Easy combinatorics

by GreekIdiot, May 16, 2025, 2:00 PM

You are given a $5 \times 5$ grid with each cell colored with an integer from $0$ to $15$. Two players take turns. On a turn, a player may increase any one cell’s value by a power of 2 (i.e., add 1, 2, 4, or 8 mod 16). The first player wins if, after their move, the sum of each row and the sum of each column is congruent to 0 modulo 16. Prove whether or not Player 1 has a forced win strategy from any starting configuration.

Simple but hard

by Lukariman, May 16, 2025, 2:47 AM

Given triangle ABC. Outside the triangle, construct rectangles ACDE and BCFG with equal areas. Let M be the midpoint of DF. Prove that CM passes through the center of the circle circumscribing triangle ABC.
Attachments:
This post has been edited 2 times. Last edited by Lukariman, 6 hours ago

Good Permutations in Modulo n

by swynca, Apr 27, 2025, 2:03 PM

An integer $n > 1$ is called $\emph{good}$ if there exists a permutation $a_1, a_2, a_3, \dots, a_n$ of the numbers $1, 2, 3, \dots, n$, such that:
$(i)$ $a_i$ and $a_{i+1}$ have different parities for every $1 \leq i \leq n-1$;
$(ii)$ the sum $a_1 + a_2 + \cdots + a_k$ is a quadratic residue modulo $n$ for every $1 \leq k \leq n$.
Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.
This post has been edited 2 times. Last edited by swynca, Apr 27, 2025, 4:15 PM

sqrt(2)<=|1+z|+|1+z^2|<=4

by SuiePaprude, Jan 23, 2025, 10:53 PM

let z be a complex number with |z|=1 show that sqrt2 <=|1+z|+|1+z^2|<=4

concyclic wanted, diameter related

by parmenides51, May 5, 2024, 1:26 AM

As shown in the figure, $AB$ is the diameter of circle $\odot O$, and chords $AC$ and $BD$ intersect at point $E$, $EF\perp AB$ intersects at point $F$, and $FC$ intersects $BD$ at point $G$. Point $M$ lies on $AB$ such that $MD=MG$ . Prove that points $F$, $M$, $D$, $G$ lies on a circle.
https://cdn.artofproblemsolving.com/attachments/2/3/614ef5b9e8c8b16a29b8b960290ef9d7297529.jpg
This post has been edited 1 time. Last edited by parmenides51, May 5, 2024, 1:27 AM

Grid combo with tilings

by a_507_bc, Apr 23, 2023, 3:40 PM

A square grid $100 \times 100$ is tiled in two ways - only with dominoes and only with squares $2 \times 2$. What is the least number of dominoes that are entirely inside some square $2 \times 2$?

<ACB=90^o if AD = BD , <ACD = 3 <BAC, AM=//MD, CM//AB,

by parmenides51, Oct 7, 2022, 8:15 PM

In the convex quadrilateral $ABCD$, $AD = BD$ and $\angle ACD  = 3 \angle BAC$. Let $M$ be the midpoint of side $AD$. If the lines $CM$ and $AB$ are parallel, prove that the angle $\angle  ACB$ is right.

Concurrency in Parallelogram

by amuthup, Jul 12, 2022, 12:25 PM

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.
This post has been edited 1 time. Last edited by amuthup, Jul 15, 2022, 4:57 PM

bulgarian concurrency, parallelograms and midpoints related

by parmenides51, May 28, 2019, 2:10 PM

In a triangle $\triangle ABC$ points $L, P$ and $Q$ lie on the segments $AB, AC$ and $BC$, respectively, and are such that $PCQL$ is a parallelogram. The circle with center the midpoint $M$ of the segment $AB$ and radius $CM$ and the circle of diameter $CL$ intersect for the second time at the point $T$. Prove that the lines $AQ, BP$ and $LT$ intersect in a point.

Reel Math

by rrusczyk, Nov 12, 2011, 11:43 PM

Quick reminder for those of you participating in the Reel Math Challenge from MATHCOUNTS: the public voting starts Tuesday. Public voting is part of the evaluation of the videos, so obviously students who get their videos in early will have an advantage! That said, if you don't have your video finished by Tuesday, you can still enter. The voting runs until February 1st of next year, so there's still plenty of time to finish your video and rally your voting forces.

MATHCOUNTS Movies

by rrusczyk, Sep 28, 2011, 11:10 PM

MATHCOUNTS has launched a new contest in which students are challenged to make movies out of MATHCOUNTS problems. (Hey, that sounds familiar!) The program is called Reel Math, and you can check it out here. Winners get a free trip to Nationals in Orlando!

Concurrency

by Omid Hatami, May 20, 2008, 9:29 AM

Suppose that $ I$ is incenter of triangle $ ABC$ and $ l'$ is a line tangent to the incircle. Let $ l$ be another line such that intersects $ AB,AC,BC$ respectively at $ C',B',A'$. We draw a tangent from $ A'$ to the incircle other than $ BC$, and this line intersects with $ l'$ at $ A_1$. $ B_1,C_1$ are similarly defined. Prove that $ AA_1,BB_1,CC_1$ are concurrent.

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