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

learn more… | top users | synonyms (2)

4
votes
3answers
49 views

Prove the triangle is equilateral

HINTS ONLY please. This is very confusing right off the bat. My guess was that we show the angle $C, M, N$ are all $60^{\text{o}}.$ But I am having difficulty doing as as none of the givens have ...
0
votes
1answer
18 views

Difference between the number of lucky numbers and medium numbers

Problem: Consider all the natural numbers from $000000$ to $999999$. Among these, those numbers with sum of first 3 digits equal to sum of last 3 digits are called lucky. And those with sum of all ...
0
votes
0answers
49 views

Mr.Smith commute word problem. Solved through logic, where is the argument unsound?

Mr. Smith commutes to the city regularly and invariably takes the same train home which arrives at the his home station at 5 pm. At this time, his chauffeur always just arrives, promptly picks him up ...
0
votes
0answers
42 views

What's the name of this problem? Interesting minimisation of a length.

There is a problem which has to do with minimising the length of a (possibly disjoint) barrier in a region of space (often a 2D circle) such that no straight line can pass through the particular ...
2
votes
4answers
31 views

How many lineups of 20 are possible where Sally is first, second or third, and Adam is somewhere in the line?

The line of 20 is created from 300 students. The next part of the question was to find how many ways there are where Sally is first, second or third. I did a permutation of 299 choose 19 for the ...
6
votes
3answers
250 views

Find the number of all subsets of $\{1, 2, \ldots,2015\}$ with $n$ elements such that the sum of the elements in the subset is divisible by 5

The problem is as in the question title. Only one addition - $n$ is not divisible by $5$. I already have a solution involving permutations, but recently I read about generating functions and I was ...
0
votes
0answers
20 views

Why is the diagonal a symmedian?

The problems asks: Let $ABCD$ be a cyclic quadrilatedral, and let $L$ and $N$ be the midpoints of its diagonals $AC$ and $BD$, respectively. Suppose that the line $BD$ bisects the angle $ANC$. Prove ...
1
vote
2answers
70 views

is $7^{101}+8^{101}$ divisible by 25? If not, what is $ 7^{101} + 8^{101}$ mod 25

What i derived is: $$\begin{align}7^{101}+8^{101} &\equiv (5+2)^{101}+ (5+3)^{101} \\ &\equiv 2^{101}+101\cdot5\cdot2^{100}+3^{101}+101\cdot 5\cdot 3^{100} \\ &\equiv ...
4
votes
1answer
46 views

Closed form expression for the number of ordered pairs $\{A, B\}$, where $A, B \subseteq \{1, 2, \dots, n\}$ such that $|A \cap B| = 1$?

What is a closed form expression for the number of ordered pairs $\{A, B\}$, where $A, B \subseteq \{1, 2, \dots, n\}$ such that $|A \cap B| = 1$?
1
vote
2answers
47 views

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, ...
1
vote
2answers
34 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 ...
0
votes
1answer
35 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 ...
0
votes
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 ...
1
vote
1answer
40 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 ...
7
votes
2answers
89 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 ...
3
votes
1answer
46 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 ...
3
votes
3answers
78 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 ...
3
votes
3answers
48 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 ...
1
vote
2answers
62 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 ...
1
vote
1answer
50 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 $a-b=-1$ ...
1
vote
1answer
427 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 ...
1
vote
1answer
38 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 ...
3
votes
0answers
48 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 ...
0
votes
2answers
79 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 ...
2
votes
2answers
62 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 ...
0
votes
1answer
58 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 ...
2
votes
1answer
62 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
40 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
58 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
36 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
108 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
66 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
33 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
49 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
33 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
158 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
58 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 ...
8
votes
2answers
125 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
39 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 $x=AB$, ...
12
votes
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
45 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 $\log_{y}{x}$ ...
1
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
30 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
48 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 ...
0
votes
2answers
32 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
111 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
46 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
102 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 ...