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Questions tagged [contest-math]

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

4
votes
4answers
61 views

If $x_1,x_2,\ldots,x_n$ are the roots for $1+x+x^2+\ldots+x^n=0$, find the value of $\frac{1}{x_1-1}+\frac{1}{x_2-1}+\ldots+\frac{1}{x_n-1}$

Let $x_1,x_2,\ldots,x_n$ be the roots for $1+x+x^2+\ldots+x^n=0$. Find the value of $$P(1)=\frac{1}{x_1-1}+\frac{1}{x_2-1}+\ldots+\frac{1}{x_n-1}$$ Source: IME entrance exam (Military Institute of ...
0
votes
0answers
11 views

Game-theory (Nash equilibrium) Vaccination Game

I'm trying to solve this question and I would like to get some help on solving it... Consider the social network of the population. The network is modeled by a graph G = (V, E) where each node ...
0
votes
4answers
49 views

Compare two fractions [on hold]

How to compare $\frac{\sin{2016°}}{\sin{2017°}}$ and $\frac{\sin{2018°}}{\sin{2019°}}$?
2
votes
3answers
29 views

System of linear equation based on Sum of the Digits of a number

My friend gave me this problem which I have not been able to solve: Let $S(k)$ denote the sum of the digits of $k$. Find all natural solutions $(x,y,z)$ to the equations: $S(x) + S(y) + S(z) = ...
-1
votes
2answers
37 views

$f(x) * f(y) = f(x-y) + f(x+y)$ [on hold]

It's a question from a Math Olympiad in Azerbaijan. There is a function $f :Z \rightarrow Z$ and $f(x) * f(y) = f(x-y) + f(x+y)$. If $f(1) = 1$, find $f(100)$
0
votes
3answers
33 views

The meaning of $(\forall M \in \mathbb{R} )( \exists B \in \mathbb{R} )( \forall x>B )( f(x)<M )$

I'm trying to understand the meaning of this: $$(\forall M \in \mathbb{R} )( \exists B \in \mathbb{R} )( \forall x>B )( f(x)<M )$$ the only thing I could figure out that if $x\rightarrow \...
-1
votes
0answers
37 views

Year to Prepare for the Putnam Exam? [on hold]

I will be taking the Putnam Exam next December and want to prepare well for it. I have never really had competition experience although I have spent time doing brain teasers and math puzzles. ...
4
votes
1answer
84 views

Coloring grid points with two colors

Let $S$ be a set of finite many grid points (points in the coordinate system with integer coordinates). Is it always possible to color them with two colors, red and blue, such that in each vertical ...
1
vote
3answers
87 views

Find all positive integers $a$ and $b$ such that $(1 + a)(8 + b)(a + b) = 27ab$.

Here's the problem I'm having difficulties with: Find all positive integers $a$ and $b$ such that $$(1 + a)(8 + b)(a + b) = 27ab\,.$$ Does anyone have an idea how to do this? Any detailed solution ...
4
votes
1answer
103 views
+50

Did I apply the Ceva's theorem correctly to this problem?

I need to confirm the following solution. I'm making a mistake somewhere. But I can't find the error. I apply the trigonometric form of the Ceva's theorem: $$\frac{\sin \angle 3}{\sin \angle 4}× \...
-3
votes
2answers
83 views

Is there an $11$-element circular permutation of $\{1,2,…,12\}$ with all $|a_i-a_{i+1}|$ distinct?

Can you choose $11$ different numbers among them so that the numbers $|a_1-a_2|, |a_2-a_3|,\ldots,|a_{10}-a_{11}|,|a_{11}-a_{1}|$ are all different. The smartest thing that my dumbest mind could ...
5
votes
1answer
104 views

find closed form for following double integral containing radicals

question: To prove : $\displaystyle\int_{0}^{1}\displaystyle \int_{0}^{1}\dfrac{dxdy}{\sqrt{1-x^2}{\sqrt{1-y^2}}{\sqrt{4x^2+y^2-x^2y^2}}}=\dfrac{3\left(\Gamma{\dfrac{1}{3}}\right)^6}{2^{\dfrac{17}{3}...
2
votes
1answer
48 views

a basic definite integration and its result used in evaluating limit

Question : $\mathbf\Omega(n)=\displaystyle\int _{0}^{2\pi}\log(n^2-2n\cos t+1)dt\ ,\ n\geq1 $ then , find : $\displaystyle \lim_{n \to \infty} \left(1+\dfrac{\mathbf \Omega(n)}{4\pi}\right)^{\log(n+...
1
vote
1answer
26 views

Prove that the $AX$ bisects $BC$ in $\triangle ABC$ where $X$ is the intersection of a side $EF$ of the contact triangle $DEF$ and $ID$

In Evan Chen's Euclidean Geometry in Mathematical Olympiads, the title of the question was reduced to proving $AX$ bisecting (using homothety) in Problem 4.16 ( Page $63$): Prove that $XB'=XC'$ ...
13
votes
2answers
256 views
+50

if such three condition find the $\sin{\angle OPA}$

Convex quadrilateral $ABCD$,and circumcenter is $O$,if Point $P$ lie on sides $AD$,and such $$\dfrac{AP}{PD}=\dfrac{8}{5},~~PA+PB=3AB,~~PB+PC=2BC,~~PC+PD=\dfrac{3}{2}CD$$find $\sin{\angle OPA}$ I ...
6
votes
3answers
340 views

Synthetic solution to this geometry problem?

Consider the following diagram. In the isosceles triangle $\triangle ABC$ with $AB=AC$, it is given that $BC=2$. Two points $M,N$ lie on $AB,AC$ respectively so that $AM=NC$. Prove: $MN$ is at least $...
0
votes
1answer
42 views

Maximum number of kings on the chessboard subject to some rules

The chess king moves one square in any direction (horizontally, vertically, or diagonally). The goal is to place as many king as possible on an r×c board subject to the following two conditions: ...
-1
votes
2answers
62 views

Is it possible to find a closed form for $x$?

To solve the problem, I followed the following steps: Is it possible to find a closed form for $x$? $$\frac{\sin(x)}{\sin(\beta-x)}=\frac{\sin(\alpha)\,\sin(\theta-\gamma)}{\sin(\gamma)\,\sin(\...
6
votes
3answers
333 views

A curious geometry problem: Find the $\angle OBC$

I can not find a Method for to solve this geometry problem.I don't even know how to start.In fact, I didn't want to add my nonsensical attempts. I looked for a similar question to this question (...
0
votes
1answer
37 views

Triple refinement of an inequality

We have the following theorem : Let $a,b,c$ be real positive numbers such that $2b\geq a+c$ then we have : $$A\geq B\geq C \geq D \geq E$$ With : $$A=3\big(\frac{\sum_{cyc}\frac{a^3}{13a^2+5b^...
0
votes
1answer
72 views

If $a+b+c=\frac 1a +\frac 1b +\frac 1c$ then prove that $ab+bc+ca \geq 3$

Let $a,b,c$ be positive real numbers. $a+b+c=\frac 1a +\frac 1b +\frac 1c$ then prove that $ab+bc+ca \geq 3$ Using CS Inequality $(a+b+c)(\frac 1a +\frac 1b +\frac 1c) >9$. then by hyp $\frac 1a +\...
3
votes
1answer
65 views

Prove that if $a_k \in [0, 1] $ then $\frac {1}{1+a_1} + \frac {1}{1+a_2} + \cdots +\frac {1}{1+a_n} \le \frac{n}{1+ \sqrt[n]{a_1a_2\cdots a_n}}$.

One of my friends gave me this problem: If $a_k \in [0, 1]$, Prove that: $$\frac {1}{1+a_1} + \frac {1}{1+a_2} + \cdots +\frac {1}{1+a_n} \le \frac{n}{1+ \sqrt[n]{a_1a_2\cdots a_n}}$$ I have ...
2
votes
2answers
68 views

Maximising the area of a triangle when the 3 sides are at most 2, 3, 4 [closed]

This is question 5 from the 2012 BMO1, please can someone provide a full solution? There is an online video solution but is incomplete as it assumes that the area of a triangle is maximised when two ...
2
votes
0answers
112 views

Technique behind solving 4 = $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$

The question: 4 = $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$ Where I have to find the minimum values for a, b and c and they have to be positive and whole? I'm slightly confused by all the ...
0
votes
1answer
57 views

Inequalities - Tangent line trick

I will first state the "trick": we fix $a=\frac{a_1+a_2+...+a_n}{n}, \ $If $f$ is not convex we can sometimes prove:$$f(x)\ge f(a)+f'(a)(x-a) $$ If this manages to hold for all x, then summing up ...
0
votes
1answer
133 views

frequencies comparison

I have a rather quiz question (sorry if this a wrong stack to ask such questions). A propeller with 3 blades makes exactly 24 spins in 1 second. Camera, that is filming it, takes 54 frames in 1 ...
0
votes
1answer
24 views

What would be the another possible answer?

It is easy to guess that 51 is the missing number. The entries in the last row are obtained by taking multiplication of entries in third and second row and subtracting entries in first row in the same ...
3
votes
3answers
125 views

Showing that $a^2+b^2+c^2+d^2+e^2+65=abcde$ has integer solutions greater than $2018$?

This question comes from a Chinese high school olympiad training program. It seems remarkably more difficult (and indeed, interesting!) than all other problems arising in the same program, especially ...
3
votes
1answer
95 views

Don't know where to use the hypothesis

Let a be a real number such that $|a| > 2 $. Prove that if $a^{4}-4a^{2}+2$ and $a^5-5a^3+5a$ are rational numbers, then $a$ is a rational number as well. My attempt is the following. $a^5-5a^3+5a ...
5
votes
2answers
67 views

Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $2n+2001≤f(f(n))+f(n)≤2n+2002$.

Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $$2n+2001≤f(f(n))+f(n)≤2n+2002\,.$$ I don't know where to start as in is there a ...
25
votes
2answers
4k views

Prove that there exists a triangle which can be cut into 2005 congruent triangles.

I thought maybe we can start with congruent triangle and try to cut it similar to how we create a Sierpinski's Triangle? However, the number of smaller triangles we get is a power of $4$ so it does ...
2
votes
4answers
450 views

Alternative proof that if $a,b,c \in \mathbb{R}$ and $(a+b+c)c<0$ then $b^2-4ac>0$?

A problem from a Moscow Olympiad states : let $a,b,c$ be real numbers such that $(a+b+c)c<0$. Show that $b^2-4ac>0$. There is a well known proof of this applying the IVT on the polynomial $f(x)...
3
votes
3answers
64 views

Find all real numbers $x,y,z\in [0,1]^3$ such that $(x^2+y^2)\sqrt{1-z^2}\ge z$…

Such that: $$(x^2+y^2)\sqrt{1-z^2}\ge z$$ and $$(z^2+y^2)\sqrt{1-x^2}\ge x$$ and $$(x^2+z^2)\sqrt{1-y^2}\ge y$$ Since $x,y,z$ $\in ]0,1[^3$ then , there are some real numbers $a,b,c$ such that $\...
1
vote
1answer
61 views

An inequality to finalize a proof

Related to this Inequality $\sum\limits_{cyc}\frac{a^3}{13a^2+5b^2}\geq\frac{a+b+c}{18}$ and my second answer I have to prove this : Let $a,b,c$ be real positive numbers then we have : $$\sum_{cyc}\...
3
votes
1answer
47 views

$\sum_{i=1}^{\infty}\frac{1}{a_i}$ converges.,prove that $\lim_{n \to \infty}\frac{b_n}{n}=0$.

Help: Let $a_1,a_2,...$ be positive integers such that $\sum_{i=1}^{\infty}\frac{1}{a_i}$ converges. For each $n$, let $b_n$ denote the number of positive integers $i$ for which $a_i \leq n$. prove ...
3
votes
1answer
73 views

Expected profit of my simple board game

How to play: Use 1 host and at least 1 player Each player has to toss fair six-sided dice to go to goal. If the player is at the 35th cell and tosses 2 or more, he can go to goal aa same as he ...
2
votes
1answer
43 views

Circular Geometry

A circle of radius $1$ is internally tangent to two circles of radius $2$ at points $A$ and $B$, where $AB$ is a diameter of the smaller circle. What is the area of the region, shaded in the picture, ...
0
votes
1answer
46 views

Number of increasing sequences alternating between odd and even involving only integers from the set $[1, 2, 3, …, n]$

I am new to this forum so do correct me if I am doing anything wrong. (Problem 6 of British Mathematics Olympiad1 - 1994) An increasing sequence of integers is said to be alternating if it starts ...
5
votes
1answer
99 views

$x_n = \left[1^{1^p}2^{2^p}\cdots (n+1)^{(n+1)^p}\right]-\left[1^{1^p}2^{2^p}\cdots n^{n^p}\right]$ converge?

This problem comes from the Titu Andreescu's book Problems in Real Analysis - Chapter 1, page 9. Let $p$ be a nonnegative real number. Study the convergence of the sequence $$x_n = \left[1^{1^p}2^{2^...
9
votes
1answer
132 views

Prove that $f(n)\ge 2^{n-1}$

This is from a Brazilian math contest for college students (OBMU): Let $f: (0,+\infty) \to (0,+\infty)$ be a infinitely differentiable function such that For all positive integer $k$ and positive ...
1
vote
1answer
54 views

Cover the plane with closed disks

Help with this Putname problem: Is it possible to find an infinite sequence of closed disks $D_1,D_2,...$ in the plane with centers $c_1,c_2,...$ such that a) the $c_1$ have no limit point in the ...
0
votes
0answers
33 views

Prove that the polynomial is $g(x,y)(x^2 + y^2 -1)^2 + c$

This is from a Brazilian math contest for college students (OBMU): Let $f(x,y)$ be a polynomial in two real variables such that the polynomials $$\frac{\partial f}{\partial x}(x,y)$$ $$\frac{\...
5
votes
1answer
124 views

Proving that there is an element common to all $35$ sets given certain set restrictions

Consider the $35$ sets $A_1,A_2,\dots,A_{35}$ such that $|A_i|=27$ for all $1\leq i \leq 35$, and every triplet of sets have one exactly one element in common to all three. Prove that there is at ...
3
votes
2answers
319 views

Given a three digit number $n$, let $f(n)$ be the sum of digits of $n$, their products in pairs, and the product of all digits. When does $n=f(n)$?

This is my first time posting so do correct me if I am doing anything wrong. Please help me with this math problem from the British Maths Olympiad (1994 British Maths Olympiad1 Q1 Number Theory). ...
0
votes
1answer
20 views

Symmetric Graphing

What is the graph of $x^2+y^2=|x|+|y|$. I tried solving this but I have don't understand how we know that the graph is symmetric to the axes. I read the solutions to this Area enclosed by the curve $...
0
votes
0answers
16 views

In the quadrilateral ABCD , the points M and N are the centers of opposite sides AB and DC. [duplicate]

In the quadrilateral ABCD , the points M and N are the centers of opposite sides AB and DC, let MD and AN intersect each other at the point Q and let MC and BN intersect each other at the point R. ...
1
vote
1answer
52 views

Three sides of a regular triangle is bicolored, are there three points with the same color forming a rectangular triangle?

The sides of a regular triangle $\triangle_1=ABC$ is bicolored(red, and blue), Do there exist three vertices on the perimeter of $\triangle_1$ three monochromatic vertices forming the corners of a ...
1
vote
2answers
93 views

First International Olympiad, 1959

The problem is: Prove that $\dfrac{21n+4}{14n+3}$ is irreducible for every natural number $n$. Can anyone please give me a hint?
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votes
1answer
50 views

Refinement of a strong inequality

It's related to this If $a+b+c=abc$ then $\sum\limits_{cyc}\frac{1}{7a+b}\leq\frac{\sqrt3}{8}$ . I make a little refinement wich could be usefull to prove the original one . Let $a,b,c$ be real ...
2
votes
0answers
46 views

Show that a certain set of points lies on a straight line [duplicate]

Let $S$ be an infinite set of points in the plane. The distance between two points of $S$ is integral. Prove that $S$ is a subset of a straight line. Here is my attempt: Suppose not, then for each ...