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

Deriving Van der Waerden Theorem

by Didier2, Mar 29, 2025, 5:18 PM

Suppose we have already proved that for any coloring of $\Large \mathbb{N}$ in $r$ colors, there exists an arithmetic progression of size $k$ . How can we derive Van der Waerden's theorem for $W(r, k)$ from this?

This post has been edited 2 times. Last edited by Didier2, an hour ago

arithmetic sequence

Van der Waerden number

Functional equations

by hanzo.ei, Mar 29, 2025, 4:33 PM

Find all $f: \mathbb R_+ \rightarrow \mathbb R_+$ such that $f(xf(y)+f(x))=yf(x+yf(x)) \: \forall \: x,y \in \mathbb R_+$

algebra

functional equation

2025 TST 22

by EthanWYX2009, Mar 29, 2025, 2:50 PM

Let $A$ be a set of 2025 positive real numbers. For a subset $T \subseteq A$ , define $M_T$ as the median of $T$ when all elements of $T$ are arranged in increasing order, with the convention that $M_\emptyset = 0$ . Define
$P(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ odd}}} M_T, \quad Q(A) = \sum_{\substack{T \subseteq A \\ |T| \text{ even}}} M_T.$ Find the smallest real number $C$ such that for any set $A$ of 2025 positive real numbers, the following inequality holds:
$P(A) - Q(A) \leq C \cdot \max(A),$ where $\max(A)$ denotes the largest element in $A$ .

combinatorics

Not so classic orthocenter problem

by m4thbl3nd3r, Mar 28, 2025, 4:59 PM

Let $O$ be circumcenter of a non-isosceles triangle $ABC$ and $H$ be a point in the interior of $\triangle ABC$ . Let $E,F$ be foots of perpendicular lines from $H$ to $AC,AB$ . Suppose that $BCEF$ is cyclic and $M$ is the circumcenter of $BCEF$ , $HM\cap AB=K,AO\cap BE=T$ . Prove that $KT$ bisects $EF$

Attachments:

This post has been edited 2 times. Last edited by m4thbl3nd3r, Yesterday at 5:00 PM

geometry

circumcircle

Geometry

by Jackson0423, Mar 28, 2025, 4:40 PM

In triangle ABC with circumcenter O, if the intersection point of lines BO and AC is N, then BO = 2ON, and BMN = 122 degrees with respect to the midpoint M of AB. Find MNB.

This post has been edited 1 time. Last edited by Jackson0423, Yesterday at 4:51 PM
Reason: D to M

geometry

A number theory about divisors which no one fully solved at the contest

by nAalniaOMliO, Jul 24, 2024, 7:40 PM

Let's call a pair of positive integers $(k,n)$ interesting if $n$ is composite and for every divisor $d<n$ of $n$ at least one of $d-k$ and $d+k$ is also a divisor of $n$
Find the number of interesting pairs $(k,n)$ with $k \leq 100$
M. Karpuk

This post has been edited 1 time. Last edited by nAalniaOMliO, Jul 27, 2024, 10:08 AM

number theory

CHKMO 2017 Q3

by noobatron3000, Dec 31, 2016, 7:59 AM

Let ABC be an acute-angled triangle. Let D be a point on the segment BC, I the incentre of ABC. The circumcircle of ABD meets BI at P and the circumcircle of ACD meets CI at Q. If the area of PID and the area of QID are equal, prove that PI*QD=QI*PD.

This post has been edited 1 time. Last edited by noobatron3000, Dec 31, 2016, 8:00 AM
Reason: provide source

geometry

incenter

circumcircle

Find a given number of divisors of ab

by proglote, Oct 24, 2013, 9:51 PM

Arnaldo and Bernaldo play the following game: given a fixed finite set of positive integers $A$ known by both players, Arnaldo picks a number $a \in A$ but doesn't tell it to anyone. Bernaldo thens pick an arbitrary positive integer $b$ (not necessarily in $A$ ). Then Arnaldo tells the number of divisors of $ab$ . Show that Bernaldo can choose $b$ in a way that he can find out the number $a$ chosen by Arnaldo.

algebra

polynomial

combinatorics proposed

combinatorics

CGMO6: Airline companies and cities

by v_Enhance, Aug 13, 2012, 10:47 PM

There are $n$ cities, $2$ airline companies in a country. Between any two cities, there is exactly one $2$ -way flight connecting them which is operated by one of the two companies. A female mathematician plans a travel route, so that it starts and ends at the same city, passes through at least two other cities, and each city in the route is visited once. She finds out that wherever she starts and whatever route she chooses, she must take flights of both companies. Find the maximum value of $n$ .

induction

combinatorics unsolved

combinatorics

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42 shouts

Pi in the Sky

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

by Didier2, Mar 29, 2025, 5:18 PM

by hanzo.ei, Mar 29, 2025, 4:33 PM

by EthanWYX2009, Mar 29, 2025, 2:50 PM

by m4thbl3nd3r, Mar 28, 2025, 4:59 PM

by Jackson0423, Mar 28, 2025, 4:40 PM

by nAalniaOMliO, Jul 24, 2024, 7:40 PM

by noobatron3000, Dec 31, 2016, 7:59 AM

by proglote, Oct 24, 2013, 9:51 PM

by v_Enhance, Aug 13, 2012, 10:47 PM