d+2 pts in R^d can partition

by EthanWYX2009, May 12, 2025, 5:12 AM

Show that: any set of $d + 2$ points in $\mathbb R^d$ can be partitioned into two sets whose convex hulls intersect.

Shortest number theory you might've seen in your life

by AlperenINAN, May 11, 2025, 7:51 PM

Let $p$ and $q$ be prime numbers. Prove that if $pq(p+1)(q+1)$ is a perfect square, then $pq + 1$ is also a perfect square.

This post has been edited 2 times. Last edited by AlperenINAN, Yesterday at 7:58 PM

number theory

prime numbers

hard inequality omg

by tokitaohma, May 11, 2025, 5:24 PM

1. Given $a, b, c > 0$ and $abc=1$
Prove that: $\sqrt{a^2+1} + \sqrt{b^2+1} + \sqrt{c^2+1} \leq \sqrt{2}(a+b+c)$

2. Given $a, b, c > 0$ and $a+b+c=1$
Prove that: $\dfrac{\sqrt{a^2+2ab}}{\sqrt{b^2+2c^2}} + \dfrac{\sqrt{b^2+2bc}}{\sqrt{c^2+2a^2}} + \dfrac{\sqrt{c^2+2ca}}{\sqrt{a^2+2b^2}} \geq \dfrac{1}{a^2+b^2+c^2}$

inequalities

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

ISI UGB 2025 P6

by SomeonecoolLovesMaths, May 11, 2025, 11:18 AM

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.

This post has been edited 1 time. Last edited by SomeonecoolLovesMaths, Yesterday at 11:19 AM

number theory

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, Yesterday at 12:00 PM

trigonometry

geometry

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

Six variables

by Nguyenhuyen_AG, May 11, 2025, 5:09 AM

Let $a,\,b,\,c,\,x,\,y,\,z$ be six positive real numbers. Prove that
$\frac{a}{b+c} \cdot \frac{y+z}{x} + \frac{b}{c+a} \cdot \frac{z+x}{y} + \frac{c}{a+b} \cdot \frac{x+y}{z} \geqslant 2+\sqrt{\frac{8abc}{(a+b)(b+c)(c+a)}}.$

inequalities

inequalities proposed

Brilliant guessing game on triples

by Assassino9931, May 10, 2025, 9:46 AM

There are $100$ cards on a table, flipped face down. Madina knows that on each card a single number is written and that the numbers are different integers from $1$ to $100$ . In a move, Madina is allowed to choose any $3$ cards, and she is told a number that is written on one of the chosen cards, but not which specific card it is on. After several moves, Madina must determine the written numbers on as many cards as possible. What is the maximum number of cards Madina can ensure to determine?

Shubin Yakov, Russia

This post has been edited 1 time. Last edited by Assassino9931, Saturday at 9:46 AM

combinatorics

Al-Khwarizmi IJMO

line JK of intersection points of 2 lines passes through the midpoint of BC

by parmenides51, Dec 11, 2018, 8:17 PM

Let $ABC$ be an acute triangle with $AC> AB$ . be $\Gamma$ the circumcircle circumscribed to the triangle $ABC$ and $D$ the midpoint of the smallest arc $BC$ of this circle. Let $E$ and $F$ points of the segments $AB$ and $AC$ respectively such that $AE = AF$ . Let $P \neq A$ be the second intersection point of the circumcircle circumscribed to $AEF$ with $\Gamma$ . Let $G$ and $H$ be the intersections of lines $PE$ and $PF$ with $\Gamma$ other than $P$ , respectively. Let $J$ and $K$ be the intersection points of lines $DG$ and $DH$ with lines $AB$ and $AC$ respectively. Show that the $JK$ line passes through the midpoint of $BC$

This post has been edited 1 time. Last edited by parmenides51, Jun 21, 2022, 1:39 AM

Submit

The most viewed blog of all time
by giangtruong13, Feb 24, 2025, 1:37 PM
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it started when I was an infant
wow and like 1.5 million views
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tbh it scares me too I still dont know where those views came from
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this blog started when i was like 2 months old
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1579363 visits rn, curious how many views (if any) this blog gets now
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EpicSkills32 reaped on Jul 27, 18:44:19 and gained 10 minutes, 57 seconds

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1.5M views pro
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BTW, lets talk about the NBA playoffs 2012
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ALONG WITH OCCASIONAL MISUSE OF CAPS
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contrib???
by El_Ectric, May 10, 2012, 7:13 PM
by EpicSkills32, May 2, 2012, 5:22 AM
by EpicSkills32, Apr 26, 2012, 5:11 PM

536 shouts

Seems