NEPAL TST DAY 2 PROBLEM 2

by Tony_stark0094, Apr 12, 2025, 8:37 AM

Kritesh manages traffic on a $45 \times 45$ grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the $45 \times 45$ grid.

Kritesh realizes that it may not always be possible for all the cars to leave the grid. Therefore, before the process begins, he will remove $k$ cars from the $45 \times 45$ grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid.

What is the minimum value of $k$ that guarantees that Kritesh's job is possible?

combinatorics

prove that any quadrilateral satisfying this inequality is a trapezoid

by mqoi_KOLA, Apr 12, 2025, 3:48 AM

Prove that any quadrilateral satisfying this inequality is a Trapezoid/trapzium $|r - p| < q + s < r + p$ where $p,r$ are lengths of parallel sides and $q,s$ are other two sides.

This post has been edited 2 times. Last edited by mqoi_KOLA, Today at 4:46 AM

Inequality with a,b,c

by GeoMorocco, Apr 10, 2025, 9:51 PM

Let $a,b,c$ be positive real numbers such that : $ab+bc+ca=3$ . Prove that : $\frac{a\sqrt{3+bc}}{b+c}+\frac{b\sqrt{3+ca}}{c+a}+\frac{c\sqrt{3+ab}}{a+b}\ge a+b+c$

inequalities

Problem 3

by SlovEcience, Apr 9, 2025, 5:37 PM

Find all real numbers $k$ such that the following inequality holds for all $a, b, c \geq 0$ :

$ab + bc + ca \leq \frac{(a + b + c)^2}{3} + k \cdot \max \{ (a - b)^2, (b - c)^2, (c - a)^2 \} \leq a^2 + b^2 + c^2$

inequalities

A and B play a game

by EthanWYX2009, Mar 29, 2025, 2:49 PM

Let $n \geq 2$ be an integer. Two players, Alice and Bob, play the following game on the complete graph $K_n$ : They take turns to perform operations, where each operation consists of coloring one or two edges that have not been colored yet. The game terminates if at any point there exists a triangle whose three edges are all colored.

Prove that there exists a positive number $\varepsilon$ , Alice has a strategy such that, no matter how Bob colors the edges, the game terminates with the number of colored edges not exceeding
$\left( \frac{1}{4} - \varepsilon \right) n^2 + n.$

This post has been edited 1 time. Last edited by EthanWYX2009, Mar 29, 2025, 2:56 PM

combinatorics

China TST

Abelkonkurransen 2025 1a

by Lil_flip38, Mar 20, 2025, 11:03 AM

Peer and Solveig are playing a game with $n$ coins, all of which show $M$ on one side and $K$ on the opposite side. The coins are laid out in a row on the table. Peer and Solveig alternate taking turns. On his turn, Peer may turn over one or more coins, so long as he does not turn over two adjacent coins. On her turn, Solveig picks precisely two adjacent coins and turns them over. When the game begins, all the coins are showing $M$ . Peer plays first, and he wins if all the coins show $K$ simultaneously at any time. Find all $n\geqslant 2$ for which Solveig can keep Peer from winning.

This post has been edited 3 times. Last edited by Lil_flip38, Mar 20, 2025, 11:05 AM

game

combinatorics

Rhombus EVAN

by 62861, Feb 23, 2017, 5:13 PM

Let $ABC$ be a triangle with altitude $\overline{AE}$ . The $A$ -excircle touches $\overline{BC}$ at $D$ , and intersects the circumcircle at two points $F$ and $G$ . Prove that one can select points $V$ and $N$ on lines $DG$ and $DF$ such that quadrilateral $EVAN$ is a rhombus.

Danielle Wang and Evan Chen

This post has been edited 2 times. Last edited by 62861, May 18, 2018, 11:58 PM

cos k theta and cos(k + 1) theta are both rational

by N.T.TUAN, Dec 8, 2007, 1:39 AM

Let $\theta$ be an angle in the interval $(0,\pi/2)$ . Given that $\cos \theta$ is irrational, and that $\cos k \theta$ and $\cos[(k + 1)\theta ]$ are both rational for some positive integer $k$ , show that $\theta = \pi/6$ .

greatest common divisor

Prove that there exists a convex 1990-gon

by orl, Nov 11, 2005, 6:55 PM

Prove that there exists a convex 1990-gon with the following two properties :

a.) All angles are equal.
b.) The lengths of the 1990 sides are the numbers $1^2$ , $2^2$ , $3^2$ , $\cdots$ , $1990^2$ in some order.

This post has been edited 1 time. Last edited by orl, Aug 15, 2008, 4:18 PM

Problem 3 IMO 2005 (Day 1)

by Valentin Vornicu, Jul 13, 2005, 6:00 PM

Let $x,y,z$ be three positive reals such that $xyz\geq 1$ . Prove that
$\frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 .$
Hojoo Lee, Korea

This post has been edited 1 time. Last edited by Valentin Vornicu, Sep 25, 2005, 12:23 AM