Inspired by Czech-Polish-Slovak 2024

by sqing, Jun 1, 2025, 1:04 PM

Let $a,b,c\geq 0, (a+1)(b+ c )=2025.$ Prove that $a+b^2+c\geq \frac{355}{4}$ Let $a,b,c\geq 0, (a-1)(b+ c )=2025.$ Prove that $a+b^2+c\geq \frac{364}{4}$ Let $a,b,c\geq 0, (a+ 1)(b- c )=2025.$ Prove that $a+b^2+c\geq \frac{135 \sqrt[3]{90}-2}{2}$

This post has been edited 1 time. Last edited by sqing, 2 hours ago

inequalities proposed

algebra

inequalities

FE i created on bijective function with x≠y

by benjaminchew13, Jun 1, 2025, 11:22 AM

Find all bijective functions $f:\mathbb{R}\to \mathbb{R}$ such that $(x-y)f(x+f(f(y)))=xf(x)+f(y)^{2}$ for all $x,y\in \mathbb{R}$ such that $x\neq y$ .

function

algebra

interesting geometry config (3/3)

by Royal_mhyasd, Jun 1, 2025, 7:06 AM

Let $\triangle ABC$ be an acute triangle, $H$ its orthocenter and $E$ the center of its nine point circle. Let $P$ be a point on the parallel through $C$ to $AB$ such that $\angle CPH = |\angle BAC-\angle ABC|$ and $P$ and $A$ are on different sides of $BC$ and $Q$ a point on the parallel through $B$ to $AC$ such that $\angle BQH = |\angle BAC - \angle ACB|$ and $C$ and $Q$ are on different sides of $AB$ . If $B'$ and $C'$ are the reflections of $H$ over $AC$ and $AB$ respectively, $S$ and $T$ are the intersections of $B'Q$ and $C'P$ respectively with the circumcircle of $\triangle ABC$ , prove that the intersection of lines $CT$ and $BS$ lies on $HE$ .

final problem for this "points on parallels forming strange angles with the orthocenter" config, for now. personally i think its pretty cool

Attachments:

geometry

orthocenter

circumcircle

interesting geo config (2/3)

by Royal_mhyasd, May 31, 2025, 11:36 PM

Let $\triangle ABC$ be an acute triangle and $H$ its orthocenter. Let $P$ be a point on the parallel through $A$ to $BC$ such that $\angle APH = |\angle ABC-\angle ACB|$ . Define $Q$ and $R$ as points on the parallels through $B$ to $AC$ and through $C$ to $AB$ similarly. If $P,Q,R$ are positioned around the sides of $\triangle ABC$ as in the given configuration, prove that $P,Q,R$ are collinear.

Attachments:

2-var inequality

by sqing, May 31, 2025, 1:35 PM

Let $a,b> 0 ,a^3+ab+b^3=3.$ Prove that
$(a+b)(a+1)(b+1) \leq 8$ $(a^2+b^2)(a+1)(b+1) \leq 8$ Let $a,b> 0 ,a^3+ab(a+b)+b^3=3.$ Prove that
$(a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$

This post has been edited 1 time. Last edited by sqing, Yesterday at 1:45 PM

inequalities proposed

inequalities

algebra

Polyline with increasing links

by NO_SQUARES, May 5, 2025, 5:30 PM

There are $100$ points on the plane, all pairwise distances between which are different. Is there always a polyline with vertices at these points, passing through each point once, in which the link lengths increase monotonously?

geometry

combinatorics

combinatorial geometry

A complex FE from Iran

by mojyla222, Aug 29, 2024, 9:23 AM

A surjective function $g: \mathbb{C} \to \mathbb C$ is given. Find all functions $f: \mathbb{C} \to \mathbb C$ such that for all $x,y\in \mathbb C$ we have
$|f(x)+g(y)| = | f(y) + g(x)|.$

Proposed by Mojtaba Zare, Amirabbas Mohammadi

This post has been edited 3 times. Last edited by mojyla222, Dec 27, 2024, 9:44 AM

Sum of divisors

by Kimchiks926, Nov 12, 2022, 4:56 PM

Let $n$ be a positive integer such that the sum of its positive divisors is at least $2022n$ . Prove that $n$ has at least $2022$ distinct prime factors.

number theory

Convex and concave functions in Real numbers -- Basic 1

by adityaguharoy, Mar 1, 2018, 1:44 PM

Convex functions
Let $f : \mathbb{R} \to \mathbb{R}$ be a function, and let $a,b$ be two real numbers with $a<b$ . Then we say that $f$ is a convex function on the interval $[a,b]$ if and only if the following is true :
Given any $t \in [0,1]$ , and , any $x_1 , x_2 \in [a,b]$ then,
$\boxed{f(tx_1 + (1-t)x_2) \le t \cdot f(x_1) + (1-t) \cdot f(x_2)}$ And we say that $f$ is strictly convex on $[a,b]$ if the above inequality is strict whenever $x_1 \ne x_2$ and $t \in (0,1)$ .

Concave functions
Let $f : \mathbb{R} \to \mathbb{R}$ be a function, and let $a,b$ be two real numbers with $a<b$ . Then we say that $f$ is a concave function on the interval $[a,b]$ if and only if the following is true :
Given any $t \in [0,1]$ , and , any $x_1 , x_2 \in [a,b]$ then,
$\boxed{f(tx_1 + (1-t)x_2) \ge t \cdot f(x_1) + (1-t) \cdot f(x_2)}$ And we say that $f$ is strictly concave on $[a,b]$ if the above inequality is strict whenever $x_1 \ne x_2$ and $t \in (0,1)$ .

Quick exercises

Let us celebrate

This post has been edited 2 times. Last edited by adityaguharoy, Mar 4, 2018, 7:25 AM

Find the number of interesting numbers

by WakeUp, May 19, 2011, 2:08 PM

A positive integer $n$ is known as an interesting number if $n$ satisfies
${\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}}$
for all $k=1,2,\ldots 9$ .
Find the number of interesting numbers.

number theory proposed

number theory

combinatorics

Find the perfect squares

by Johann Peter Dirichlet, Mar 18, 2006, 4:47 AM

The sequence $(a_n)_{n \in\mathbb{N}}$ is defined by $a_1 = 8, a_2 = 18, a_{n+2} = a_{n+1}a_{n}$ . Find all terms which are perfect squares.

Submit

hi guys $~~~$
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2025 shout!
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by Morrigan_Black, Jan 28, 2022, 1:05 PM
still waiting for the mathlinks camp lol
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hello
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Right below the shout box it says how many it has.
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by adityaguharoy, Mar 21, 2021, 2:58 PM
Nice blog!
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by bobert1, Jun 30, 2016, 9:29 PM
yeah sure we must and will discuss real numbers.
by adityaguharoy, Jun 20, 2016, 4:11 AM
Why not real numbers
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very interesting topic !! ?? !!
by adityaguharoy, Jun 12, 2016, 6:52 AM
Hello!!!!!!! Very interesting topic, here!
by MinionLA, Jun 9, 2016, 11:43 PM
Should it not start by the discussion of Set Theory.
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by skipiano, Jun 9, 2016, 4:52 AM
first shout >
by doitsudoitsu, Apr 26, 2016, 8:03 PM

118 shouts

An Introduction to the Theory of Mathematics

by sqing, Jun 1, 2025, 1:04 PM

by benjaminchew13, Jun 1, 2025, 11:22 AM

by Royal_mhyasd, Jun 1, 2025, 7:06 AM

by Royal_mhyasd, May 31, 2025, 11:36 PM

by sqing, May 31, 2025, 1:35 PM

by NO_SQUARES, May 5, 2025, 5:30 PM

by mojyla222, Aug 29, 2024, 9:23 AM

by Kimchiks926, Nov 12, 2022, 4:56 PM

by adityaguharoy, Mar 1, 2018, 1:44 PM

by WakeUp, May 19, 2011, 2:08 PM

by Johann Peter Dirichlet, Mar 18, 2006, 4:47 AM