User Hamid Reza Ebrahimi

10 votes

2 answers

576 views

Show that among any consecutive $16$ natural numbers one is coprime to all others

elementary-number-theory divisibility

asked May 11, 2016 at 14:17

9 votes

3 answers

183 views

For function $f:[0,1]\rightarrow \Bbb R$ we know : $f(x+y)\geq f(x)+f(y)$.Prove that $f(x)\leq 2x$

real-analysis

asked Nov 4, 2017 at 9:41

9 votes

1 answer

189 views

Verify surjectivity of the function $f$ satisfying: $f(x)+f(x^2)=x$

functions functional-equations

asked Jan 1, 2018 at 5:14

7 votes

3 answers

396 views

Find all positive integers satisfying: $x^5+y^6=z^7$

elementary-number-theory diophantine-equations

asked Jul 12, 2016 at 15:24

6 votes

2 answers

191 views

What's $\alpha+\beta$ if we have: $\alpha^3-6\alpha^2+13\alpha=1$ and $\beta^3-6\beta^2+13\beta=19$ ($\alpha$ and $\beta$ are Real)

algebra-precalculus

asked Feb 18, 2016 at 10:30

6 votes

1 answer

641 views

Find all polynomials with real coefficients $P(x),Q(x)$ such that for every real number $x$ we have: $P(Q(x))=P(x)^{2017}$

algebra-precalculus polynomials contest-math

asked Aug 14, 2017 at 6:01

5 votes

1 answer

238 views

Find all positive integers $k,m,n$ satisfying: $\frac1k+\frac1m+\frac1n=\frac{1}{1996}$

elementary-number-theory diophantine-equations

asked Jul 11, 2016 at 15:26

5 votes

6 answers

1k views

Solve the following nonlinear system of equations:$ xy+xz+yz=12 , xyz=2+x+y+z$

algebra-precalculus polynomials systems-of-equations nonlinear-system a.m.-g.m.-inequality

asked Sep 13, 2017 at 3:28

5 votes

3 answers

110 views

Show that $1^7+7^7+13^7+19^7+23^7\equiv{0}\pmod{63}$

elementary-number-theory modular-arithmetic divisibility chinese-remainder-theorem perfect-powers

asked Jun 4, 2016 at 10:54

5 votes

5 answers

1k views

Find all integer roots of: $x^2(y-1)+y^2(x-1)=1$

elementary-number-theory diophantine-equations

asked Jul 14, 2016 at 5:24

5 votes

2 answers

572 views

"Good" subsets of natural numbers

elementary-number-theory

asked Aug 5, 2016 at 7:58

5 votes

6 answers

450 views

Find the rightmost digit of: $1^n+2^n+3^n+4^n+5^n+6^n+7^n+8^n+9^n$

elementary-number-theory modular-arithmetic number-systems decimal-expansion

asked Jun 6, 2016 at 3:03

5 votes

2 answers

196 views

Find $I$ in $\frac{\overline{SIX}}{\overline{NINE}}=\frac23$

algebra-precalculus puzzle

asked Jun 15, 2016 at 14:17

4 votes

5 answers

2k views

Find all positive integers $n$ such that $n^2+n+43$ becomes a perfect square

elementary-number-theory diophantine-equations

asked May 13, 2016 at 16:07

4 votes

3 answers

878 views

Show that among every consecutive 5 integers one is coprime to the others

elementary-number-theory divisibility

asked Apr 16, 2016 at 15:19

4 votes

4 answers

1k views

Perfect square product among 17 integers

combinatorics discrete-mathematics elementary-set-theory pigeonhole-principle square-numbers

asked Dec 15, 2017 at 16:12

4 votes

5 answers

168 views

Find minimum of $a+b+c+\frac1a+\frac1b+\frac1c$ given that: $a+b+c\le \frac32$

algebra-precalculus inequality maxima-minima a.m.-g.m.-inequality sum-of-squares-method

asked Jul 16, 2017 at 19:11

3 votes

2 answers

152 views

Find all positive integers such that: $n^2-3|3n^3-5n^2+7$

elementary-number-theory divisibility

asked Aug 10, 2017 at 19:53

3 votes

2 answers

64 views

Find the maximum of positive integer $k$ so that for all positive real numbers $x$ we have: $x^6+x^5+x^4+x^2+x+4>kx^3$

algebra-precalculus inequality

asked Feb 9, 2017 at 15:10

3 votes

2 answers

103 views

In the following figure prove that $AE||EF$ (without using similar triangles)...

euclidean-geometry

asked Sep 22, 2016 at 11:46

3 votes

2 answers

113 views

Find sum of all positive real numbers $x$ such that $\sqrt[3]{2+x} + \sqrt[3]{2-x}$ becomes an integer.

algebra-precalculus

asked Nov 3, 2017 at 16:24

3 votes

1 answer

488 views

Among any $11$ integers, sum of $6$ of them is divisible by $6$

elementary-number-theory divisibility pigeonhole-principle

asked Dec 15, 2017 at 5:39

3 votes

3 answers

958 views

How many ways are there to go through all intersections and come back to starting point in the following street map...

combinatorics

asked Aug 25, 2017 at 11:20

3 votes

5 answers

2k views

Calculate the area of center square in the following figure [duplicate]

geometry euclidean-geometry area

asked Sep 30, 2017 at 16:04

3 votes

2 answers

367 views

Show that if $ \gcd(a,b)=d,\gcd(a,c)=f,\gcd(b,c)=1 \ \Rightarrow\gcd(a,bc)=df$

elementary-number-theory divisibility

asked Apr 7, 2016 at 21:07

3 votes

2 answers

150 views

Show that :$89|2^{44}-1$ [duplicate]

elementary-number-theory divisibility

asked Jun 5, 2016 at 10:45

3 votes

1 answer

93 views

Inequality in a triangle: $BC > \frac 12AB$ if $∠A > ∠B$ (self-answered)

geometry inequality euclidean-geometry triangles

asked Sep 28, 2015 at 21:48

3 votes

4 answers

154 views

Minimum number of fractions to be summed up to $\frac45$

algebra-precalculus diophantine-equations

asked Oct 16, 2015 at 13:54

3 votes

7 answers

206 views

Find all positive integer roots of $\,5xy=19x+96y$

elementary-number-theory diophantine-equations

asked Jun 16, 2016 at 14:01

3 votes

2 answers

187 views

Upper bound for $\gcd(a,b)$ if $\frac{a+1}{b}+\frac{b+1}{a}\in\Bbb{N}$

elementary-number-theory optimization divisibility gcd-and-lcm

asked Jun 17, 2016 at 3:36

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158 Questions

Show that among any consecutive $16$ natural numbers one is coprime to all others

For function $f:[0,1]\rightarrow \Bbb R$ we know : $f(x+y)\geq f(x)+f(y)$.Prove that $f(x)\leq 2x$

Verify surjectivity of the function $f$ satisfying: $f(x)+f(x^2)=x$

Find all positive integers satisfying: $x^5+y^6=z^7$

What's $\alpha+\beta$ if we have: $\alpha^3-6\alpha^2+13\alpha=1$ and $\beta^3-6\beta^2+13\beta=19$ ($\alpha$ and $\beta$ are Real)

Find all polynomials with real coefficients $P(x),Q(x)$ such that for every real number $x$ we have: $P(Q(x))=P(x)^{2017}$

Find all positive integers $k,m,n$ satisfying: $\frac1k+\frac1m+\frac1n=\frac{1}{1996}$

Solve the following nonlinear system of equations:$ xy+xz+yz=12 , xyz=2+x+y+z$

Show that $1^7+7^7+13^7+19^7+23^7\equiv{0}\pmod{63}$

Find all integer roots of: $x^2(y-1)+y^2(x-1)=1$

"Good" subsets of natural numbers

Find the rightmost digit of: $1^n+2^n+3^n+4^n+5^n+6^n+7^n+8^n+9^n$

Find $I$ in $\frac{\overline{SIX}}{\overline{NINE}}=\frac23$

Find all positive integers $n$ such that $n^2+n+43$ becomes a perfect square

Show that among every consecutive 5 integers one is coprime to the others

Perfect square product among 17 integers

Find minimum of $a+b+c+\frac1a+\frac1b+\frac1c$ given that: $a+b+c\le \frac32$

Find all positive integers such that: $n^2-3|3n^3-5n^2+7$

Find the maximum of positive integer $k$ so that for all positive real numbers $x$ we have: $x^6+x^5+x^4+x^2+x+4>kx^3$

In the following figure prove that $AE||EF$ (without using similar triangles)...

Find sum of all positive real numbers $x$ such that $\sqrt[3]{2+x} + \sqrt[3]{2-x}$ becomes an integer.

Among any $11$ integers, sum of $6$ of them is divisible by $6$

How many ways are there to go through all intersections and come back to starting point in the following street map...

Calculate the area of center square in the following figure [duplicate]

Show that if $ \gcd(a,b)=d,\gcd(a,c)=f,\gcd(b,c)=1 \ \Rightarrow\gcd(a,bc)=df$

Show that :$89|2^{44}-1$ [duplicate]

Inequality in a triangle: $BC > \frac 12AB$ if $∠A > ∠B$ (self-answered)

Minimum number of fractions to be summed up to $\frac45$

Find all positive integer roots of $\,5xy=19x+96y$

Upper bound for $\gcd(a,b)$ if $\frac{a+1}{b}+\frac{b+1}{a}\in\Bbb{N}$