polynomial

by hanzo.ei, Mar 30, 2025, 3:12 PM

Given a polynomial $P(x)$ satisfying $P(0) = 0$ , $P(1) = 1$ , and for $n$ ( $n \geq 2, n \in \mathbb{N}$ ) positive real numbers $k_1, k_2, \dots, k_n$ . Prove that there exists a strictly increasing sequence of real numbers $(a_i)_{i=1}^{n} \subset (0,1)$ such that
$\sum_{i=1}^{n} \frac{k_i}{P'(a_i)} = \sum_{i=1}^{n} k_i.$

polynomial

algebra

possible triangle inequality

by sunshine_12, Mar 30, 2025, 2:12 PM

a, b, c are real numbers
|a| + |b| + |c| − |a + b| − |b + c| − |c + a| + |a + b + c| ≥ 0
hey everyone, so I came across this inequality, and I did make some progress:
Let (a+b), (b+c), (c+a) be three sums T1, T2 and T3. As there are 3 sums, but they can be of only 2 signs, by pigeon hole principle, atleast 2 of the 3 sums must be of the same sign.
But I'm getting stuck on the case work. Can anyone help?
Thnx a lot

This post has been edited 2 times. Last edited by sunshine_12, Today at 2:41 PM

inequalities

Geo challenge on finding simple ways to solve it

by Assassino9931, Mar 30, 2025, 12:35 PM

Let $ABC$ be an acute scalene triangle inscribed in a circle $\Gamma$ . The angle bisector of $\angle BAC$ intersects $BC$ at $L$ and $\Gamma$ at $S$ . The point $M$ is the midpoint of $AL$ . Let $AD$ be the altitude in $\triangle ABC$ , and the circumcircle of $\triangle DSL$ intersects $\Gamma$ again at $P$ . Let $N$ be the midpoint of $BC$ , and let $K$ be the reflection of $D$ with respect to $N$ . Prove that the triangles $\triangle MPS$ and $\triangle ADK$ are similar.

This post has been edited 1 time. Last edited by Assassino9931, Today at 1:09 PM

nice problem

by hanzo.ei, Mar 29, 2025, 5:58 PM

Let triangle $ABC$ be inscribed in the circumcircle $(O)$ and circumscribed about the incircle $(I)$ , with $AB < AC$ . The incircle $(I)$ touches the sides $BC$ , $CA$ , and $AB$ at $D$ , $E$ , and $F$ , respectively. A line through $I$ , perpendicular to $AI$ , intersects $BC$ , $CA$ , and $AB$ at $X$ , $Y$ , and $Z$ , respectively. The line $AI$ meets $(O)$ at $M$ (distinct from $A$ ). The circumcircle of triangle $AYZ$ intersects $(O)$ at $N$ (distinct from $A$ ). Let $P$ be the midpoint of the arc $BAC$ of $(O)$ . The line $AI$ cuts segments $DF$ and $DE$ at $K$ and $L$ , respectively, and the tangents to the circle $(DKL)$ at $K$ and $L$ intersect at $T$ . Prove that $AT \perp BC$ .

geometry unsolved

geometry

Finding big a_i a_i+1

by nAalniaOMliO, Mar 28, 2025, 8:36 PM

Positive real numbers $a_1>a_2>\ldots>a_n$ with sum $s$ are such that the equation $nx^2-sx+1=0$ has a positive root $a_{n+1}$ smaller than $a_n$ .
Prove that there exists a positive integer $r \leq n$ such that the inequality $a_ra_{r+1} \geq \frac{1}{r}$ holds.

algebra

A positive integer changes every second and becomes a power of two

by nAalniaOMliO, Mar 28, 2025, 8:19 PM

A positive integer with three digits is written on the board. Each second the number $n$ on the board gets replaced by $n+\frac{n}{p}$ , where $p$ is the largest prime divisor of $n$ .
Prove that either after 999 seconds or 1000 second the number on the board will be a power of two.

number theory

fifth power

by mathbetter, Mar 25, 2025, 1:11 PM

$\text{Find all prime numbers } (p, q) \text{ such that } p^2 + 3pq + q^2 \text{ is a fifth power of an integer.}$

number theory

prime numbers

Special students

by BR1F1SZ, Dec 24, 2024, 5:47 PM

In a school with double schooling, in the morning the language teacher divided the students into $200$ groups for an activity. In the afternoon, the math teacher divided the same students into $300$ groups for another activity. A student is considered special if the group they belonged to in the afternoon is smaller than the group they belonged to in the morning. Find the minimum number of special students that can exist in the school.

Note: Each group has at least one student.

combinatorics

Prove that AY is tangent to (AEF)

by geometry6, Aug 11, 2021, 9:03 AM

Let $P$ be an arbitrary interior point of $\triangle ABC$ , and $AP$ , $BP$ , $CP$ intersect $BC$ , $CA$ , $AB$ at $D$ , $E$ , $F$ , respectively. Suppose that $M$ be the midpoint of $BC$ , $\odot(AEF)$ and $\odot(ABC)$ intersect at $S$ , $SD$ intersects $\odot(ABC)$ at $X$ , and $XM$ intersects $\odot(ABC)$ at $Y$ . Show that $AY$ is tangent to $\odot(AEF)$ .

geometry

tangent

Equal radius

by FabrizioFelen, Jun 20, 2016, 7:09 PM

Let $\triangle ABC$ be triangle with incenter $I$ and circumcircle $\Gamma$ . Let $M=BI\cap \Gamma$ and $N=CI\cap \Gamma$ , the line parallel to $MN$ through $I$ cuts $AB$ , $AC$ in $P$ and $Q$ . Prove that the circumradius of $\odot (BNP)$ and $\odot (CMQ)$ are equal.

This post has been edited 1 time. Last edited by FabrizioFelen, Jun 20, 2016, 7:10 PM

geometry

incenter

circumcircle

Submit

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by pupitrethebean, Mar 19, 2025, 2:05 PM
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pupitre

by hanzo.ei, Mar 30, 2025, 3:12 PM

by sunshine_12, Mar 30, 2025, 2:12 PM

by Assassino9931, Mar 30, 2025, 12:35 PM

by hanzo.ei, Mar 29, 2025, 5:58 PM

by nAalniaOMliO, Mar 28, 2025, 8:36 PM

by nAalniaOMliO, Mar 28, 2025, 8:19 PM

by mathbetter, Mar 25, 2025, 1:11 PM

by BR1F1SZ, Dec 24, 2024, 5:47 PM

by geometry6, Aug 11, 2021, 9:03 AM

by FabrizioFelen, Jun 20, 2016, 7:09 PM