1
vote
2answers
72 views

Tough probability problem

Two numbers $x$ and $y$ are chosen at random without replacement from the set $\{1,2,3,\cdots,100\}$. Find the probability that $x^4 - y^4$ is divisible by $5$. I don't know how to proceed with this ...
0
votes
3answers
55 views

Logic Question with who has a key and truth

Four people are standing infront of a treasure chest, each makes a statement. One statement is false, the other three are true. Ann: "I do not have the key and Cal does not have the key." Ben: "I do ...
3
votes
3answers
53 views

$\operatorname{lcm}(n,m,p)\times \gcd(m,n) \times \gcd(n,p) \times \gcd(n,p)= nmp \times \gcd(n,m,p)$, solve for $n,m,p$?

$\newcommand{\lcm}{\operatorname{lcm}}$ I saw this in the first Moscow Olympiad of Mathematics (1935), the equation was : $$\lcm(n,m,p)\times \gcd(m,n) \times \gcd(n,p)^2 = nmp \times \gcd(n,m,p)$$ ...
2
votes
1answer
58 views

Given $|f(x) - f(y)| \le \frac{1}{2}|x-y|$ what are the points of intersection of the graph of $y = f(x)$ and the line $y = x$?

Let $f(x)$ be a real-valued function, defined for all real numbers $x$ such that $$|f(x) - f(y)| \le \frac{1}{2}|x-y|$$ for all $x,y$. Then the number of points of intersection of the graph of $y = ...
0
votes
0answers
35 views

Ratio of area of triangle to that formed by its medians

What is the ratio of the area of a triangle $ABC$ to the area of the triangle whose sides are equal in length to the medians of triangle $ABC$? I see an obvious method of brute-force wherein I can ...
1
vote
4answers
88 views

Areas in a rectangle

Suppose $P,Q, R$, and $S$ are the midpoints of the sides $AB, BC, CD$, and $DA$, respectively of rectangle $ABCD$. If the area of the rectangle is $\delta$, then prove that the area of the figure ...
1
vote
1answer
29 views

Orthocentre of triangle and related ratio

$ABC$ is a triangle with $AB = 13$, $BC = 14$ and $CA = 15$. $AD$ and $BE$ are the altitudes from $A$ to $B$ to $BC$ and $AC$ respectively. $H$ is the point of intersection of $AD$ and $BE$. Then the ...
4
votes
1answer
188 views

Binomial Summation

The sum $$ 1 + {n \choose 1}\cos \theta + {n \choose 2}\cos 2\theta + \cdots+ {n \choose n}\cos n\theta $$ is? I try to write this as the real part of $(1 + \cos \theta + i\sin \theta)^n$ but then ...
2
votes
1answer
48 views

Algebraic maximum and minimum based on a constraint

Suppose $a,b,c$ are real numbers such that $a^2b^2 + b^2c^2 + c^2a^2 = k$, where $k$ is a constant. Then the set of all possible values of $abc(a+b+c)$ is? I attempted writing the constraint in the ...
0
votes
1answer
28 views

Algebra Value based on condition provided

Let $a, b, c$ be distinct real numbers such that $a^2 - b = b^2 - c = c^2 - a$ Then $(a+b)(b+c)(c+a)$ equals? I attempted manipulations with that condition provided, but then I'm unable to go ...
1
vote
1answer
51 views

Conic Sections and Complex numbers

If $\omega$ is a complex number such that |$\omega$| does not equal 1, then the complex number $$z = \omega + \frac{1}{\omega}$$ describes a conic. The distance between the foci of the conic described ...
1
vote
3answers
72 views

Infinite Sum of products

What is the infinite sum $$S = {1 + \frac{1}{3} + \frac{1\cdot 3}{3\cdot 6} + \frac{1\cdot 3\cdot 5}{3\cdot 6\cdot 9}+ ....}?$$ I attempted messing around with the $n$ th term in the series but didnt ...
1
vote
2answers
82 views

Given perimeter of triangle and one side, find other two sides

In triangle ABC, all three sides have integer lengths. If AB = 21, the perimeter is 54, and the area is a positive integer, what are the lengths of BC and AC? I tried using Heron's Formula, but I ...
0
votes
1answer
65 views

Series Summation involving factorials, and powers.

What is the value of $\dfrac{1.2}{3!} + \dfrac{2.2^2}{4!} + \dfrac{3.2^3}{5!} + ...... + \dfrac{15.2^{15}}{17!}$ How would you proceed with this? I attempted writing the general term and tried some ...
-3
votes
1answer
57 views

Why there are two different values of θ for same quadrant?

Let Sin θ = 1/2 is function. Let us find its solution set. sine is +ve in I and II quadrant with reference angle π/6 θ = π/6 (I quadrant) Now here is my problem. We can use π-θ and (π/2)+θ to find ...
1
vote
1answer
85 views

What are some of the more efficient ways of studying for an Olympiad?

This September I am participating in a competition called the Australian Intermediate Mathematics olympiad, and you may not have heard of it but it's very similar to the AIME. Could you please tell me ...
0
votes
1answer
21 views

Help with distance question points A and B

Ok. I had no idea how to do the question but I tried fiddling with the triangles to see if I can get any value but only managed to get $MN$. I read the solution to this question, and it said that I ...
5
votes
4answers
786 views

Solve without a calculator: What is the possible value of 2*((1+1/100)^100)?

What is the possible value of $2·((1+\tfrac{1}{100})^{100})$? Google will give $2·((1+\tfrac{1}{100})^{100}) = 5.40962765884$. How can I find the possible value without Google or a calculator? How ...
1
vote
1answer
60 views

math contest ranking problem?

A math contest is held among 4 middle schools. Each school enters a team of 3 students. The 12 contestants are ranked from 1 (best performance) to 12 (worst performance). The team that has the overall ...
1
vote
2answers
35 views

Locating a point based on a condition

C and D are two points on the same side of straight line AB. Find a point X on AB such that the angles CXA and DXB are equal. How would you go about this?
1
vote
1answer
79 views

Diophantine Equation.

How many solutions are there in $\mathbb{N} \times \mathbb{N}$ to the equation $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{1995}?$ How would you solve this? I have tried but am not sure how I should ...
4
votes
2answers
221 views

Logic Puzzle of Diamonds and sons

I came across a math problem and I need a solution for this. An old man has 49 diamonds. Each one has a different worth as $1, $2, $3, ….. $49. He has 7 sons and he ...
5
votes
1answer
186 views

Tricky Puzzle!! Please help.

I stumbled upon a puzzle I can't crack. It goes like this: In a certain Code language: 7321=6 5342=3 8645=15 Then 9312=? The Answer is 9. But I can't seem to find the logic behind it??
3
votes
3answers
95 views

Manipulating Algebraic Expression

$a + b + c = 7$ and $\dfrac{1}{a+b} + \dfrac{1}{b+c} + \dfrac{1}{c+a} = \dfrac{7}{10}$. Find the value of $\dfrac{a}{b+c} + \dfrac{b}{c+a} + \dfrac{c}{a+b}$. I algebraically manipulated the ...
0
votes
0answers
41 views

Question based on counting. [duplicate]

Given two integers $N$ and $M$, find how many permutations of $1, 2, ..., N$ (first $N$ natural numbers) are there where the sum of every two adjacent numbers is at most $M$
1
vote
1answer
88 views

3 complex-variable equation

Moderator Note: This is a current contest question on Brilliant.org. $x,y,z$ are complex numbers satisfying $$ \begin{align} x+y+z & =1\\ x^2+y^2+z^2 & =2\\ x^3+y^3+z^3 & =3 ...
1
vote
2answers
154 views

Ordered triples solution to system of equations

How many ordered triples $(x,y,z)$ of integer solutions are there to the following system of equations? $$ \begin{align} x^2+y^2+z^2&=194 \\ x^2z^2+y^2z^2&=4225 \end{align} $$
3
votes
2answers
155 views

Finding the number of different ordered quadruples $(a,b,c,d)$ of complex numbers

Find the number of different ordered quadruples $(a,b,c,d)$ of complex numbers such that: $$a^2=1$$ $$b^3=1$$ $$c^4=1$$ $$d^6=1$$ $$a+b+c+d=0$$
1
vote
1answer
332 views

Let $N$ = $11^2 \times 13^4 \times 17^6$. How many positive factors of $N^2$ are less than $N$ but not a factor of $N$?

Let $N$ = $11^2 \times 13^4 \times 17^6$. How many positive factors of $N^2$ are less than $N$ but not a factor of $N$? $Approach$: $N$=$11^2$.$13^4$.$17^6$ $N^2$=$11^4$.$13^8$.$17^{12}$ This ...
6
votes
1answer
182 views

how to prove $\sum _{|k|\lt\sqrt m}\binom{2m}{m+k}\ge2^{2m-1}$

how to prove $$\sum _{|k|\lt\sqrt m}\binom{2m}{m+k}\ge2^{2m-1},\forall m\ge1$$ Thanks in advance .
4
votes
2answers
203 views

Prove that function is bijective

Let $n \in \mathbb{N} \setminus \{ 0 \} $ and $A \in M_n(\mathbb{R})$ with $m \in \mathbb{N} \setminus \{ 0 \}$ as $A^m= \alpha \times I_n$, with $ \alpha \in \mathbb{R} \setminus \{ -1,1 \}$. ...
5
votes
3answers
209 views

Show that $\operatorname{rank}(A^2+A+I_3)=1$

If $A \in M_3(\mathbb{R}), A \ne I_3 $ and $A^3=I_3$ Show that $\operatorname{rank}(A^2+A+I_3)=1$. What I have reached so far is that $\operatorname{rank}(A-I_3)+\operatorname{rank}(A^2+A+I_3)\le ...
3
votes
3answers
79 views

how prove $\sum_{m=0}^{n}\left(\frac{n!}{m!(n-m)!}\right)^2=\frac{(2n)!}{(n!)^2}$?

How to prove $\forall n \in \mathbb N$ $$\sum_{m = 0}^{n} \left(\frac{n!}{m!(n-m)!}\right)^2=\frac{(2n)!}{(n!)^2}$$
1
vote
1answer
125 views

confusing question please help

Moderator Note: this is a question from the Federal Mathematics Competition 2013. so these two people are playing a game, a boy and a girl. They both take turns writing numbers on the white board ...
2
votes
1answer
117 views

$\exists a,b\in \mathbb R^+ $such that $|f(x)|\le a|x|+b$

Assuming $f:\mathbb R\to\mathbb R $ be an uniform continuous function, how to prove $$\exists a,b\in \mathbb R^+~~~~\text{such that}~~~~|f(x)|\le a|x|+b.$$ Thanks in advance!
7
votes
1answer
98 views

How to prove that exists distinct $x_1,x_2 \in(a,b)$ such that $f '(x_1)f '(x_2)=1$?

Assume $f:[a,b]\to[a,b]$ be continuous and differentiable on $(a,b)$ and $f(a)=a$, $f(b)=b$. How to prove that exists distinct $x_1,x_2 \in(a,b)$ such that $f '(x_1)f '(x_2)=1$? Thanks in advance.
0
votes
1answer
52 views

how prove $\exists a,b$ that satisfied in following conditions $0 <a\leq b\leq 1, b-a=\frac12, \text{ and }f(a)=f(b)$ [duplicate]

Possible Duplicate: Universal Chord Theorem let $f:[0,1]\mapsto\mathbb R$ be continuous and $f(0)=f(1)$how prove $\exists a,b$ that satisfied in following conditions $$1)0<a\leq b\leq ...
0
votes
2answers
196 views

Find square root of non-rational fraction

If we have to compute this without using calculator, is there a quick way to find the answer approximately of the following problem: which one is smaller ? $$ A = ...
0
votes
1answer
125 views

Proving that $ f(1)=\frac{1-\sqrt{5}}{2}$ for this function

Let $f:(0,+\infty)\mapsto R$ be a strictly increasing function such that $\forall x\ge0,$ $$f(x)+\frac{1}{x}\ge0, \qquad f(x)f\left(f(x)+\frac{1}{x}\right)=1.$$ Show that ...
6
votes
2answers
109 views

let $A,B\in M_{n}(C)$ such that c is complex field and $AB^2-B^2A=B$ how prove $B^n=0$

Let $A,B\in M_{n}(C)$ such that $C$ is complex field and $AB^2-B^2A=B$. How prove $B^n=0$. thanks in advance
2
votes
1answer
92 views

how prove $A_1$+$(-1)^nA_n$ is scalar matrix with following condition

let $A_i\in M_n (\mathbb{R})$ ,$i=1,2,...,n$ $$A_1\cdot A_2 \cdot...\cdot A_n=I\hspace{5pt}\&\hspace{5pt}\det A_1=...=\det A_n=1$$ Assume that $A_1-A_k$ for $k=1,2,..,n-1$ are none zero and ...
0
votes
2answers
99 views

probability( central limit theory)

A seed manufacturer sells seeds in packets of 50. Assume that each seed germinates with probability .99 independently of all the others.The manufacturer promises to replace, at no cost to the buyer, ...
1
vote
4answers
464 views

Geometry Prove - two perpendicular lines in a circle

In a circle of radius r, two lines (AB and CD) are perpendicular to each other and meet at X. Show that:
1
vote
4answers
2k views

Equation of a circle that touches a line and both x and y-axes

As shown in the graph below, a circle touches the $x$-axis, the $y$-axis and a line that has equation $y = x/2 +2$. How to find the equation of the circle? Thanks very much!
0
votes
1answer
108 views

Finding a diagonal of a trapezoid that touches 3 points on a circle

In the image below: - AB and AD are tangent to the circle - BC and AD are parallel What is the length of AC? Thank you very much in advance!
2
votes
3answers
965 views

Finding a diagonal in a trapezoid given the other diagonal and three sides

The figure below is a trapezoid, what is the length of the red line? Thank you very much in advance!
8
votes
1answer
679 views

Hard math contest trigonometry type problem

How to solve this problem: Also, most people would use trigonometry, but is there a way to use derivative to solve this too?
3
votes
2answers
296 views

Modification of 5th question from BMO'81

First of all I will introduce original problem (Question 5 from British Mathematical Olympiad). You can find complete list of BMO'81 there BMO'81. Find, with proof, the smallest possible value ...
0
votes
1answer
179 views

finding final state of numbers after certain operations

There are $N$ children sitting along a circle, numbered $1,2,\dots,n$ clockwise. The ith child has a piece of paper with number $a_i$ written on it. They play the following game: In the first round, ...
2
votes
2answers
603 views

Euclidean Geometry Intersection of Circles

Two circles intersect in the Cartesian Coordinate system at points $A$ and $B$. Point $A$ lies on the line $y=3$. Point $B$ lies on the line $y=12$. These two circles are also tangent to the x-axis at ...