Taking antipode on isosceles triangle's circumcenter

by Nuran2010, May 11, 2025, 11:46 AM

In isosceles triangle, the condition $AB=AC>BC$ is satisfied. Point $D$ is taken on the circumcircle of $ABC$ such that $\angle CAD=90^{\circ}$.A line parallel to $AC$ which passes from $D$ intersects $AB$ and $BC$ respectively at $E$ and $F$.Show that circumcircle of $ADE$ passes from circumcenter of $DFC$.
geometry
antipode
Circumcenter
circumcircle
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ISI UGB 2025 P4

by SomeonecoolLovesMaths, May 11, 2025, 11:24 AM

Let $S^1 = \{ z \in \mathbb{C} \mid |z| =1 \}$ be the unit circle in the complex plane. Let $f \colon S^1 \longrightarrow S^2$ be the map given by $f(z) = z^2$. We define $f^{(1)} \colon = f$ and $f^{(k+1)} \colon = f \circ f^{(k)}$ for $k \geq 1$. The smallest positive integer $n$ such that $f^{(n)}(z) = z$ is called the period of $z$. Determine the total number of points in $S^1$ of period $2025$.
(Hint : $2025 = 3^4 \times 5^2$)
algebra
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ISI UGB 2025 P2

by SomeonecoolLovesMaths, May 11, 2025, 11:16 AM

If the interior angles of a triangle $ABC$ satisfy the equality, $$\sin ^2 A + \sin ^2 B + \sin^2 C = 2 \left( \cos ^2 A + \cos ^2 B + \cos ^2 C \right),$$prove that the triangle must have a right angle.
This post has been edited 1 time. Last edited by SomeonecoolLovesMaths, 2 hours ago
trigonometry
geometry
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ISI UGB 2025 P5

by SomeonecoolLovesMaths, May 11, 2025, 11:15 AM

Let $a,b,c$ be nonzero real numbers such that $a+b+c \neq 0$. Assume that $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{a+b+c}$$Show that for any odd integer $k$, $$\frac{1}{a^k} + \frac{1}{b^k} + \frac{1}{c^k} = \frac{1}{a^k+b^k+c^k}.$$
algebra
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R to R, with x+f(xy)=f(1+f(y))x

by NicoN9, May 11, 2025, 8:52 AM

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that\[ x+f(xy)=f(1+f(y))x \]for all $x, y\in \mathbb{R}$.
This post has been edited 1 time. Last edited by NicoN9, 4 hours ago
fe
rational
algebra
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Radiant sets

by BR1F1SZ, May 10, 2025, 11:12 PM

A finite set $\mathcal S$ of distinct positive real numbers is called radiant if it satisfies the following property: if $a$ and $b$ are two distinct elements of $\mathcal S$, then $a^2 + b^2$ is also an element of $\mathcal S$.
  1. Does there exist a radiant set with a size greater than or equal to $4$?
  2. Determine all radiant sets of size $2$ or $3$.
algebra
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find angle

by TBazar, May 8, 2025, 6:57 AM

Given $ABC$ triangle with $AC>BC$. We take $M$, $N$ point on AC, AB respectively such that $AM=BC$, $CM=BN$. $BM$, $AN$ lines intersect at point $K$. If $2\angle AKM=\angle ACB$, find $\angle ACB$
geometry
Triangle
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Knights NOT crowded on the chessboard

by mshtand1, Mar 13, 2025, 10:45 PM

What is the maximum number of knights that can be placed on a chessboard of size \(8 \times 8\) such that any knight, after making 1 or 2 arbitrary moves, does not land on a square occupied by another knight?

Proposed by Bogdan Rublov
combinatorics
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Asymmetric FE

by sman96, Feb 8, 2025, 5:11 PM

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that$$f(xf(y)-y) + f(xy-x) + f(x+y) = 2xy$$for all $x, y \in \mathbb{R}$.
algebra
functional equation
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Synthetic Division for linear and higher degree divisors

by luimichael, Jan 23, 2020, 3:12 PM

The synthetic division for linear divisor is basically the same as the Horner's Method.
It can be extended to the general case with divisor of higher degrees.
Polynomials

hard inequality

by moldovan, Jul 8, 2009, 6:44 PM

Let $ x_1,...,x_n$ be positive real numbers. Prove that:

$ \displaystyle\sum_{k=1}^{n}x_k+\sqrt{\displaystyle\sum_{k=1}^{n}x_k^2} \le \frac{n+\sqrt{n}}{n^2} \left( \displaystyle\sum_{k=1}^{n} \frac{1}{x_k} \right) \left( \displaystyle\sum_{k=1}^{n} x_k^2 \right).$
inequalities
inequalities unsolved
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a