Interior point of ABC

by Jackson0423, Jun 5, 2025, 2:17 PM

Let D be an interior point of the acute triangle ABC with AB > AC so that ∠DAB = ∠CAD. The point E on the segment AC satisfies ∠ADE = ∠BCD, the point F on the segment AB satisfies ∠F DA = ∠DBC, and the point X on the line AC satisfies CX = BX. Let O1 and O2 be the circumcenters of the triangles ADC and EXD, respectively. Prove that the lines BC, EF, and O1O2 are concurrent

geometry

Bisectors in BHC,... Find \alpha+\beta+\gamma

by NO_SQUARES, Jun 5, 2025, 2:12 PM

The altitudes $AA_1$ , $BB_1$ , $CC_1$ of an acute-angled triangle $ABC$ intersect at point $H$ . The bisectors of angles $B$ and $C$ of triangle $BHC$ meet the segments $CH$ and $BH$ at points $X$ and $Y$ respectively. Denote the value of the angle $XA_1Y$ by $\alpha$ . Define $\beta$ and $\gamma$ similarly. Find the sum $\alpha+\beta+\gamma$ .
A. Doledenok

Attachments:

This post has been edited 1 time. Last edited by NO_SQUARES, Today at 2:15 PM

geometry

Tricky FE

by Rijul saini, Jun 4, 2025, 6:58 PM

Let $\mathbb{R}$ denote the set of all real numbers. Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that
$f(xy) + f(f(y)) = f((x + 1)f(y))$ for all real numbers $x$ , $y$ .

Proposed by MV Adhitya and Kanav Talwar

This post has been edited 1 time. Last edited by Rijul saini, Yesterday at 7:27 PM

functional equation

algebra

lmao

Write down sum or product of two numbers

by Rijul saini, Jun 4, 2025, 6:56 PM

Suppose Alice's grimoire has the number $1$ written on the first page and $n$ empty pages. Suppose in each of the next $n$ seconds, Alice can flip to the next page, and write down the sum or product of two numbers (possibly the same) which are already written in her grimoire.

Let $F(n)$ be the largest possible number such that for any $k < F(n)$ , Alice can write down the number $k$ on the last page of her grimoire. Prove that there exists a positive integer $N$ such that for all $n>N$ , we have that $n^{0.99n}\leqslant F(n)\leqslant n^{1.01n}.$
Proposed by Rohan Goyal and Pranjal Srivastava

This post has been edited 1 time. Last edited by Rijul saini, Yesterday at 7:26 PM

combinatorics

Inspired by current year (2025)

by Rijul saini, Jun 4, 2025, 6:46 PM

Let $k>2$ be an integer. We call a pair of integers $(a,b)$ $k-$ good if $0\leqslant a<k,\hspace{0.2cm} 0<b \hspace{1cm} \text{and} \hspace{1cm} (a+b)^2=ka+b$ Prove that the number of $k-$ good pairs is a power of $2$ .

Proposed by Prithwijit De and Rohan Goyal

This post has been edited 1 time. Last edited by Rijul saini, Yesterday at 7:24 PM

number theory

modular arithmetic

Might be slightly generalizable

by Rijul saini, Jun 4, 2025, 6:39 PM

Let $ABC$ be an acute angled triangle with orthocenter $H$ and $AB<AC$ . Let $T(\ne B,C, H)$ be any other point on the arc $\stackrel{\LARGE\frown}{BHC}$ of the circumcircle of $BHC$ and let line $BT$ intersect line $AC$ at $E(\ne A)$ and let line $CT$ intersect line $AB$ at $F(\ne A)$ . Let the circumcircles of $AEF$ and $ABC$ intersect again at $X$ ( $\ne A$ ). Let the lines $XE,XF,XT$ intersect the circumcircle of $(ABC)$ again at $P,Q,R$ ( $\ne X$ ). Prove that the lines $AR,BC,PQ$ concur.

geometry

"all of the stupid geo gets sent to tst 2/5" -allen wang

by pikapika007, Dec 11, 2023, 5:01 PM

Let $ABC$ be a triangle with incenter $I$ . Let segment $AI$ intersect the incircle of triangle $ABC$ at point $D$ . Suppose that line $BD$ is perpendicular to line $AC$ . Let $P$ be a point such that $\angle BPA = \angle PAI = 90^\circ$ . Point $Q$ lies on segment $BD$ such that the circumcircle of triangle $ABQ$ is tangent to line $BI$ . Point $X$ lies on line $PQ$ such that $\angle IAX = \angle XAC$ . Prove that $\angle AXP = 45^\circ$ .

Luke Robitaille

This post has been edited 3 times. Last edited by v_Enhance, Dec 14, 2023, 6:46 PM
Reason: author

Length Condition on Circumcenter Implies Tangency

by ike.chen, Jul 9, 2023, 4:35 AM

Let $ABC$ be an acute-angled triangle with $AC > AB$ , let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$ . The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$ .
Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$

This post has been edited 4 times. Last edited by ike.chen, Jul 9, 2023, 5:14 PM

geometry

IMO Shortist

IMO Shortlist 2022

IMO ShortList 2008, Number Theory problem 2

by April, Jul 9, 2009, 10:26 PM

Let $a_1$ , $a_2$ , $\ldots$ , $a_n$ be distinct positive integers, $n\ge 3$ . Prove that there exist distinct indices $i$ and $j$ such that $a_i + a_j$ does not divide any of the numbers $3a_1$ , $3a_2$ , $\ldots$ , $3a_n$ .

Proposed by Mohsen Jamaali, Iran

Iranian tough nut: AA', BN, CM concur in Gergonne picture

by grobber, Dec 29, 2003, 7:59 PM

Let $ABC$ be a triangle. The incircle of triangle $ABC$ touches the side $BC$ at $A^{\prime}$ , and the line $AA^{\prime}$ meets the incircle again at a point $P$ . Let the lines $CP$ and $BP$ meet the incircle of triangle $ABC$ again at $N$ and $M$ , respectively. Prove that the lines $AA^{\prime}$ , $BN$ and $CM$ are concurrent.