Question

Solve $\sqrt{1 + \sqrt{1-x^{2}}}\left(\sqrt{(1+x)^{3}} + \sqrt{(1-x)^{3}} \right) = 2 + \sqrt{1-x^{2}} $

score 305 · Answer 1 · 2019-05-25 09:37:59

305

Views

Solve $\sqrt{1 + \sqrt{1-x^{2}}}\left(\sqrt{(1+x)^{3}} + \sqrt{(1-x)^{3}} \right) = 2 + \sqrt{1-x^{2}} $

Published on 25 May 2019 - 9:37

score 299 · Answer 2 · 2019-05-25 12:33:29

299

Views

Let $m,n\in \mathbb{Z}$ and $p(x)=x^3+mx+n$ be such that if $107\mid p(x)-p(y)\implies 107\mid x-y$. Prove that $107\mid m$.

Published on 25 May 2019 - 12:33

score 34 · Answer 3 · 2019-05-25 12:41:07

34

Views

Show $\sum_{i \in V(G)}(out(i)-in(i))^3 \geq 0$ for a complete directed graph G with more than 3 vertices

Published on 25 May 2019 - 12:41

score 142 · Answer 4 · 2026-03-27 01:48:11

142

Views

If $ x(9^{\sqrt{x^{2}-3}} + 3^{\sqrt{x^{2}-3}}) = (3^{2\sqrt{x^{2}-3}+1} - 3^{\sqrt{x^{2}-3} + 1} - 18) \sqrt{x}+ 6x $, what is the maximum $x$?

Published on 27 Mar 2026 - 1:48

score 2.6k · Answer 5 · 2019-05-26 08:40:48

2.6k

Views

Compute without calculator, $ \frac{1}{\cos^{2}(10)} + \frac{1}{\sin^{2}(20)} + \frac{1}{\sin^{2}(40)} - \frac{1}{\cos^{2}(45)} $

Published on 26 May 2019 - 8:40

score 150 · Answer 6 · 2026-03-25 06:11:03

150

Views

Prove $a^2b(a-b)+ b^2c(b-c)+ c^2a(c-a)\geq 0$

Published on 25 Mar 2026 - 6:11

score 317 · Answer 7 · 2026-03-25 12:45:45

317

Views

Difficult limits question

Published on 25 Mar 2026 - 12:45

score 74 · Answer 8 · 2026-03-26 17:50:21

74

Views

$ \lfloor (2^{2})/3 \rfloor +... + \lfloor (2^{1000})/3 \rfloor = \frac{2^{A}-B}{C}$, minimum of $A+B+C$?

Published on 26 Mar 2026 - 17:50

score 177 · Answer 9 · 2019-05-27 09:15:54

177

Views

In concert there are 20 singers will perform. For each singer there is a set of other singers that he wishes to perform after them..

Published on 27 May 2019 - 9:15

score 157 · Answer 10 · 2019-05-27 17:06:09

157

Views

All-ukrainian math Olympiad task

Published on 27 May 2019 - 17:06

score 168 · Answer 11 · 2019-05-28 04:11:12

168

Views

An another Monovariant for IMO 1986(Problem–3)

Published on 28 May 2019 - 4:11

score 170 · Answer 12 · 2026-03-25 04:42:59

170

Views

We have a sequence $a_0,a_1,a_2,...,a_9$ so that each member is $1$ or $-1$. Is it possible: $a_0a_1+a_1a_2+...+a_8a_9+a_9a_0=0$

Published on 25 Mar 2026 - 4:42

score 385 · Answer 13 · 2019-05-29 05:31:44

385

Views

Better method to solve a geometric problem.

Published on 29 May 2019 - 5:31

score 195 · Answer 14 · 2026-03-26 02:25:24

195

Views

Square valued integer polynomial

Published on 26 Mar 2026 - 2:25

score 94 · Answer 15 · 2019-05-30 06:36:15

94

Views

How can a junior in high school start preparing for the Putnam?

Published on 30 May 2019 - 6:36

List Question

Solve $\sqrt{1 + \sqrt{1-x^{2}}}\left(\sqrt{(1+x)^{3}} + \sqrt{(1-x)^{3}} \right) = 2 + \sqrt{1-x^{2}} $

Let $m,n\in \mathbb{Z}$ and $p(x)=x^3+mx+n$ be such that if $107\mid p(x)-p(y)\implies 107\mid x-y$. Prove that $107\mid m$.

Show $\sum_{i \in V(G)}(out(i)-in(i))^3 \geq 0$ for a complete directed graph G with more than 3 vertices

If $ x(9^{\sqrt{x^{2}-3}} + 3^{\sqrt{x^{2}-3}}) = (3^{2\sqrt{x^{2}-3}+1} - 3^{\sqrt{x^{2}-3} + 1} - 18) \sqrt{x}+ 6x $, what is the maximum $x$?

Compute without calculator, $ \frac{1}{\cos^{2}(10)} + \frac{1}{\sin^{2}(20)} + \frac{1}{\sin^{2}(40)} - \frac{1}{\cos^{2}(45)} $

Prove $a^2b(a-b)+ b^2c(b-c)+ c^2a(c-a)\geq 0$

Difficult limits question

$ \lfloor (2^{2})/3 \rfloor +... + \lfloor (2^{1000})/3 \rfloor = \frac{2^{A}-B}{C}$, minimum of $A+B+C$?

In concert there are 20 singers will perform. For each singer there is a set of other singers that he wishes to perform after them..

All-ukrainian math Olympiad task

An another Monovariant for IMO 1986(Problem–3)

We have a sequence $a_0,a_1,a_2,...,a_9$ so that each member is $1$ or $-1$. Is it possible: $a_0a_1+a_1a_2+...+a_8a_9+a_9a_0=0$

Better method to solve a geometric problem.

Square valued integer polynomial

How can a junior in high school start preparing for the Putnam?

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