High School Math geometry

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High School Math Grades 9-12, Ages 13-18

Grades 9-12, Ages 13-18

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High School Math Grades 9-12, Ages 13-18

Grades 9-12, Ages 13-18

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Six variables

Nguyenhuyen_AG 2

N 26 minutes ago by arqady

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

2 replies

Nguyenhuyen_AG

Yesterday at 5:09 AM

arqady

26 minutes ago

Brilliant guessing game on triples

Assassino9931 2

N 28 minutes ago by Mirjalol

Source: Al-Khwarizmi Junior International Olympiad 2025 P8

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

2 replies

Assassino9931

Saturday at 9:46 AM

Mirjalol

28 minutes ago

ISI UGB 2025 P5

SomeonecoolLovesMaths 4

N an hour ago by Shiny_zubat

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 $k$ , $\frac{1}{a^k} + \frac{1}{b^k} + \frac{1}{c^k} = \frac{1}{a^k+b^k+c^k}.$

4 replies

SomeonecoolLovesMaths

Yesterday at 11:15 AM

Shiny_zubat

an hour ago

ISI UGB 2025 P2

SomeonecoolLovesMaths 6

N an hour ago by quasar_lord

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.

6 replies

SomeonecoolLovesMaths

Yesterday at 11:16 AM

quasar_lord

an hour ago

ISI UGB 2025 P6

SomeonecoolLovesMaths 3

N an hour ago by Shiny_zubat

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.

3 replies

SomeonecoolLovesMaths

Yesterday at 11:18 AM

Shiny_zubat

an hour ago

Shortest number theory you might've seen in your life

AlperenINAN 5

N an hour ago by Royal_mhyasd

Source: Turkey JBMO TST 2025 P4

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.

5 replies

AlperenINAN

Yesterday at 7:51 PM

Royal_mhyasd

an hour ago

d+2 pts in R^d can partition

EthanWYX2009 0

2 hours ago

Source: Radon's Theorem

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

0 replies

EthanWYX2009

2 hours ago

0 replies

hard inequality omg

tokitaohma 4

N 3 hours ago by arqady

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: $\dfrac{\sqrt{a^2+2ab}}{\sqrt{b^2+2c^2}} + \dfrac{\sqrt{b^2+2bc}}{\sqrt{c^2+2a^2}} + \dfrac{\sqrt{c^2+2ca}}{\sqrt{a^2+2b^2}} \geq \dfrac{1}{a^2+b^2+c^2}$

4 replies

tokitaohma

Yesterday at 5:24 PM

arqady

3 hours ago

ISI UGB 2025 P4

SomeonecoolLovesMaths 6

N 3 hours ago by Atmadeep

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 : $2025 = 3^4 \times 5^2$ )

6 replies

SomeonecoolLovesMaths

Yesterday at 11:24 AM

Atmadeep

3 hours ago

An innocent-looking inequality

Bryan0224 0

3 hours ago

Source: Idk

If $\{a_i\}_{1\le i\le n }$ and $\{b_i\}_{1\le i\le n}$ are two sequences between $1$ and $2$ and they satisfy $\sum_{i=1}^n a_i^2=\sum_{i=1}^n b_i^2$ , prove that $\sum_{i=1}^n\frac{a_i^3}{b_i}\leq 1.7\sum_{i=1}^{n} a_i^2$ , and determine when does equality hold
Please answer this @sqing :trampoline:

0 replies

Bryan0224

3 hours ago

0 replies

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