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Triangle is similar to two others
gghx   3
N 2 minutes ago by LeYohan
Source: SMO junior 2024 Q2
Let $ABCD$ be a parallelogram and points $E,F$ be on its exterior. If triangles $BCF$ and $DEC$ are similar, i.e. $\triangle BCF \sim \triangle DEC$, prove that triangle $AEF$ is similar to these two triangles.
3 replies
gghx
Oct 12, 2024
LeYohan
2 minutes ago
Parallel lines lead to similar triangles
a1267ab   30
N 6 minutes ago by Ilikeminecraft
Source: USA TST for EGMO 2020, Problem 2, by Andrew Gu
Let $ABC$ be a triangle and let $P$ be a point not lying on any of the three lines $AB$, $BC$, or $CA$. Distinct points $D$, $E$, and $F$ lie on lines $BC$, $AC$, and $AB$, respectively, such that $\overline{DE}\parallel \overline{CP}$ and $\overline{DF}\parallel \overline{BP}$. Show that there exists a point $Q$ on the circumcircle of $\triangle AEF$ such that $\triangle BAQ$ is similar to $\triangle PAC$.

Andrew Gu
30 replies
a1267ab
Dec 16, 2019
Ilikeminecraft
6 minutes ago
Circumcircle of ADM
v_Enhance   68
N 23 minutes ago by Giant_PT
Source: USA TSTST 2012, Problem 7
Triangle $ABC$ is inscribed in circle $\Omega$. The interior angle bisector of angle $A$ intersects side $BC$ and $\Omega$ at $D$ and $L$ (other than $A$), respectively. Let $M$ be the midpoint of side $BC$. The circumcircle of triangle $ADM$ intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$), respectively. Let $N$ be the midpoint of segment $PQ$, and let $H$ be the foot of the perpendicular from $L$ to line $ND$. Prove that line $ML$ is tangent to the circumcircle of triangle $HMN$.
68 replies
v_Enhance
Jul 19, 2012
Giant_PT
23 minutes ago
Weird locus problem
Sedro   1
N Today at 4:20 PM by sami1618
Points $A$ and $B$ are in the coordinate plane such that $AB=2$. Let $\mathcal{H}$ denote the locus of all points $P$ in the coordinate plane satisfying $PA\cdot PB=2$, and let $M$ be the midpoint of $AB$. Points $X$ and $Y$ are on $\mathcal{H}$ such that $\angle XMY = 45^\circ$ and $MX\cdot MY=\sqrt{2}$. The value of $MX^4 + MY^4$ can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
1 reply
Sedro
Today at 3:12 AM
sami1618
Today at 4:20 PM
Inequalities
sqing   4
N Today at 3:35 PM by sqing
Let $ a,b,c\geq 0 , (a+8)(b+c)=9.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq  \frac{38}{23}$$Let $ a,b,c\geq 0 , (a+2)(b+c)=3.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq  \frac{2(2\sqrt{3}+1)}{5}$$
4 replies
sqing
Yesterday at 12:50 PM
sqing
Today at 3:35 PM
Find the range of 'f'
agirlhasnoname   1
N Today at 2:46 PM by Mathzeus1024
Consider the triangle with vertices (1,2), (-5,-1) and (3,-2). Let Δ denote the region enclosed by the above triangle. Consider the function f:Δ-->R defined by f(x,y)= |10x - 3y|. Then the range of f is in the interval:
A)[0,36]
B)[0,47]
C)[4,47]
D)36,47]
1 reply
agirlhasnoname
May 14, 2021
Mathzeus1024
Today at 2:46 PM
Function equation
hoangdinhnhatlqdqt   1
N Today at 1:52 PM by Mathzeus1024
Find all functions $f:\mathbb{R}\geq 0\rightarrow \mathbb{R}\geq 0$ satisfying:
$f(f(x)-x)=2x\forall x\geq 0$
1 reply
hoangdinhnhatlqdqt
Dec 17, 2017
Mathzeus1024
Today at 1:52 PM
Inequality with function.
vickyricky   3
N Today at 1:51 PM by SpeedCuber7
If x satisfies the inequalit$ |x - 1| + |x - 2| + |x - 3| \ge 6$, then
$(a) 0 \le x \le 4. (b) x \le 0 or x \ge 4. (c) x \le -2 or x \ge 4$. (d) None of these.
3 replies
vickyricky
May 28, 2018
SpeedCuber7
Today at 1:51 PM
Writing/Evaluating Exponential Functions
Samarthsshah   1
N Today at 1:47 PM by Mathzeus1024
Rewrite the function and determine if the function represents exponential growth or decay. Identify the percent rate of change.

y=2(9)^-x/2
1 reply
Samarthsshah
Jan 30, 2018
Mathzeus1024
Today at 1:47 PM
Functional equation
TuZo   1
N Today at 1:37 PM by Mathzeus1024
My question is, if we can determinate or not, all $f:R\to R$ continuous function with $sin(f(x+y))=sin(f(x)+f(y))$ for all real $x,y$.
Thank you!
1 reply
TuZo
Oct 23, 2018
Mathzeus1024
Today at 1:37 PM
Real parameter equation
L.Lawliet03   1
N Today at 1:12 PM by Mathzeus1024
For which values of the real parameter $a$ does only one solution to the equation $(a+1)x^{2} -(a^{2} + a + 6)x +6a = 0$ belong to the interval (0,1)?
1 reply
L.Lawliet03
Nov 3, 2019
Mathzeus1024
Today at 1:12 PM
a inequality problem
Polus425   1
N Today at 12:36 PM by Mathzeus1024
$x_1,x_2\; are\; such\; two\; different\; real\; numbers:\; $
$(x_1 ^2 -2x_1 +4ln\, x_1)+(x_2 ^2 -2x_2 +4ln\, x_2)- x_1 ^2 x_2 ^2=0$
$prove\; that:\; x_1+x_2\ge 3$
1 reply
Polus425
Dec 19, 2019
Mathzeus1024
Today at 12:36 PM
Function of Common Area [China HS Mathematics League 2021]
HamstPan38825   1
N Today at 11:35 AM by Mathzeus1024
Define the regions $M, N$ in the Cartesian Plane as follows:
\begin{align*}
M &= \{(x, y) \in \mathbb R^2 \mid 0 \leq y \leq \text{min}(2x, 3-x)\} \\
N &= \{(x, y) \in \mathbb R^2 \mid t \leq x \leq t+2 \}
\end{align*}for some real number $t$. Denote the common area of $M$ and $N$ for some $t$ be $f(t)$. Compute the algebraic form of the function $f(t)$ for $0 \leq t \leq 1$.

(Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 5)
1 reply
HamstPan38825
Jun 29, 2021
Mathzeus1024
Today at 11:35 AM
Squares on height in right triangle
Miquel-point   1
N Apr 20, 2025 by LiamChen
Source: Romanian NMO 2025 7.4
Consider the right-angled triangle $ABC$ with $\angle A$ right and $AD\perp BC$, $D\in BC$. On the ray $[AD$ we take two points $E$ and $H$ so that $AE=AC$ and $AH=AB$. Consider the squares $AEFG$ and $AHJI$ containing inside $C$ and $B$, respectively. If $K=EG\cap AC$ and $L=IH\cap AB$, $N=IL\cap GK$ and $M=IB\cap GC$, prove that $LK\parallel BC$ and that $A$, $N$ and $M$ are collinear.
1 reply
Miquel-point
Apr 19, 2025
LiamChen
Apr 20, 2025
Squares on height in right triangle
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G H BBookmark kLocked kLocked NReply
Source: Romanian NMO 2025 7.4
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Miquel-point
479 posts
#1
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Consider the right-angled triangle $ABC$ with $\angle A$ right and $AD\perp BC$, $D\in BC$. On the ray $[AD$ we take two points $E$ and $H$ so that $AE=AC$ and $AH=AB$. Consider the squares $AEFG$ and $AHJI$ containing inside $C$ and $B$, respectively. If $K=EG\cap AC$ and $L=IH\cap AB$, $N=IL\cap GK$ and $M=IB\cap GC$, prove that $LK\parallel BC$ and that $A$, $N$ and $M$ are collinear.
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LiamChen
33 posts
#2
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My solution:
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