High School Math ratio

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Cute NT Problem

M11100111001Y1R 4

N 2 hours ago by RANDOM__USER

Source: Iran TST 2025 Test 4 Problem 1

A number $n$ is called lucky if it has at least two distinct prime divisors and can be written in the form:
$n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}$ where $p_1, \dots, p_k$ are distinct prime numbers that divide $n$ . (Note: it is possible that $n$ has other prime divisors not among $p_1, \dots, p_k$ .) Prove that for every prime number $p$ , there exists a lucky number $n$ such that $p \mid n$ .

4 replies

M11100111001Y1R

Today at 7:20 AM

RANDOM__USER

2 hours ago

J

H

USAMO 2003 Problem 4

MithsApprentice 72

N 2 hours ago by endless_abyss

Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$ , respectively. Lines $AB$ and $DE$ intersect at $F$ , while lines $BD$ and $CF$ intersect at $M$ . Prove that $MF = MC$ if and only if $MB\cdot MD = MC^2$ .

72 replies

MithsApprentice

Sep 27, 2005

endless_abyss

2 hours ago

J

H

Easy but unusual junior ineq

Maths_VC 1

N 2 hours ago by blug

Source: Serbia JBMO TST 2025, Problem 2

Real numbers $x, y$ $\ge$ $0$ satisfy $1$ $\le$ $x^2 + y^2$ $\le$ $5$ . Determine the minimal and the maximal value of the expression $2x + y$

1 reply

Maths_VC

3 hours ago

blug

2 hours ago

J

H

Bosnia and Herzegovina JBMO TST 2009 Problem 1

gobathegreat 1

N 2 hours ago by FishkoBiH

Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2009

Lengths of sides of triangle $ABC$ are positive integers, and smallest side is equal to $2$ . Determine the area of triangle $P$ if $v_c = v_a + v_b$ , where $v_a$ , $v_b$ and $v_c$ are lengths of altitudes in triangle $ABC$ from vertices $A$ , $B$ and $C$ , respectively.

1 reply

gobathegreat

Sep 17, 2018

FishkoBiH

2 hours ago

J

H

USAMO 2001 Problem 2

MithsApprentice 53

N 3 hours ago by lksb

Let $ABC$ be a triangle and let $\omega$ be its incircle. Denote by $D_1$ and $E_1$ the points where $\omega$ is tangent to sides $BC$ and $AC$ , respectively. Denote by $D_2$ and $E_2$ the points on sides $BC$ and $AC$ , respectively, such that $CD_2=BD_1$ and $CE_2=AE_1$ , and denote by $P$ the point of intersection of segments $AD_2$ and $BE_2$ . Circle $\omega$ intersects segment $AD_2$ at two points, the closer of which to the vertex $A$ is denoted by $Q$ . Prove that $AQ=D_2P$ .

53 replies

MithsApprentice

Sep 30, 2005

lksb

3 hours ago

J

H

A=b

k2c901_1 89

N 3 hours ago by reni_wee

Source: Taiwan 1st TST 2006, 1st day, problem 3

Let $a$ , $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$ . Prove that $a=b$ .

Proposed by Mohsen Jamali, Iran

89 replies

k2c901_1

Mar 29, 2006

reni_wee

3 hours ago

J

H

Strange angle condition and concyclic points

lminsl 129

N 3 hours ago by Aiden-1089

Source: IMO 2019 Problem 2

In triangle $ABC$ , point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$ . Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$ , respectively, such that $PQ$ is parallel to $AB$ . Let $P_1$ be a point on line $PB_1$ , such that $B_1$ lies strictly between $P$ and $P_1$ , and $\angle PP_1C=\angle BAC$ . Similarly, let $Q_1$ be the point on line $QA_1$ , such that $A_1$ lies strictly between $Q$ and $Q_1$ , and $\angle CQ_1Q=\angle CBA$ .

Prove that points $P,Q,P_1$ , and $Q_1$ are concyclic.

Proposed by Anton Trygub, Ukraine

129 replies

lminsl

Jul 16, 2019

Aiden-1089

3 hours ago

J

H

Simple inequality

sqing 12

N 3 hours ago by Rayvhs

Source: MEMO 2018 T1

Let $a,b$ and $c$ be positive real numbers satisfying $abc=1.$ Prove that $\frac{a^2-b^2}{a+bc}+\frac{b^2-c^2}{b+ca}+\frac{c^2-a^2}{c+ab}\leq a+b+c-3.$

12 replies

sqing

Sep 2, 2018

Rayvhs

3 hours ago

J

H

Random concyclicity in a square config

Maths_VC 2

N 3 hours ago by Maths_VC

Source: Serbia JBMO TST 2025, Problem 1

Let $M$ be a random point on the smaller arc $AB$ of the circumcircle of square $ABCD$ , and let $N$ be the intersection point of segments $AC$ and $DM$ . The feet of the tangents from point $D$ to the circumcircle of the triangle $OMN$ are $P$ and $Q$ , where $O$ is the center of the square. Prove that points $A$ , $C$ , $P$ and $Q$ lie on a single circle.

2 replies

Maths_VC

4 hours ago

Maths_VC

3 hours ago

J

H

Serbian selection contest for the IMO 2025 - P3

OgnjenTesic 3

N 3 hours ago by atdaotlohbh

Source: Serbian selection contest for the IMO 2025

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that:
- $f$ is strictly increasing,
- there exists $M \in \mathbb{N}$ such that $f(x+1) - f(x) < M$ for all $x \in \mathbb{N}$ ,
- for every $x \in \mathbb{Z}$ , there exists $y \in \mathbb{Z}$ such that
$f(y) = \frac{f(x) + f(x + 2024)}{2}.$ Proposed by Pavle Martinović

3 replies

OgnjenTesic

May 22, 2025

atdaotlohbh

3 hours ago

J