Do not try to case bash lol

by ItzsleepyXD, Apr 30, 2025, 9:08 AM

Let $n,d\geqslant 6$ be a positive integer such that $d\mid 6^{n!}+1$ .
Prove that $d>2n+6$ .

Short combi omg

by Davdav1232, Feb 3, 2025, 7:50 PM

Let $n$ be a positive integer. A graph on $2n - 1$ vertices is given such that the size of the largest clique in the graph is $n$ . Prove that there exists a vertex that is present in every clique of size $n$

combinatorics

graph theory

Tilted Students Thoroughly Splash Tiger part 2

by DottedCaculator, Jun 21, 2024, 4:17 PM

In triangle $ABC$ with $AB<AC$ and $AB+AC=2BC$ , let $M$ be the midpoint of $\overline{BC}$ . Choose point $P$ on the extension of $\overline{BA}$ past $A$ and point $Q$ on segment $\overline{AC}$ such that $M$ lies on $\overline{PQ}$ . Let $X$ be on the opposite side of $\overline{AB}$ from $C$ such that $\overline{AX} \parallel \overline{BC}$ and $AX=AP=AQ$ . Let $\overline{BX}$ intersect the circumcircle of $BMQ$ again at $Y \neq B$ , and let $\overline{CX}$ intersect the circumcircle of $CMP$ again at $Z \neq C$ . Prove that $A$ , $Y$ , and $Z$ are collinear.

Tiger Zhang

This post has been edited 1 time. Last edited by DottedCaculator, Jun 21, 2024, 4:17 PM

Monochromatic bipartite subgraphs

by L567, Jan 8, 2023, 2:40 PM

For a positive integer $n$ , let $f(n)$ denote the largest integer such that for any coloring of a $K_{n,n}$ with two colors, there exists a monochromatic subgraph of $K_{n,n}$ isomorphic to $K_{f(n), f(n)}$ . Is it true that for each positive integer $m$ we can find a natural $N$ such that for any integer $n \geqslant N$ , $f(n) \geqslant m$ ?

Proposed by Suchir

combinatorics

Nordic squares!

by mathisreaI, Jul 13, 2022, 2:56 AM

Let $n$ be a positive integer. A Nordic square is an $n \times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only to cells containing larger numbers is called a valley. An uphill path is a sequence of one or more cells such that:

(i) the first cell in the sequence is a valley,

(ii) each subsequent cell in the sequence is adjacent to the previous cell, and

(iii) the numbers written in the cells in the sequence are in increasing order.

Find, as a function of $n$ , the smallest possible total number of uphill paths in a Nordic square.

Author: Nikola Petrović

This post has been edited 1 time. Last edited by elitza, Jul 15, 2022, 11:59 AM
Reason: Added credit to problem's author.

Easy integer functional equation

by MarkBcc168, Jun 11, 2019, 12:20 AM

Let $\mathbb{Z}^+$ be the set of positive integers. Determine all functions $f : \mathbb{Z}^+\to\mathbb{Z}^+$ such that $a^2+f(a)f(b)$ is divisible by $f(a)+b$ for all positive integers $a,b$ .

Isi 2016 geometry

by zizou10, May 8, 2016, 12:33 PM

Prove that there exists a right angle triangle with rational sides and area $d$ if and only if $x^2,y^2$ and $z^2$ are squares of rational numbers and are in Arithmetic Progression

Here $d$ is an integer.

This post has been edited 3 times. Last edited by zizou10, May 29, 2016, 6:41 AM

geometry

Find area!

by ComplexPhi, Feb 4, 2015, 4:08 PM

Let $O_1$ be a point in the exterior of the circle $\omega$ of center $O$ and radius $R$ , and let $O_1N$ , $O_1D$ be the tangent segments from $O_1$ to the circle. On the segment $O_1N$ consider the point $B$ such that $BN=R$ .Let the line from $B$ parallel to $ON$ intersect the segment $O_1D$ at $C$ . If $A$ is a point on the segment $O_1D$ other than $C$ so that $BC=BA=a$ , and if the incircle of the triangle $ABC$ has radius $r$ , then find the area of $\triangle ABC$ in terms of $a ,R ,r$ .

geometry

The product of two p-pods is a p-pod

by MellowMelon, Jul 26, 2011, 9:17 PM

Let $p$ be a prime. We say that a sequence of integers $\{z_n\}_{n=0}^\infty$ is a $p$ -pod if for each $e \geq 0$ , there is an $N \geq 0$ such that whenever $m \geq N$ , $p^e$ divides the sum
$\sum_{k=0}^m (-1)^k {m \choose k} z_k.$
Prove that if both sequences $\{x_n\}_{n=0}^\infty$ and $\{y_n\}_{n=0}^\infty$ are $p$ -pods, then the sequence $\{x_ny_n\}_{n=0}^\infty$ is a $p$ -pod.

function

algebra

polynomial

number theory unsolved

number theory

-2 belongs to S

by WakeUp, Mar 19, 2011, 1:47 PM

Let $S$ be a set of integers containing the numbers $0$ and $1996$ . Suppose further that any integer root of any non-zero polynomial with coefficients in $S$ also belongs to $S$ . Prove that $-2$ belongs to $S$ .

algebra

polynomial

algebra proposed

Submit

how do you have so many posts
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PLEASE CONTRIB

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Enjoy. (no offense)
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by asf, Sep 11, 2010, 5:18 PM

91 shouts

math_exploration

by ItzsleepyXD, Apr 30, 2025, 9:08 AM

by Davdav1232, Feb 3, 2025, 7:50 PM

by DottedCaculator, Jun 21, 2024, 4:17 PM

by L567, Jan 8, 2023, 2:40 PM

by mathisreaI, Jul 13, 2022, 2:56 AM

by MarkBcc168, Jun 11, 2019, 12:20 AM

by zizou10, May 8, 2016, 12:33 PM

by ComplexPhi, Feb 4, 2015, 4:08 PM

by MellowMelon, Jul 26, 2011, 9:17 PM

by WakeUp, Mar 19, 2011, 1:47 PM