Counting Numbers

by steven_zhang123, Mar 30, 2025, 12:12 AM

Let the decimal representations of numbers $A$ and $B$ be given as: $A = 0.a_1a_2\cdots a_k > 0$ , $B = 0.b_1b_2\cdots b_k > 0$ (where $a_k, b_k$ can be 0), and let $S$ be the count of numbers $0.c_1c_2\cdots c_k$ such that $0.c_1c_2\cdots c_k < A$ and $0.c_kc_{k-1}\cdots c_1 < B$ ( $c_k, c_1$ can also be 0). (Here, $0.c_1c_2\cdots c_r (c_r \neq 0)$ is considered the same as $0.c_1c_2\cdots c_r0\cdots0$ ).

Prove: $\left| S - 10^k AB \right| \leq 9k.$

China TST

combinatorics

Perfect Numbers

by steven_zhang123, Mar 30, 2025, 12:09 AM

If the sum of all positive divisors (including itself) of a positive integer $n$ is $2n$ , then $n$ is called a perfect number. For example, the sum of the positive divisors of 6 is $1 + 2 + 3 + 6 = 2 \times 6$ , hence 6 is a perfect number.
Prove: There does not exist a perfect number of the form $p^a q^b r^c$ , where $a, b, c$ are positive integers, and $p, q, r$ are odd primes.

China TST

number theory

An alien statement I came across

by GreekIdiot, Feb 15, 2025, 4:31 PM

Let $\mathbb{P} \subset \mathbb{N}$ be a set that intersects all non-finite integer arithmetic progressions, $\mathbb {A}$ be the set of prime divisors of $a^n-1$ and $\mathbb {B}$ be the set of prime divisors of $b^n-1$ . Suppose $\mathbb {B} \subset \mathbb {A} \hspace{2 mm} \forall \hspace{2mm} n \in \mathbb{P}$ . Prove that $b=a^k$ , $k \in \mathbb {N}$

Arithmetic Progression

Cyclotomic Polynomials

very hard

number theory

A projectional vision in IGO

by Shayan-TayefehIR, Nov 14, 2024, 7:59 PM

In the triangle $\bigtriangleup ABC$ let $D$ be the foot of the altitude from $A$ to the side $BC$ and $I$ , $I_A$ , $I_C$ be the incenter, $A$ -excenter, and $C$ -excenter, respectively. Denote by $P\neq B$ and $Q\neq D$ the other intersection points of the circle $\bigtriangleup BDI_C$ with the lines $BI$ and $DI_A$ , respectively. Prove that $AP=AQ$ .

Proposed Michal Jan'ik - Czech Republic

geometry

complex bash oops

by megahertz13, Nov 5, 2024, 2:33 AM

On a cyclic quadrilateral $ABCD$ , let $M$ and $N$ denote the midpoints of $\overline{AB}$ and $\overline{CD}$ . Let $E$ be the projection of $C$ onto $\overline{AB}$ and let $F$ be the reflection of $N$ over the midpoint of $\overline{DE}$ . Assume $F$ lies in the interior of quadrilateral $ABCD$ . Prove that $\angle BMF = \angle CBD$ .

geometry

complex bash

Circles tangent to BC at B and C

by MarkBcc168, Jun 22, 2024, 3:46 PM

Let $ABC$ be a triangle, and let $\omega_1,\omega_2$ be centered at $O_1$ , $O_2$ and tangent to line $BC$ at $B$ , $C$ respectively. Let line $AB$ intersect $\omega_1$ again at $X$ and let line $AC$ intersect $\omega_2$ again at $Y$ . If $Q$ is the other intersection of the circumcircles of triangles $ABC$ and $AXY$ , then prove that lines $AQ$ , $BC$ , and $O_1O_2$ either concur or are all parallel.

Advaith Avadhanam

geometry

Elmo

BAMO Geo

by jsdd_, Aug 11, 2019, 6:07 PM

Let $O = (0,0), A = (0,a), and B = (0,b)$ , where $0<b<a$ are reals. Let $\Gamma$ be a circle with diameter $\overline{AB}$ and let $P$ be any other point on $\Gamma$ . Line $PA$ meets the x-axis again at $Q$ . Prove that angle $\angle BQP = \angle BOP$ .

BAMO

geometry

Perpendicular following tangent circles

by buzzychaoz, Mar 21, 2016, 5:53 AM

The diagonals of a cyclic quadrilateral $ABCD$ intersect at $P$ , and there exist a circle $\Gamma$ tangent to the extensions of $AB,BC,AD,DC$ at $X,Y,Z,T$ respectively. Circle $\Omega$ passes through points $A,B$ , and is externally tangent to circle $\Gamma$ at $S$ . Prove that $SP\perp ST$ .

Attachments:

geometry

cyclic quadrilateral

harmonic division

Cono Sur Olympiad 2011, Problem 6

by Leicich, Aug 23, 2014, 2:37 AM

Let $Q$ be a $(2n+1) \times (2n+1)$ board. Some of its cells are colored black in such a way that every $2 \times 2$ board of $Q$ has at most $2$ black cells. Find the maximum amount of black cells that the board may have.

combinatorics proposed

combinatorics

Iran TST 2009-Day3-P3

by khashi70, May 16, 2009, 5:22 PM

In triangle $ABC$ , $D$ , $E$ and $F$ are the points of tangency of incircle with the center of $I$ to $BC$ , $CA$ and $AB$ respectively. Let $M$ be the foot of the perpendicular from $D$ to $EF$ . $P$ is on $DM$ such that $DP = MP$ . If $H$ is the orthocenter of $BIC$ , prove that $PH$ bisects $EF$ .

This post has been edited 1 time. Last edited by khashi70, May 16, 2009, 6:07 PM

geometry

geometric transformation

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I am now able to make clickable images in my posts!
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43 shouts

Pi in the Sky

by steven_zhang123, Mar 30, 2025, 12:12 AM

by steven_zhang123, Mar 30, 2025, 12:09 AM

by GreekIdiot, Feb 15, 2025, 4:31 PM

by Shayan-TayefehIR, Nov 14, 2024, 7:59 PM

by megahertz13, Nov 5, 2024, 2:33 AM

by MarkBcc168, Jun 22, 2024, 3:46 PM

by jsdd_, Aug 11, 2019, 6:07 PM

by buzzychaoz, Mar 21, 2016, 5:53 AM

by Leicich, Aug 23, 2014, 2:37 AM

by khashi70, May 16, 2009, 5:22 PM