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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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Nordic 2025 P2

anirbanbz 7

N 2 hours ago by Mathdreams

Source: Nordic 2025

Let $p$ be a prime and suppose $2^{2p} \equiv 1 (\text{mod}$ $2p+1)$ is prime. Prove that $2p+1$ is prime $^{1}$

$^{1}$ This is a special case of Pocklington's theorem. A proof of this special case is required.

7 replies

anirbanbz

Yesterday at 12:35 PM

Mathdreams

2 hours ago

Lines AD, BE, and CF are concurrent

orl 45

N 2 hours ago by Mapism

Source: IMO Shortlist 2000, G3

Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$ . Show that there exist points $D$ , $E$ , and $F$ on sides $BC$ , $CA$ , and $AB$ respectively such that $OD + DH = OE + EH = OF + FH$ and the lines $AD$ , $BE$ , and $CF$ are concurrent.

45 replies

orl

Aug 10, 2008

Mapism

2 hours ago

Find f such that $f(f(f(x)))=x : \forall x \in R $

Lang_Tu_Mua_Bui 3

N 2 hours ago by jasperE3

Find f such that $f(f(f(x)))=x : \forall x \in R$

3 replies

Lang_Tu_Mua_Bui

Dec 2, 2015

jasperE3

2 hours ago

AM-GM problem from a handout

prtoi 1

N 2 hours ago by Primeniyazidayi

Prove that:
$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{3(abc)^{1/3}}{a+b+c}\ge3+n$

1 reply

prtoi

4 hours ago

Primeniyazidayi

2 hours ago

Cauchy Schwarz 4

prtoi 1

N 2 hours ago by Primeniyazidayi

Source: Zhautykov Olympiad 2008

Let a, b, c be positive real numbers such that abc = 1.
Show that
$\frac{1}{b(a+b)}+\frac{1}{b(a+b)}+\frac{1}{b(a+b)}\ge\frac{3}{2}$

1 reply

prtoi

4 hours ago

Primeniyazidayi

2 hours ago

Cauchy-Schwarz 1

prtoi 2

N 2 hours ago by Primeniyazidayi

Source: Handout by Samin Riasat

$\frac{a}{b+c}+\frac{b}{c+d}+\frac{c}{d+a}+\frac{d}{a+b}\ge2$

2 replies

prtoi

4 hours ago

Primeniyazidayi

2 hours ago

Parallel lines and angle congruences

math154 36

N 2 hours ago by ErTeeEs06

Source: ELMO Shortlist 2012, G5; also ELMO #5

Let $ABC$ be an acute triangle with $AB<AC$ , and let $D$ and $E$ be points on side $BC$ such that $BD=CE$ and $D$ lies between $B$ and $E$ . Suppose there exists a point $P$ inside $ABC$ such that $PD\parallel AE$ and $\angle PAB=\angle EAC$ . Prove that $\angle PBA=\angle PCA$ .

Calvin Deng.

36 replies

math154

Jul 2, 2012

ErTeeEs06

2 hours ago

Do you have any idea why they all call their problems' characters "Mykhailo"???

mshtand1 1

N 3 hours ago by ravengsd

Source: Ukrainian Mathematical Olympiad 2025. Day 2, Problem 10.7

In a row, $1000$ numbers $2$ and $2000$ numbers $-1$ are written in some order.
Mykhailo counted the number of groups of adjacent numbers, consisting of at least two numbers, whose sum equals $0$ .
(a) Find the smallest possible value of this number.
(b) Find the largest possible value of this number.

Proposed by Anton Trygub

1 reply

mshtand1

Mar 14, 2025

ravengsd

3 hours ago

Fridolin just can't get enough from jumping on the number line

Tintarn 1

N 3 hours ago by EmersonSoriano

Source: Bundeswettbewerb Mathematik 2025, Round 1 - Problem 1

Fridolin the frog jumps on the number line: He starts at $0$ , then jumps in some order on each of the numbers $1,2,\dots,9$ exactly once and finally returns with his last jump to $0$ . Can the total distance he travelled with these $10$ jumps be a) $20$ , b) $25$ ?

1 reply

Tintarn

Mar 17, 2025

EmersonSoriano

3 hours ago

function

MuradSafarli 3

N 3 hours ago by pco

Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying the equation for all real numbers $x, y$ :
$f(x^2 + y + f(y)) = f(x)^2$

3 replies

MuradSafarli

Today at 1:26 PM

pco

3 hours ago

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