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

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3
votes
1answer
48 views

Let $x_1,x_2,\dots ,x_{50}$ be $50$ integers such that the sum of any $6$ of them is 24, then:

Let $x_1,x_2,\dots,x_{50}$ be $50$ integers such that the sum of any $6$ of them is $24$, then which option is true the largest of $x_i$ equals $6$. the smallest of $x_i$ equals $3$. $x_{16}=x_{34}$....
2
votes
0answers
95 views

Prove that $a_{n+1}a_{n-2}-a_{n}a_{n-1}=1$ is always an integer [duplicate]

We are given the sequence $a_1, ... , a_n$ defined by $a_1=a_2=a_3=1$, and $$a_{n+1}a_{n-2}-a_{n}a_{n-1}=1.$$ Prove that $a_k$ is an integer for all positive integers $k$. The most obvious idea to me ...
2
votes
2answers
39 views

Prove that the area of their union is greater than $\frac{2}{9}S$

A finite set of unit circles is given in a plane such that the area of their union $U$ is $S$. Prove that there exists a subset of mutually disjoint circles such that the area of their union is ...
6
votes
5answers
778 views

Do there exist several positive real numbers such that their sum is $1$ and sum of their squares is less than $0.01$

Do there exist several positive real numbers such that their sum is $1$ and sum of their squares is less than $0.01$? My Attempt: Let there are $n$ real numbers and we call them $x_{1},x_{2},..,x_{n}$...
2
votes
1answer
106 views

Divisibility of a summation

Let $n , l, k, p$ be positive integers, and $1\leq p\leq n$. Let $B(n, l, k, p)$ be the cardinality of the following set \begin{eqnarray} \{(a_1, a_2, \cdots, a_n)\in\mathbb{Z}^{\oplus n}|\ \ 0\leq ...
1
vote
1answer
86 views

Find the largest integer $n$ such that $n^2$ is the difference of two consecutive cubes and $2n +79$ is a perfect square.

Find the largest integer $n$ such that $n^2$ is the difference of two consecutive cubes and $2n +79$ is a perfect square. This is an AIME problem. I have been trying and have been going round in ...
4
votes
1answer
109 views

Problem PUTNAM of the day - Harvard Mathematics department [closed]

Let $f$ be a twice-differentiable real-valued function satisfying $f(x)+f''(x)= -xg(x)f'(x)$, where $g(x) \geq 0$ for all real $x$. Prove that $|f(x)|$ is bound. Honnestly I worked on this problem ...
1
vote
3answers
126 views

What's the ratio between these two lengths? plane geometry problem

I'm thinking following plane geometry problem. Question: There is a parallelogram $ABCD$ such that $\overline{AC}:\overline{BD}=2:1$ and $\overline{AB}\neq\overline{BC}$. Draw a line which is ...
4
votes
1answer
67 views

Math Problem: Forty-nine points

49 points are marked on a sheet of paper in a square. Adjacent points horizontally or vertically are separated by exactly 1 centimetre. How many straight lines of length 5 centimetres can be ...
2
votes
1answer
64 views

How can I get better at algorithmic thinking?

I have been practising for a an upcoming algorithmic thinking competition but have always found that when doing the past papers, I have never had enough time left to finish. I can do basically all of ...
1
vote
2answers
94 views

If $\det(A+B)=\det(A+2B)=\det(A+3B)=1$ and $AB=BA$ then $B^2=0$

Prove that if $\det(A+B)=\det(A+2B)=\det(A+3B)=1$ and $AB=BA$ then $B^2=0$. A problem from a math competition. $A$, $B$ are 2 by 2 complex matrices. I've tried using Cayley Hamilton theorem, on $A+B$...
6
votes
1answer
91 views

$A^2$ $B=A^2-B$ then $AB=BA$

If for $2$ real $n$ by $n$ matrices we have $A^2B=A^2-B$ then prove that the two matrices commute. This is a problem from a competition. I've tried several manipulations but none of them work. Can'...
9
votes
0answers
173 views

solve in positive integers sum of squares of sines equation

Find all positive integer triples $(l,m,n)$ such that $\sin^2\frac{\pi}{n}+\sin^2\frac{\pi}{m}=\sin^2\frac{\pi}{l}$. I have found the solutions $(m,m,1)$ for any $m\in\mathbb{Z}^+$, and also $(2,3,6),...
4
votes
0answers
37 views

Functional division $\max(f(x+y),f(x-y))\mid \min(xf(y)-yf(x), xy)$

As the title suggests, the problem here is: Find all functions $f:\mathbb{Z}\to\mathbb{N}$ such that, for every $x,y\in\mathbb{Z}$, we have $$\max(f(x+y),f(x-y))\mid \min(xf(y)-yf(x), xy)$$ I ...
2
votes
2answers
343 views

Indistinguishable pairs, distinguishable triples of metal circles in key-ring jumble.

The following problem was part of a $\pi$-day contest sponsored by Pizza Hut and written by John H. Conway: My key-rings are metal circles of diameter about two inches. They are all linked ...
0
votes
2answers
160 views

Guess the number. Maximizing expected winnings? [closed]

A man in a trench coat approaches you and pulls an envelope from his pocket. He tells you that it contains a sum of money in bills, anywhere from 1 dollar up to 1,000 dollars. He says that if you can ...
2
votes
4answers
102 views

Prove $x^2+y^4=1994$

Let $x$ and $y$ positive integers with $y>3$, and $$x^2+y^4=2(x-6)^2+2(y+1)^2$$ Prove that $x^2+y^4=1994$. I've tried finding an upper bound on the value of $x$ or $y$, but without sucess. Can ...
2
votes
1answer
46 views

The Functional Equation $f(mn)=f(m)f(n)$ where $f:\mathbb{N}\rightarrow \mathbb{R}$, $f(2)=2$, and $f(m) > fn)$ if $m>n$.

The following is exercise 3.3 from Terence Tao's "Solving Mathematical Problems." Emphasis added. Suppose $f$ is a function on the positive integers which takes real values with the following ...
3
votes
1answer
37 views

An infimum of a double integral on the unit disk

The following question comes from Arnold's Trivium of $1991$ and it is problem $68$. I do not have a solution neither can I come up with something. Find $$\inf \iint \limits_{x^2+y^2 \leq 1} \left[ ...
-3
votes
1answer
31 views

Get rate in Excel without using “RATE” command [closed]

I want your help to calculate rate of interest in excel without using RATE command. (so that I can try to convert the formula in php) This is the code we use in Excel B5= 60 B4= 16070 B3= 750000 B6= ...
0
votes
7answers
157 views

How many real roots does the equation $e^x-x^2=0$ have?

How many real roots does the equation $e^x-x^2=0$ have? I can see from wolfram that the curve cuts X-axis only once. How do you go about solving it?
2
votes
2answers
37 views

What are the valid deductions of a congruence equation?

So I was just sitting here, doing math, and I came over this: $9+16a\equiv 12 \pmod 5$ Obviously, through some simple manipulations: $9+16a-15a-9\equiv 12-9 \pmod 5$ $a\equiv 3 $ And that is a ...
-1
votes
1answer
39 views

Write the set of all positive integers in triangular array as

1 3 6 10 15 . . 2 5 9 14 . . . 4 8 13 . . . . 7 12 . . . . . 11 Find the row number and column number where 20096 occurs. For example 8 appears in the third row and second column.
0
votes
8answers
169 views

The number of real roots of $x^5 + 2x^3 + x^2 + 2 = 0 $ is

The number of real roots of $x^5 + 2x^3 + x^2 + 2 = 0 $ is A. 0; B. 3; C. 5; D. 1. I don't know how to solve this.
3
votes
2answers
57 views

Number of common roots of $x^3 + 2 x^2 +2x +1 = 0$ and $x^{200} + x^{130} + 1 = 0 $

The equations $x^3 + 2 x^2 +2x +1 = 0$ and $x^{200} + x^{130} + 1 = 0 $ have exactly one common root; no common root; exactly three common roots; exactly two common roots. I factored the first ...
-2
votes
1answer
49 views

Let $a_1 = 2$ and for all natural number n, define $a_{n+1}= a_{n}(a_{n}+1)$. Then as $n\rightarrow \infty$, the number of prime factors of $a_{n}$ [closed]

Let $a_1 = 2$ and for all natural number n, define $a_{n+1}= a_{n}(a_{n}+1)$. Then as $n\rightarrow \infty$, the number of prime factors of $a_{n}$: goes to infinity. goes to a finite limit. ...
4
votes
0answers
149 views

(2016 China team selection Test) with a complex inequality

Let $z_{1},z_{2},z_{3}$ be complex numbers, such that: $z_{1}+z_{2}+z_{3}=0,|z_{i}|<1,i=1,2,3$. Find the minimum of the positive $A$ such that: $$|z_{1}z_{2}+z_{2}z_{3}+z_{3}z_{1}|^2+|z_{1}z_{2}z_{...
1
vote
1answer
67 views

Clever way to sum these angles? [closed]

In the image, is there a nice way to write down the sum of a+b+c?
6
votes
5answers
231 views

On equations $m^2+1=5^n$

I am looking for integer solutions of Diophantine equation $m^2+1=5^n$. I found that $m=0,n=0$ and $m=2,n=1$. I could not find any other solutions. I try to prove this but I could not. Could anyone ...
5
votes
4answers
156 views

How to simplify the nested radical $\sqrt{1 - \frac{\sqrt{3}}{2}}$ by hand?

I was solving a Mock Mathcounts Contest Mock contest (.pdf) written by a user on the Art of Problem Solving Forums. In problem #24 the only thing I couldn't do by hand was simplify the radical ...
21
votes
2answers
410 views

Prove that $f(1999)=1999$

A function $f$ maps from the positive integers to the positive integers, with the following properties: $$f(ab)=f(a)f(b)$$ where $a$ and $b$ are coprime, and $$f(p+q)=f(p)+f(q)$$ for all prime numbers ...
0
votes
6answers
69 views

How many digits are in the integer representation of 2 to the 30th power?

How many digits are in the integer representation of 2 to the 30th power? Since I didn't really know any 'expert' way to approach this, I just started out by listing the powers of 2, like 2, 4, 8, 16,...
0
votes
1answer
27 views

Centre of Invariant Circle under Inversion

Given an inversion of the plane, and a circle invariant under this inversion, what information do we know about the inverse of the centre this circle? (I know that an invariant circle must be ...
5
votes
1answer
101 views

If $a^{2}+84a+2008=b^{2}$ what is $a+b$

let $a, b$ are two positive integer satisfy the condition $a^{2}+84a+2008=b^{2}$. Find out $a+b$ My Solution $a^{2}+84a+2008=b^{2} \implies (a+42)^{2}+244=b^{2} \implies (b+a+42)(b-a-42)=2^{2}61$. ...
1
vote
0answers
29 views

Show lines through circumcenters of triangles concurrent with complex numbers

Cyclic quadrilateral $ABCD$ has circumcenter $O$. Point $O$ does not lie on any of the sides of the quadrilateral. Let $O_1,O_2,O_3,O_4$ denote the circumcenters of $\triangle OAB, \triangle OBC, \...
0
votes
1answer
21 views

Squares, midpoints and heights

Let $ABC$ be a traingle, we draw squares on the sides $AB$ and $AC$, now we draw a segment from the vertexes of the square which are closer and then it forms a triangle, so prove that the line throw A ...
5
votes
5answers
568 views

Find a polynomial with integer coefficients

Find a polynomial $p$ with integer coefficients for which $a = \sqrt{2} + \sqrt[3]{2}$ is a root. That is find $p$ such that for some non-negative integer $n$, and integers $a_0$, $a_1$, $a_2$, ..., $...
2
votes
0answers
60 views

2015 Mathcounts State Sprint #30 [duplicate]

NOTE: This question is not a duplicate. It is actually the other way around. This question was posted before the other question that this question was marked as a duplicate of. Please mark this ...
6
votes
0answers
73 views

Game, stealing edges in a graph.

I was inventing a problem for a math contest, I was really pleased with it, but then I found a mistake in my solution and have not been able to solve it. It is as follows: Alice and Bob play a game. ...
1
vote
1answer
14 views

Areal Co-ordinate Geometry Question

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 to $...
1
vote
0answers
26 views

Filling a grid square with 0,1,2 [duplicate]

Each of the 25 cells in a five-by-five grid square is filled with a 0, 1, or 2 in such a way that the numbers written in neighboring cells differ from the number in that cell by 1. Two cells are ...
2
votes
0answers
41 views

Absolute difference and probability [closed]

Fifty tickets numbered with consecutive integers are in a jar. Two are drawn at random and without replacement. What is the probability that the absolute difference between the two numbers is 10 or ...
0
votes
1answer
36 views

Pentagon Problem

In a regular pentagon ABCDE, point M is the midpoint of side AE, and segments AC and BM intersect at point Z. If ZA = 3, what is the value of AB? (The answer is supposed to be in simplest radical form....
1
vote
1answer
124 views

Shortest distance between two circles

What is the shortest distance, in units, between the circles $(x - 9)^2 + (y - 5)^2 = 6.25$ and $(x + 6)^2 + (y + 3)^2 = 49$? Express your answer as a decimal to the nearest tenth. So I know that ...
-2
votes
1answer
35 views

heart rate problem [closed]

The average heart rate of a shrew is 800 beats per minute, while an elephant has a heart rate of 25 beats per minute. If 1 billion heartbeats is a natural life span for each animal, on average, how ...
4
votes
3answers
98 views

How many different paths from top to bottom spell ALGEBRA?

Starting with the A on top and only moving one letter at a time down to the left or down to the right, how many different paths from top to bottom spell ALGEBRA? ...
3
votes
3answers
42 views

How many tokens would person A have under these conditions?

Persons A and B each have a positive integer number of tokens, and the number of tokens B has is a square number less than 100. B says to A, "If you give me all of your tokens, my total number of ...
1
vote
3answers
76 views

Three-digit numbers whose digits and digit sum are all prime

How many 3$$-digit numbers are there such that each of the digits is prime, and the sum of the digits is prime? Shouldn't it be $0$, because the only one digit primes are $2,3,5,7$, and so the ...
2
votes
4answers
114 views

Find the minimum value of $\frac{4}{4-x^2} + \frac{9}{9-y^2} $

Let $x, y ∈ (−2, 2)$ and $xy = −1$. Find the minimum value of $\frac{4}{4-x^2} + \frac{9}{9-y^2} $ ? My Attempt let $t=\frac{4}{4-x^2} + \frac{9}{9-y^2} $ , replacing $y$ by $- \frac{1}{x}$ we get $t=\...
0
votes
2answers
54 views

Perimeter of Quadrilateral

The lengths of two sides of a quadrilateral are equal to 1 and 4. One of the diagonals has a lengths of 2 and divide the quadrilateral into two isosceles triangles. What is the perimeter of the ...