User richrow

8 votes

Accepted

How to find this $\prod_{k=-\infty}^{+\infty}\frac{x^2+(4k+1-y)^2}{x^2+(4k+3-y)^2}$

answered Aug 22, 2020 at 12:56

6 votes

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Show that the elements of the sequence are divisible by $2^n$

answered Jul 25, 2020 at 13:36

6 votes

Show that $(\binom{p^2}{p} -p ) $ is divisible by $p^5$, for every prime number $p, p\ge 5$

answered Apr 15, 2019 at 19:36

6 votes

Prove that every tournament contains at least one Hamiltonian path.

answered Mar 12, 2020 at 18:19

5 votes

Zolotarev's Lemma and Quadratic Reciprocity

answered Apr 3, 2020 at 17:49

5 votes

Can the number of terms of sequence $p_1$, $2p_1+1$, $2(2p_1+1)+1$, . . . be more than 3?

answered Oct 26, 2020 at 17:34

5 votes

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Prove that the midpoint of $AH$ lies on the radical axis of $(REC)$ and $(QFB)$.

answered Jan 29, 2021 at 9:48

4 votes

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Find minimum of $\sqrt{a}+\sqrt{b}+\sqrt{c}$, given that $a,b,c \ge 0$ and $ab+bc+ca+abc=4$

answered Apr 26, 2022 at 19:31

4 votes

Can anyone explain why expressions of the form $\sqrt[3]{x-\sqrt{y}}+\sqrt[3]{x+\sqrt{y}}$ can be rational?

answered Nov 8, 2020 at 19:48

4 votes

A geometry problem with the reflection of the incenter

answered Nov 13, 2020 at 16:49

4 votes

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Positive integer solutions to $y^2=a(1+xy-x^2)$

answered Jun 24, 2021 at 7:38

4 votes

Accepted

Show that $\frac{46!}{48}+1$ is not a power of 47

answered Jan 17, 2021 at 17:38

4 votes

Let $a$ and $b$ be positive integers such that $an + 1$ s a cube if and only if $ bn + 1$ is a cube. Prove that $a = b.$

answered Jan 20, 2022 at 16:13

4 votes

Prove that $\sum \frac{a^3}{a^2+b^2}\le \frac12 \sum \frac{b^2}{a}$

answered Feb 3, 2022 at 20:47

4 votes

Accepted

can incentre lie on the Euler line for an obtuse triangle?

answered Jul 31, 2020 at 18:23

4 votes

Accepted

Prove that $n$ is a power of 2 in the following sets of sums

answered Sep 19, 2020 at 20:17

4 votes

Proving inequality $(a+\frac{1}{a})^2 + (b+\frac{1}{b})^2 \geq \frac{25}{2}$ for $a+b=1$

answered Sep 13, 2019 at 18:29

4 votes

Accepted

$f(x,y) - f(x,z) = g(y,z)$ implies $f(x,y) = a(x) + b(y)$

answered May 31, 2019 at 17:40

3 votes

L^2 function also bounded almost everywhere?

answered Apr 29, 2022 at 14:43

3 votes

Accepted

Find all triples $(a, b, c)$ of real numbers such that $a + 4b + 18c =\frac{a^2+b^2+c^2}{6}=2022$

answered May 13, 2022 at 10:14

3 votes

Accepted

Help in solving this geometry problem

answered Mar 25, 2019 at 15:37

3 votes

$(x^{2022}+1)(1+x^2+x^4+...+x^{2020})=2022\cdot x^{2021}$

answered Aug 11, 2019 at 12:00

3 votes

$[ABC]=s \cdot r$ (where s is the semi-perimeter and r is the inradius)

answered Jan 23, 2019 at 18:39

3 votes

Counting, Probability and Binomial Coefficients

answered Jan 13, 2019 at 13:10

3 votes

Accepted

Combinatorial Proof (Argument) for $\sum_{j=0}^{k} \binom{n}{j} = \sum_{j=0}^{k} \binom{n-1-j}{k-j} 2^j$

answered Mar 24, 2019 at 8:14

3 votes

Accepted

$(1+x)^d \leq 1 + x^d$ for $x \geq 0$ and $d \in (0,1]$

answered Sep 18, 2020 at 4:14

3 votes

Number of real roots $x^8-x^5+x^2-x+1=0$

answered Sep 6, 2020 at 13:29

3 votes

Proving $\frac{a^3+b^3+c^3}{3}-abc\ge \frac{3}{4}\sqrt{(a-b)^2(b-c)^2(c-a)^2}$

answered Aug 21, 2020 at 15:11

3 votes

Prove that $(A,N;P,B)=(A,M;Q,C)=-1$ .

answered Aug 15, 2020 at 12:55

3 votes

How can I calculate this Riemann sum?

answered May 31, 2019 at 17:51

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125 Answers

How to find this $\prod_{k=-\infty}^{+\infty}\frac{x^2+(4k+1-y)^2}{x^2+(4k+3-y)^2}$

Show that the elements of the sequence are divisible by $2^n$

Show that $(\binom{p^2}{p} -p ) $ is divisible by $p^5$, for every prime number $p, p\ge 5$

Prove that every tournament contains at least one Hamiltonian path.

Zolotarev's Lemma and Quadratic Reciprocity

Can the number of terms of sequence $p_1$, $2p_1+1$, $2(2p_1+1)+1$, . . . be more than 3?

Prove that the midpoint of $AH$ lies on the radical axis of $(REC)$ and $(QFB)$.

Find minimum of $\sqrt{a}+\sqrt{b}+\sqrt{c}$, given that $a,b,c \ge 0$ and $ab+bc+ca+abc=4$

Can anyone explain why expressions of the form $\sqrt[3]{x-\sqrt{y}}+\sqrt[3]{x+\sqrt{y}}$ can be rational?

A geometry problem with the reflection of the incenter

Positive integer solutions to $y^2=a(1+xy-x^2)$

Show that $\frac{46!}{48}+1$ is not a power of 47

Let $a$ and $b$ be positive integers such that $an + 1$ s a cube if and only if $ bn + 1$ is a cube. Prove that $a = b.$

Prove that $\sum \frac{a^3}{a^2+b^2}\le \frac12 \sum \frac{b^2}{a}$

can incentre lie on the Euler line for an obtuse triangle?

Prove that $n$ is a power of 2 in the following sets of sums

Proving inequality $(a+\frac{1}{a})^2 + (b+\frac{1}{b})^2 \geq \frac{25}{2}$ for $a+b=1$

$f(x,y) - f(x,z) = g(y,z)$ implies $f(x,y) = a(x) + b(y)$

L^2 function also bounded almost everywhere?

Find all triples $(a, b, c)$ of real numbers such that $a + 4b + 18c =\frac{a^2+b^2+c^2}{6}=2022$

Help in solving this geometry problem

$(x^{2022}+1)(1+x^2+x^4+...+x^{2020})=2022\cdot x^{2021}$

$[ABC]=s \cdot r$ (where s is the semi-perimeter and r is the inradius)

Counting, Probability and Binomial Coefficients

Combinatorial Proof (Argument) for $\sum_{j=0}^{k} \binom{n}{j} = \sum_{j=0}^{k} \binom{n-1-j}{k-j} 2^j$

$(1+x)^d \leq 1 + x^d$ for $x \geq 0$ and $d \in (0,1]$

Number of real roots $x^8-x^5+x^2-x+1=0$

Proving $\frac{a^3+b^3+c^3}{3}-abc\ge \frac{3}{4}\sqrt{(a-b)^2(b-c)^2(c-a)^2}$

Prove that $(A,N;P,B)=(A,M;Q,C)=-1$ .

How can I calculate this Riemann sum?