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High School Olympiads Regional, national, and international math olympiads

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High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

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China Northern MO 2009 p4 CNMO

parkjungmin 1

N an hour ago by WallyWalrus

Source: China Northern MO 2009 p4 CNMO P4

The problem is too difficult.

1 reply

parkjungmin

Apr 30, 2025

WallyWalrus

an hour ago

Polynomial Squares

zacchro 26

N an hour ago by Mathandski

Source: USA December TST for IMO 2017, Problem 3, by Alison Miller

Let $P, Q \in \mathbb{R}[x]$ be relatively prime nonconstant polynomials. Show that there can be at most three real numbers $\lambda$ such that $P + \lambda Q$ is the square of a polynomial.

Alison Miller

26 replies

zacchro

Dec 11, 2016

Mathandski

an hour ago

Mmo 9-10 graders P5

Bet667 8

N an hour ago by User141208

Let $a,b,c,d$ be real numbers less than 2.Then prove that $\frac{a^3}{b^2+4}+\frac{b^3}{c^2+4}+\frac{c^3}{d^2+4}+\frac{d^3}{a^2+4}\le4$

8 replies

Bet667

Apr 3, 2025

User141208

an hour ago

Tangent to two circles

Mamadi 1

N an hour ago by ricarlos

Source: Own

Two circles $w_1$ and $w_2$ intersect each other at $M$ and $N$ . The common tangent to two circles nearer to $M$ touch $w_1$ and $w_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$ . The circumcircle of the triangle $DCM$ intersect circles $w_1$ and $w_2$ respectively at points $E$ and $F$ (both distinct from $M$ ). Show that the line $EF$ is the second tangent to $w_1$ and $w_2$ .

1 reply

Mamadi

Today at 7:01 AM

ricarlos

an hour ago

China Northern MO 2009 p4 CNMO

parkjungmin 2

N an hour ago by WallyWalrus

Source: China Northern MO 2009 p4 CNMO

China Northern MO 2009 p4 CNMO

The problem is too difficult.
Is there anyone who can help me?

2 replies

parkjungmin

Apr 30, 2025

WallyWalrus

an hour ago

Problem 4

codyj 86

N an hour ago by Mathgloggers

Source: IMO 2015 #4

Triangle $ABC$ has circumcircle $\Omega$ and circumcenter $O$ . A circle $\Gamma$ with center $A$ intersects the segment $BC$ at points $D$ and $E$ , such that $B$ , $D$ , $E$ , and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$ , such that $A$ , $F$ , $B$ , $C$ , and $G$ lie on $\Omega$ in this order. Let $K$ be the second point of intersection of the circumcircle of triangle $BDF$ and the segment $AB$ . Let $L$ be the second point of intersection of the circumcircle of triangle $CGE$ and the segment $CA$ .

Suppose that the lines $FK$ and $GL$ are different and intersect at the point $X$ . Prove that $X$ lies on the line $AO$ .

Proposed by Greece

86 replies

codyj

Jul 11, 2015

Mathgloggers

an hour ago

Israeli Mathematical Olympiad 1995

YanYau 24

N an hour ago by bjump

Source: Israeli Mathematical Olympiad 1995

Let $PQ$ be the diameter of semicircle $H$ . Circle $O$ is internally tangent to $H$ and tangent to $PQ$ at $C$ . Let $A$ be a point on $H$ and $B$ a point on $PQ$ such that $AB\perp PQ$ and is tangent to $O$ . Prove that $AC$ bisects $\angle PAB$

24 replies

YanYau

Apr 8, 2016

bjump

an hour ago

P(x), integer, integer roots, P(0) =-1,P(3) = 128

parmenides51 3

N an hour ago by Rohit-2006

Source: Nordic Mathematical Contest 1989 #1

Find a polynomial $P$ of lowest possible degree such that
(a) $P$ has integer coefficients,
(b) all roots of $P$ are integers,
(c) $P(0) = -1$ ,
(d) $P(3) = 128$ .

3 replies

parmenides51

Oct 5, 2017

Rohit-2006

an hour ago

2017 CGMO P1

smy2012 9

N an hour ago by Bardia7003

Source: 2017 CGMO P1

(1) Find all positive integer $n$ such that for any odd integer $a$ , we have $4\mid a^n-1$
(2) Find all positive integer $n$ such that for any odd integer $a$ , we have $2^{2017}\mid a^n-1$

9 replies

smy2012

Aug 13, 2017

Bardia7003

an hour ago

Euler's function

luutrongphuc 1

N 2 hours ago by luutrongphuc

Find all real numbers $\alpha$ such that for every positive real $c$ , there exists an integer $n>1$ satisfying
$\frac{\varphi(n!)}{n^\alpha\,(n-1)!} \;>\; c.$

1 reply

luutrongphuc

2 hours ago

luutrongphuc

2 hours ago

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