Ah, easy one

by irregular22104, May 14, 2025, 4:01 PM

In the number series $1,9,9,9,8,...,$ every next number (from the fifth number) is the unit number of the sum of the four numbers preceding it. Is there any cases that we get the numbers $1234$ and $5678$ in this series?
combinatorics
L

truncated cone box packing problem

by chomk, May 14, 2025, 3:44 PM

box : 48*48*32
truncated cone: upper circle(radius=2), lower circle(radius=8), height=12

how many truncated cones are packed in a box?
geometry
3D geometry
L

Hard Inequality

by Asilbek777, May 14, 2025, 3:21 PM

Waits for Solution
Attachments:
This post has been edited 2 times. Last edited by Asilbek777, an hour ago
inequalities
algebra
Trigonometric inequality
trigonometry
geometry
geometric transformation
reflection
L

Proving that these are concyclic.

by Acrylic3491, May 14, 2025, 9:06 AM

In $\bigtriangleup ABC$, points $P$ and $Q$ are isogonal conjugates. The tangent to $(BPC)$ at $P$ and the tangent to $(BQC)$ at Q, meet at $R$. $AR$ intersects $(ABC)$ at $D$. Prove that points $P$,$Q$, $R$ and $D$ are concyclic.

Any hints on this ?
This post has been edited 2 times. Last edited by Acrylic3491, 3 hours ago
geometry
L

Interesting inequality

by sealight2107, May 6, 2025, 4:53 PM

Let $a,b,c>0$ such that $a+b+c=3$. Find the minimum value of:
$Q=\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{1}{a^3+b^3+abc}+\frac{1}{b^3+c^3+abc}+\frac{1}{c^3+a^3+abc}$
inequalities
inequalities unsolved
L

student that has at least 10 friends

by parmenides51, May 17, 2024, 8:16 PM

A class has $24$ students. Each group consisting of three of the students meet, and choose one of the other $21$ students, A, to make him a gift. In this case, A considers each member of the group that offered him a gift as being his friend. Prove that there is a student that has at least $10$ friends.
combinatorics
L

Integer polynomial commutes with sum of digits

by cjquines0, Jul 19, 2017, 4:38 PM

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
Proposed by Warut Suksompong, Thailand
This post has been edited 1 time. Last edited by Amir Hossein, Jul 19, 2017, 5:11 PM
Reason: Added the proposer.
number theory
IMO Shortlist
polynomial
functional equation
sum of digits
L

Dwarfes and river

by RagvaloD, May 3, 2017, 12:07 PM

There are $100$ dwarfes with weight $1,2,...,100$. They sit on the left riverside. They can not swim, but they have one boat with capacity 100. River has strong river flow, so every dwarf has power only for one passage from right side to left as oarsman. On every passage can be only one oarsman. Can all dwarfes get to right riverside?
number theory
combinatorics
L

Concurrent Gergonnians in Pentagon

by numbertheorist17, Jul 16, 2014, 12:09 PM

Consider a convex pentagon circumscribed about a circle. We name the lines that connect vertices of the pentagon with the opposite points of tangency with the circle gergonnians.
(a) Prove that if four gergonnians are conncurrent, the all five of them are concurrent.
(b) Prove that if there is a triple of gergonnians that are concurrent, then there is another triple of gergonnians that are concurrent.
trigonometry
geometry
circumcircle
analytic geometry
projective geometry
complex numbers
L

PA = QB

by zhaoli, Sep 2, 2005, 3:37 PM

$AB$ is a chord of a circle with center $O$, $M$ is the midpoint of $AB$. A non-diameter chord is drawn through $M$ and intersects the circle at $C$ and $D$. The tangents of the circle from points $C$ and $D$ intersect line $AB$ at $P$ and $Q$, respectively. Prove that $PA$ = $QB$.
trigonometry
geometry proposed
geometry
L

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