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

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Proving that four lines (which are perpendicular bisectors of chords) meet a point

In the diagram above, each of the four lines is a perpendicular bisector of one of the circles' chord. There are two pairs of circles which touch each other, and of course, as shown in the diagram, ...
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2answers
37 views

Proving sum of product forms a pattern in n * nnnnnn…

I am consider a problem regarding numbers which are, in decimal, one digit repeated - for instance, $88888888$ is such a number. In particular, I am looking at the following problem: The sum of ...
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1answer
37 views

What is the largest prime number in the denominator of a fraction that creates a decimal that repeats every 4 digits?

I was studying a Target question for Math League competitions, and after a few hours of pondering, I finally came up with the following method of solving the mentioned problem: For any decimal, it is ...
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1answer
20 views

Showing there exists a sequence that majorizes another

The exact quantity of gas needed for a car to complete a single loop around a track is distrubuted among $n$ containers placed along the track. Show that there exists a point from which the car can ...
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1answer
53 views

How do you find the sum: $\sum_{r=1}^6 \tan^2\left(\frac{r \pi}{n}\right)$

How do you find the sum: $$\sum_{r=1}^6 \tan^2\left(\frac{r \pi}{n}\right)$$ I managed to solve this question using complex numbers so I thought I'd share the solution. If you know of any better ...
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2answers
107 views

How do you find the value of $\sum_{r=0}^{44} \tan^2(2r+1)$?

Problem: Find the value of $$\sum_{r=0}^{44} \tan^2(2r+1)$$ Note: The angles here are in degrees. I don't know how to solve this question because trigonometric simplifications didn't get me ...
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1answer
49 views

How do you find the value of $N$ given $P(N) = N+51$ and other information about the polynomial $P(x)$?

Problem: Let $P(x)$ be a polynomial with integer coefficients such that $P(21)=17$, $P(32)=-247$, $P(37)=33$. If $P(N) = N + 51$ for some positive integer $N$, then find $N$. I can't think of ...
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3answers
83 views

Computing $2016$ using basic operations on the fewest integers, in sequence

Using the operators $$+,-,\div,\times,\exp,(,),!$$ what is the least $n$ to come up with the number $2016$ using the sequence of numbers $1,2,3,\ldots,n$ in that order. You cannot combine numbers, so ...
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3answers
53 views

Finding the number of sequences with $0 \leq a_m \leq 3m$

Problem: Let $\alpha, \beta$ be non-negative numbers. Suppose the number of strictly increasing sequences $a_0, a_1, a_2 \cdots a_{2014}$ satisfying $0 \leq 3m$ is $2^{\alpha}(2\beta+1)$. Find ...
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68 views

Labelling the edges of a cube with {1, 2, 3,…,12}

I did the following problem: a) Is it possible to label the edges of a cube by $1, 2, \cdots 12$ (using each number only once) so that at each vertex, the labels of the edges leaving that vertex ...
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1answer
51 views

Determine all positive rational numbers $r \neq 1$ such that $r^{\frac{1}{r-1}}$ is rational?

Here's what I've got so far: Let $r = \frac{a}{b}$, where $a$ and $b$ are integers. We then have $$r^{\frac{1}{r-1}} = \frac{a^{\frac{b}{a-b}}}{b^{\frac{b}{a-b}}}$$ Clearly, $a-b=1$ and $a-b=-1$ ...
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1answer
445 views

Powerball Mass Quickpick Odds

The odds of picking the right powerball numbers for the jackpot are 1:292,201,338. Right now, because the powerball has reached 1.4 billion dollars, many people are claiming that you could buy every ...
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1answer
42 views

Determine what when multiplied with $180$ gives a perfect cube

Recently, at a math competition, I was given the following question: Determine the smallest number that gives a perfect cube when multiplied by $180$ . I had thirty seconds to solve this question and ...
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1answer
81 views

Match off points into $N$ red/blue pairs with straight lines connecting pairs, so that none of lines we draw intersect

Suppose we are given $2N$ points in the plane (we may assume that no $3$ are collinear). Assume that $N$ of these points are colored red, and $N$ points are colored blue. Can we match off the points ...
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2answers
83 views

Showing two lines on a triangle coincide

Let $M$ be the midpoint of (the smaller) arc $BC$ in circumcircle of triangle $ABC$. Suppose that the altitude drawn from $A$ intersects the circle at $N$. Draw two lines through circumcenter $O$ of ...
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2answers
67 views

Find all positive integers $n$ such that $n+2008$ divides $n^2 + 2008$ and $n+2009$ divides $n^2 + 2009$

I wrote $$ \begin{align} n^2 + 2008 &= (n+2008)^2 - 2 \cdot 2008n - 2008^2 + 2008 \\ &= (n+2008)^2 - 2 \cdot 2008(n+2008) + 2008^2 + 2008 \\ &= (n+2008)^2 - 2 \cdot 2008(n+2008) + 2008 ...
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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 ...
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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 - ...
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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 ...
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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 ...
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38 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 ...
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2answers
122 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: $\int_{0}^{a} f(x)dx$ = ...
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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?
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1answer
42 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 ...
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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 ...
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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 ...
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1answer
35 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
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1answer
202 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 ...
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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 ...
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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 ...
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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 ...
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1answer
45 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 $x=AB$, ...
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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, ...
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1answer
51 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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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 ...
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1answer
33 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 ...
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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 ...
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2answers
116 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 ...
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1answer
47 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 ...
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3answers
107 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 ...
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2answers
73 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 ...
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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
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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 ...
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1answer
149 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)= ...
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1answer
75 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 ...
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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
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1answer
106 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
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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?