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

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0
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1answer
65 views

Using Radical Axis to prove Concurrence

Let $BB',CC'$ be altitudes in $\triangle ABC$, and assume $AB\neq AC$. Let $M$ be the midpoint of $BC$, $H$ the orthocenter of $\triangle ABC$, and define $D$ as the intersection of lines $BC$ and ...
4
votes
1answer
83 views

How do you find the value of $m$ and $n$ if $x+y+z=\frac{m}{\sqrt n}$ given certain conditions on x,y,z?

Problem: Let $x,y$ and $z$ be real numbers satisfying: $$x=\sqrt{y^2 - \frac{1}{16}} + \sqrt{z^2 - \frac{1}{16}}$$ $$y=\sqrt{z^2 - \frac{1}{25}} + \sqrt{x^2 - \frac{1}{25}}$$ $$z=\sqrt{x^2 - ...
3
votes
1answer
45 views

Find the minimum roots of $f'(x)\cdot f'''(x)+(f''(x))^2 =0$ given certain conditions on $f(x)$.

Problem: Let $f(x)$ be a thrice differentiable function satisfying: $$|f(x) - f(4-x)| + |f(4-x)-f(4+x)| = 0, \forall x \in R$$ If $f'(1)=0$, then find the minimum number of roots of $f'(x)\cdot ...
5
votes
1answer
60 views

If $f(0)=f(1)=1$ and $|f(a)-f(b)| < |a-b|$ then $|f(a)-f(b)| < \frac{1}{2}$

Problem: $f$ be a function on $[0,1]$ such that $f(0)=f(1)=1$ and $f(a)-f(b) < |a-b|$ for all $a$ not equal to $b$. Prove that $|f(a)-f(b)| < \frac{1}{2}$. My attempt: Things I observed are ...
0
votes
0answers
40 views

Tiling a Rectangle with integer length horizontal/vertical strips

Source: Bay Area Math Circle 1999 (I think) Let $m$ and $n$ be positive integers. Suppose that a given rectangle can be tiled by a combination of horizontal $1\times m$ strips and vertical $n\times ...
8
votes
2answers
125 views

How do you evaluate $\int_{0}^{\frac{\pi}{2}} \frac{(\sec x)^{\frac{1}{3}}}{(\sec x)^{\frac{1}{3}}+(\tan x)^{\frac{1}{3}}} \, dx ?$

Problem: $$\int_{0}^{\frac{\pi}{2}} \frac{(\sec x)^{\frac{1}{3}}}{(\sec x)^{\frac{1}{3}}+(\tan x)^{\frac{1}{3}}} dx$$ My attempt: I tried applying the property: $\int_{0}^{a} f(x)dx$ = ...
3
votes
0answers
71 views

Math competition for school

I am trying to find a math competition where a 10 year old kid can participate. Can someone suggest a competition in USA?
1
vote
1answer
45 views

Solving a Chessboard problem using the Invariance principle

Problem Statement There is an integer in each square of an 8 x 8 chessboard. In one move, you may choose any 4 x 4 or ...
0
votes
1answer
51 views

Find $a$ such that $p(x)\geq 0$

The problem is: Let $p(x)=x^4-2x^3+ax^2-2x+1$, let a and x be real numbers, find a such that $p(x)\ge0$. My intent to solve it: We see that $(x^2-x+1)^2-3x^2+ax^2\ge0$ then ...
1
vote
3answers
48 views

For which a there exists a non-constant function $a+f(x+y-xy)+f(x)f(y) \leq f(x)+f(y)$

I came across the following problem: Find for which $a \in \mathbb{R}$ there exists a non-constant function $f:(0, 1] \rightarrow \mathbb{R}$ $a+f(x+y-xy)+f(x)f(y) \leq f(x)+f(y)$ for each $x, y \in ...
0
votes
1answer
36 views

How to generalize C from A and B.

I have Two matrix $A=\left( \begin{array}{ccc} \text a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{array} \right)$ and $B=\left( ...
9
votes
1answer
213 views

Which is larger, $\sqrt[2015]{2015!}$ or $\sqrt[2016]{2016!}$?

This was a question in a maths contest, where no calculator was allowed. Also, note that only a (>,< or =) relationship is being searched for and not the value of the numbers itself. Which is ...
5
votes
0answers
60 views

Smallest $n$-digit number $x$ with cyclic permutations multiples of $1989$

Suppose $x=a_1...a_n$, where $a_1...a_n$ are the digits in decimal of $x$ and $x$ is a positive integer. We define $x_1=x$, $x_2=a_na_1...a_{n-1}$, and so on until $x_n=a_2...a_na_1$. Find the ...
9
votes
2answers
132 views

Prove that there is only one sequence which meets the following conditions

Problem statement is as follows: Given $n\geq 2$, prove that you can choose $1 \lt a_1 \lt a_2 \lt ... \lt a_n$ such that $$a_i | 1 + a_1a_2...a_{i-1}a_{i+1}...a_n$$ Prove that if and only if $n \in ...
1
vote
1answer
34 views

Combinatoral Geometry with Distances

The following problem is from Stars of Mathematics Senior P4 Let $S$ be a finite set of points in the plane,situated in general position (any three points in $S$ are not collinear), and let ...
2
votes
1answer
48 views

Centroid of a Triangle and Cosine Law

In $\triangle ABC$, $M$ and $N$ are midpoints of $BC$ and $CA$ respectively such that $AM=14$ and $BN=8$. If $\angle C= 60^{\circ}$, find the length of $AB$. For simplicity sake, let $x=AB$, ...
13
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10answers
1k views

Small Representations of $2016$

It's the new year at least in my timezone, and to welcome it in, I ask for small representations of the number $2016$. Rules: Choose a single decimal digit ($1,2,\dots,9$), and use this chosen digit, ...
5
votes
1answer
53 views

two symmetric functions, when they have only one solution

My Question: For what $y$ is the equation $\log_{y}{x}=y^x$, does there exist only one solution. Some thoughts of mine: What I noticed was that for almost any $a$, both functions $\log_{y}{x}$ ...
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vote
2answers
61 views

Maximizing the sum of the products of endpoints of edges in a graph

Let $G$ be a graph with vertex set $V=\{v_1,v_2\dots v_n\}$ and edge set $E$. Let $f:V\rightarrow \mathbb [0,\infty)$ be a real valued function such that $\sum\limits_{i=1}^n f(v_i)=A$. What is the ...
0
votes
1answer
35 views

Applying invariance principle on a problem on sequence of positive integers

The problem statement: Start with the positive integers 1,...,4n-1. In one move you may replace any two integers by their difference. Prove that an even integer ...
3
votes
1answer
50 views

Left handed or Right handed?

Happy New Year! The following question is abstracted from Singapore Mathematical Olympiad 2015 Junior Round 1. Question 2: Adrian, Billy, Christopher, David and Eric are the five starters of a ...
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2answers
37 views

How do we compare fraction without changing to a similar denominator?

This is Singapore Mathematical Olympiad 2015 Grade 8/Secondary 2 Junior Round 1 Question 1. 1.Among the five numbers, $\frac{5}{9},\frac{4}{7},\frac{3}{5},\frac{6}{11}$ and $\frac{13}{21}$, which ...
2
votes
2answers
117 views

Diophantine Equation with 2017th powers: $a^{2017}+a-2=(a-1)(b^{11})$

This problem stems from a recent student-created olympiad contest. Find all integer (not simply positive) solutions to $a^{2017}+a-2=(a-1)(b^{11})$. My multiple attempts modulo many small primes ...
0
votes
1answer
49 views

Can circles drawn on a sphere (under specific conditions) intersect?

Gave the SAT exam recently and almost aced the Maths section. Almost because there was this one question I couldn't wrap my head around to solve. I don't remember the exact question, but it went ...
5
votes
3answers
109 views

Prove that $\frac{a_1^2}{a_1+b_1}+\cdots+\frac{a_n^2}{a_n+b_n} \geq \frac{1}{2}(a_1+\cdots+a_n).$

Let $a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n$ be positive numbers with $a_1+a_2+\cdots+a_n = b_1+b_2+\cdots+b_n$. $$\text{Prove that} \dfrac{a_1^2}{a_1+b_1}+\cdots+\dfrac{a_n^2}{a_n+b_n} \geq ...
1
vote
2answers
74 views

Proving result in inscribed triangles.

ABC is a triangle inscribed in a circle, and E is the mid-point of the arc subtended by BC different from the arc A on which A lies. If through E a diameter ED is drawn, show that $$\angle ...
1
vote
0answers
42 views

Geometric inequality: $(\text{sum of distances to vertices})>2(\text{sum of distances to sides})$ [closed]

Let $P$ be an interior point of $\triangle ABC$, and let $A^\prime$, $B^\prime$, $C^\prime$ be the projections of $P$ onto respective edge-lines $\overleftrightarrow{BC}$, $\overleftrightarrow{CA}$, ...
9
votes
2answers
222 views

Contest math problem: $\sum_{n=1}^\infty \frac{\{H_n\}}{n^2}$

$$\sum_{n=1}^\infty \frac{\{H_n\}}{n^2}$$ I have managed to prove that it converges, but am having trouble with a closed form. This came from a school contest from last year, but can't really figure ...
14
votes
1answer
180 views

Prove that ${x^7-1 \over x-1}=y^5-1$ has no integer solutions

I want to show that $${x^7-1 \over x-1}=y^5-1$$ cannot have any integer solutions. The only observation I have made so far is that the left hand side is the $7$th cyclotomic polynomial $$\Phi_7(x)= ...
4
votes
1answer
76 views

Inequality, Cauchy Schwarz and Schur

For $a,b, c>0$, prove that $$\frac{a^3}{a^3+b^3+abc}+\frac{b^3}{b^3+c^3+abc}+\frac{c^3}{c^3+a^3+abc}\geq 1$$ I tried the following $$\sum_{cyc}\frac{a^3}{a^3+b^3+abc}\cdot ...
-1
votes
1answer
25 views

Chinese Remainder Theorem for infinite system

I have a trouble understanding p.7 of the following article: http://www.edb.gov.hk/attachment/en/curriculum-development/kla/ma/IMO/Nov20155-4online.pdf which says the folllowing: By the same ...
5
votes
1answer
108 views

Inequality olympiad

For all positive numbers $a,b,c$, prove that $$\frac{a^3}{b^2-bc+c^2}+\frac{b^3}{a^2-ac+c^2}+\frac{c^3}{a^2-ab+b^2}\geq 3 \frac{(ab+bc+ac)}{a+b+c}$$ Note that both side are homogeneous of degree 1, ...
2
votes
1answer
62 views

Prove that $f \left(\lambda x + (1- \lambda )x' , \lambda y +(1- \lambda )y' \right) > \min \{f(x,y), f(x',y')\}$

Let $f(x,y)=xy$ where $x,y\geq 0$. Prove that the function $f$ satisfies the following property: $$f \left(\lambda x + (1- \lambda )x' , \lambda y +(1- \lambda )y' \right) \geq \min \{f(x,y), ...
0
votes
1answer
60 views

Calculating the area of a triplet of circles.

I have an image of the problem which is quite self-explanatory. Any ideas?
20
votes
5answers
362 views

How to solve $\sqrt {1+\sqrt {4+\sqrt {16+\sqrt {64+\sqrt {256\ldots }}}}}$

How to solve this equation? $$x=\sqrt {1+\sqrt {4+\sqrt {16+\sqrt {64+\sqrt {256\ldots }}}}}.$$ Answer: $x=2$
3
votes
0answers
37 views

Three-gap problem, easy version.

Let $N$ be a positive integer and $\theta$ an angle in $(0, 2\pi)$. Consider the map$$f: \{0, 1, 2, \dots, N-1, N\} \to \text{unit circle}, \text{ }f(k) = k\theta \text{ }(\text{mod } 2\pi).$$Show ...
4
votes
1answer
111 views

Inequality exercise (olympiad)

For positive $a$, $b$, $c$ such that $abc=1$. Show that $$(ab+bc+ca)(a+b+c)+6\geq 5(a+b+c).$$ From the LHS, using AM-GM, we see that $(ab+bc+ca)(a+b+c)+6\geq 3(abc)^{2/3}3(abc)^{1/3}+6=15$. But ...
4
votes
1answer
107 views

If $a,b,c>0$ and $a+b+c=1$, prove $\frac{a}{a+bc}+\frac{b}{b+ca}+\frac{\sqrt{abc}}{a+ba}\le 1+\frac{3\sqrt{3}}{4}$

If $a,b,c>0$ and $a+b+c=1$, then prove $$\frac{a}{a+bc}+\frac{b}{b+ca}+\frac{\sqrt{abc}}{a+ba}\le 1+\frac{3\sqrt{3}}{4}$$
0
votes
3answers
68 views

palindrome 4th grade

This is a 4th grade question. A palindrome is an integer number that does not change when read backwards. E.g. 123321 is a 6 digit palindrome. How many 9-digit palindromes are there that use only the ...
10
votes
2answers
527 views

Sum of factors of a huge number.

I recently appeared in a math olympiad and it had this one question which had me stumped. This was a few weeks back and I have been looking for a way to find its answer ever since, but with no ...
4
votes
2answers
186 views

A contest math integral: $\int_1^\infty \frac{\text{d}x}{\pi^{nx}-1}$

My school holds a math contest that has problems that vary level to level. Nobody managed to solve this particular one: $$\int_1^\infty \frac{\text{d}x}{\pi^{nx}-1}$$ In terms of $n$ I was ...
4
votes
3answers
70 views

Proving $\cot { A+\cot { B+\cot { C=\frac { { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } }{ 4K } } } } $ [closed]

For any acute $\triangle ABC$, prove that $\cot { A+\cot { B+\cot { C=\frac { { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } }{ 4K } } } } $, where $K$ is the area of $\triangle ABC$. Unfortunately I'm ...
5
votes
1answer
33 views

Manipulation with strings riddle.

Starting with the "string" $PI$, can I or not transform it into the "string" $PK$ by applying the following rules (each rule can be used any number of times, in any order, and $x$ and $y$ represents a ...
0
votes
0answers
34 views

Number of ways to choose colored points equality

Let $n$ be a positive integer and $S$ the set of points $(x,y)$ in the plane, where $x$ and $y$ are non-negative integers such that $x +y<n$. The points of $S$ are colored in red and blue so that ...
6
votes
1answer
175 views

Number of solutions to exceedingly contrived congruence.

Let $a$ be the number of solutions to$$x^{2011}-96x^{728}-x^{24}+67 \equiv y^{2011}+12718253987182795172957215781251235234235y \pmod{2^{57885161}-1}$$where $x$ and $y$ are integers in-between $0$ and ...
1
vote
1answer
64 views

How do you evaluate this summation: $S=\sum\limits_{r=0}^{15} (-1)^r \frac{\binom{15}{r}}{\binom{r+3}{r}}$

Find S: $$S=\sum_{r=0}^{15} (-1)^r \frac{\binom{15}{r}}{\binom{r+3}{r}}$$ My attempt: I tried writing the summation as: $$S=3!(15!)\sum_{r=0}^{15} (-1)^r \frac{1}{(15-r)!(r+3)!}$$ and tried to ...
7
votes
2answers
109 views

Finding the maximum value of $ab+ac+ad+bc+bd+3cd$

If $a,b,c,d>0$ satisfy the condition ${ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }+{ d }^{ 2 }=1$, find the maximum value of $ab+ac+ad+bc+bd+3cd$. I'm not progress in this inequality problem. Please ...
1
vote
0answers
104 views

Strategies for solving rational Diophantine equations

Are there any strategies for solving Diophantine equations where the solutions can be any rational number, not just an integer, besides substituting $x=p/q$ and $y=r/s$, with $p,q,r,s$ integers with ...
1
vote
1answer
54 views

Writing a summation as the ratio of polynomial with integer coefficients

Write the sum $\sum _{ k=0 }^{ n }{ \frac { { (-1) }^{ k }\left( \begin{matrix} n \\ k \end{matrix} \right) }{ { k }^{ 3 }+9{ k }^{ 2 }+26k+24 } } $ in the form $\frac { p(n) }{ q(n) }$, where ...
2
votes
1answer
74 views

Coloring diagonals in a regular polygon

Each side and diagonal of a regular $n$-gon ($n\geq 3$) is colored blue or green. A move consists of choosing a vertex and switching the color of each segment incident to that vertex (from blue to ...