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ISI UGB 2025 P7
SomeonecoolLovesMaths   2
N 27 minutes ago by Primeniyazidayi
Source: ISI UGB 2025 P7
Consider a ball that moves inside an acute-angled triangle along a straight line, unit it hits the boundary, which is when it changes direction according to the mirror law, just like a ray of light (angle of incidence = angle of reflection). Prove that there exists a triangular periodic path for the ball, as pictured below.

IMAGE
2 replies
SomeonecoolLovesMaths
an hour ago
Primeniyazidayi
27 minutes ago
ISI UGB 2025 P5
SomeonecoolLovesMaths   2
N 41 minutes ago by SomeonecoolLovesMaths
Source: ISI UGB 2025 P5
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 $k$, $$\frac{1}{a^k} + \frac{1}{b^k} + \frac{1}{c^k} = \frac{1}{a^k+b^k+c^k}.$$
2 replies
SomeonecoolLovesMaths
2 hours ago
SomeonecoolLovesMaths
41 minutes ago
A bit tricky invariant with 98 numbers on the board.
Nuran2010   0
an hour ago
Source: Azerbaijan Al-Khwarizmi IJMO TST 2025
The numbers $\frac{50}{1},\frac{50}{2},...\frac{50}{97},\frac{50}{98}$ are written on the board.In each step,two random numbers $a$ and $b$ are chosen and deleted.Then,the number $2ab-a-b-1$ is written instead.What will be the number remained on the board after the last step.
0 replies
Nuran2010
an hour ago
0 replies
Taking antipode on isosceles triangle's circumcenter
Nuran2010   0
an hour ago
Source: Azerbaijan Al-Khwarizmi IJMO TST 2025
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 $DFC$.
0 replies
Nuran2010
an hour ago
0 replies
R to R, with x+f(xy)=f(1+f(y))x
NicoN9   3
N an hour ago by EeEeRUT
Source: Own.
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that\[
x+f(xy)=f(1+f(y))x
\]for all $x, y\in \mathbb{R}$.
3 replies
NicoN9
4 hours ago
EeEeRUT
an hour ago
find angle
TBazar   7
N an hour ago by TBazar
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 $\angle ACB$
7 replies
TBazar
May 8, 2025
TBazar
an hour ago
ISI UGB 2025 P4
SomeonecoolLovesMaths   0
an hour ago
Source: ISI UGB 2025 P4
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 : $2025 = 3^4 \times 5^2$)
0 replies
SomeonecoolLovesMaths
an hour ago
0 replies
ISI UGB 2025 P8
SomeonecoolLovesMaths   0
an hour ago
Source: ISI UGB 2025 P8
Let $n \geq 2$ and let $a_1 \leq a_2 \leq \cdots \leq a_n$ be positive integers such that $\sum_{i=1}^{n} a_i = \prod_{i=1}^{n} a_i$. Prove that $\sum_{i=1}^{n} a_i \leq 2n$ and determine when equality holds.
0 replies
SomeonecoolLovesMaths
an hour ago
0 replies
ISI UGB 2025 P6
SomeonecoolLovesMaths   0
an hour ago
Source: ISI UGB 2025 P6
Let $\mathbb{N}$ denote the set of natural numbers, and let $\left( a_i, b_i \right)$, $1 \leq i \leq 9$, be nine distinct tuples in $\mathbb{N} \times \mathbb{N}$. Show that there are three distinct elements in the set $\{ 2^{a_i} 3^{b_i} \colon 1 \leq i \leq 9 \}$ whose product is a perfect cube.
0 replies
SomeonecoolLovesMaths
an hour ago
0 replies
ISI UGB 2025 P2
SomeonecoolLovesMaths   0
2 hours ago
Source: ISI UGB 2025 P2
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.
0 replies
SomeonecoolLovesMaths
2 hours ago
0 replies
Six variables
Nguyenhuyen_AG   1
N 2 hours ago by TNKT
Let $a,\,b,\,c,\,x,\,y,\,z$ be six positive real numbers. Prove that
$$\frac{a}{b+c} \cdot \frac{y+z}{x} + \frac{b}{c+a} \cdot \frac{z+x}{y} + \frac{c}{a+b} \cdot \frac{x+y}{z} \geqslant 2+\sqrt{\frac{8abc}{(a+b)(b+c)(c+a)}}.$$
1 reply
Nguyenhuyen_AG
Today at 5:09 AM
TNKT
2 hours ago
Perspective Triangles
MNJ2357   2
N Jan 26, 2020 by Pathological
Source: Korea Winter Program Practice Test 2 P4
$\triangle ABC$ and $\triangle A_1B_1C_1$ are perspective triangles. $(ABB_1)$ and $(ACC_1)$ meet at $A_2 (\neq A)$. Define $B_2,C_2$ analogously. Prove that $AA_2, BB_2,CC_2$ are concurrent.
2 replies
MNJ2357
Jan 23, 2020
Pathological
Jan 26, 2020
Perspective Triangles
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Source: Korea Winter Program Practice Test 2 P4
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MNJ2357
644 posts
#1 • 2 Y
Y by stroller, Adventure10
$\triangle ABC$ and $\triangle A_1B_1C_1$ are perspective triangles. $(ABB_1)$ and $(ACC_1)$ meet at $A_2 (\neq A)$. Define $B_2,C_2$ analogously. Prove that $AA_2, BB_2,CC_2$ are concurrent.
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stroller
894 posts
#2 • 3 Y
Y by MNJ2357, Adventure10, Mango247
mOvInG pOiNtS

why am I such a troll
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Pathological
578 posts
#4 • 3 Y
Y by stroller, Pluto1708, Adventure10
Easy Trig
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