2+1}\ge 1}$$
0
votes
0answers
24 views

BMO1 2007/08 Question 5 Geometry Problem

$5.$ Let $P$ be an internal point of triangle $ABC$. The line through $P$ parallel to $AB$ meets $BC$ at $L$, the line through $P$ parallel to $BC$ meets $CA$ at $M$, and the line through $P$ parallel ...
2
votes
2answers
83 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 ...
1
vote
2answers
65 views

Analog clock with same hands - sometimes one can't tell time [duplicate]

There is an accurate analog clock, however both hands are the same size and shape. How many moments during a day a person can not conclude current time from the position of the hands? This is from a ...
1
vote
1answer
35 views

Real Numbers are Roots $r, s$.

Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$, and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$. Find the sum of all possible values of $|b|$. Using Vieta's Formulas, $r+s+x_1$ $=0$ ...
8
votes
2answers
50 views

Given $n$ points, the difference of $2$ of them is $1/n$ close to an integer

From today's ENS Ulm Math D exam Let $x_1,\ldots,x_n$ be real numbers Prove there exists $i\neq j $ and $h\in \mathbb Z$ such that $|x_i-x_j-h|\leq \frac{1}{n}$ I tried contradiction and ...
0
votes
1answer
99 views

Solve the system $ x \lfloor y \rfloor = 7 $ and $ y \lfloor x \rfloor = 8 $.

Solve the following system for $ x,y \in \mathbb{R} $: \begin{align} x \lfloor y \rfloor & = 7, \\ y \lfloor x \rfloor & = 8. \end{align} It could be reducing to one variable, but it is ...
15
votes
4answers
200 views

A Triangle Determinant

How do we prove, without actually expanding, that $$\begin{vmatrix} \sin {2A}& \sin {C}& \sin {B}\\ \sin{C}& \sin{2B}& \sin {A}\\ \sin{B}& \sin{A}& \sin{2C} \end{vmatrix}=0$$ ...
1
vote
3answers
564 views

How to find Bitwise AND of all numbers for a given range?

How can I find Bitwise AND of all numbers for a given range say from A to B, including both? I found a beautiful answer for finding XOR for such range. http://stackoverflow.com/a/10670524/2046703How ...
0
votes
1answer
68 views

incorrect rejection of a true null hypothesis? [closed]

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 ...
0
votes
0answers
33 views

Challenge Problems [closed]

This question might be better fit for meta, but how might I find a list of challenge problems similar to the following. In addition this question may have already been asked. ...
1
vote
3answers
313 views

Geometry problem on circles from a competition

Triangle $\triangle ABC$ is an equilateral triangle whose side is $16$. A circle meets the sides of the triangle at $6$ points: it intersects $AC$ at $G$ and $F$ and $|AG|=2$, $|GF|=13$, $|FC|=1$. ...
9
votes
2answers
288 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 $x^2(x+1)=16(2^z+1)$, ...
0
votes
0answers
17 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 ...
0
votes
1answer
34 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 ...
2
votes
2answers
107 views

A question about 4 concyclic points

In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles ...
5
votes
2answers
125 views

Three circles having centres on the three sides of a triangle

NOTE: I would appreciate it if you provided a hint and not the whole solution. BdMO 2014 Nationals: In acute angled triangle ABC, considering a portion of side BC as diameter a circle is drawn ...
9
votes
3answers
148 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
71 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 ...
2
votes
1answer
22 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
22 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
34 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
56 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
27 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 ...
2
votes
1answer
34 views

Reflection to get within convex polygon

Let $P$ be a convex polygon, and let $A_1$ be a point on the same plane as $P$. Prove that we can find an integer $n$, and points $A_2,A_3,\ldots,A_n$, such that $A_{i+1}$ is a reflection of $A_i$ ...
-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
236 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?? ...
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 ...
-1
votes
0answers
28 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
58 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.
-1
votes
2answers
748 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 ...
12
votes
5answers
240 views

Show that $({\sqrt{2}\!+\!1})^{1/n} \!+ ({\sqrt{2}\!-\!1})^{1/n}\!\not\in\mathbb Q$

How could we prove that for every positive integer $n$, the number $$({\sqrt{2}+1})^{1/n} + ({\sqrt{2}-1})^{1/n}$$ is irrational? I think it could be done inductively from a more general ...
2
votes
1answer
48 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
35 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
56 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 ...
1
vote
1answer
100 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 ...