Find all such primes

by Entrepreneur, Apr 12, 2025, 6:29 PM

Find all primes $p,q\;\&\;r$ such that $\color{blue}{pq=r^2+r+1.}$

NEPAL TST 2025 DAY 2

by Tony_stark0094, Apr 12, 2025, 8:40 AM

Consider an acute triangle $\Delta ABC$ . Let $D$ and $E$ be the feet of the altitudes from $A$ to $BC$ and from $B$ to $AC$ respectively.

Define $D_1$ and $D_2$ as the reflections of $D$ across lines $AB$ and $AC$ , respectively. Let $\Gamma$ be the circumcircle of $\Delta AD_1D_2$ . Denote by $P$ the second intersection of line $D_1B$ with $\Gamma$ , and by $Q$ the intersection of ray $EB$ with $\Gamma$ .

If $O$ is the circumcenter of $\Delta ABC$ , prove that $O$ , $D$ , and $Q$ are collinear if and only if quadrilateral $BCQP$ can be inscribed within a circle.

geometry

altitudes

cyclic quadrilateral

TST Junior Romania 2025

by ant_, Apr 11, 2025, 5:01 PM

Consider the isosceles triangle $ABC$ , with $\angle BAC > 90^\circ$ , and the circle $\omega$ with center $A$ and radius $AC$ . Denote by $M$ the midpoint of side $AC$ . The line $BM$ intersects the circle $\omega$ for the second time in $D$ . Let $E$ be a point on the circle $\omega$ such that $BE \perp AC$ and $DE \cap AC = {N}$ . Show that $AN = 2AB$ .

geometry

Inspired by giangtruong13

by sqing, Apr 11, 2025, 2:57 AM

Let $a,b,c,d\geq 0 ,a-b+d=21$ and $a+3b+4c=101$ . Prove that
$61\leq a+b+2c+d\leq \frac{265}{3}$ $- \frac{2121}{2}\leq ab+bc-2cd+da\leq \frac{14045}{12}$ $\frac{519506-7471\sqrt{7471}}{27}\leq ab+bc-2cd+3da\leq 33620$

This post has been edited 2 times. Last edited by sqing, Yesterday at 3:13 AM

inequalities proposed

algebra

Inspired by Ruji2018252

by sqing, Apr 10, 2025, 2:00 AM

Let $a,b,c$ be reals such that $a^2+b^2+c^2-2a-4b-4c=7.$ Prove that
$-4\leq 2a+b+2c\leq 20$ $5-4\sqrt 3\leq a+b+c\leq 5+4\sqrt 3$ $11-4\sqrt {14}\leq a+2b+3c\leq 11+4\sqrt {14}$

inequalities proposed

algebra

inequalities

Solllllllvvve

by youochange, Jan 12, 2025, 7:18 PM

Suppose n is a composite positive integer. Let $1 = a_1 < a_2 < · · · < a_k = n$ be all the divisors of $n$ . It is known, that $a_1+1, . . . , a_k+1$ are all divisors for some $m$ (except $1, m$ ). Find all such $n.$

number theory

cricket jumping in dominoes

by YLG_123, Jan 29, 2024, 8:15 PM

A cricket wants to move across a $2n \times 2n$ board that is entirely covered by dominoes, with no overlap. He jumps along the vertical lines of the board, always going from the midpoint of the vertical segment of a $1 \times 1$ square to another midpoint of the vertical segment, according to the rules:

$(i)$ When the domino is horizontal, the cricket jumps to the opposite vertical segment (such as from $P_2$ to $P_3$ );

$(ii)$ When the domino is vertical downwards in relation to its position, the cricket jumps diagonally downwards (such as from $P_1$ to $P_2$ );

$(iii)$ When the domino is vertically upwards relative to its position, the cricket jumps diagonally upwards (such as from $P_3$ to $P_4$ ).

The image illustrates a possible covering and path on the $4 \times 4$ board.
Considering that the starting point is on the first vertical line and the finishing point is on the last vertical line, prove that, regardless of the covering of the board and the height at which the cricket starts its path, the path ends at the same initial height.

Attachments:

combinatorics

table

dominoes

Radical Condition Implies Isosceles

by peace09, Aug 10, 2023, 4:54 PM

Prove that any triangle with
$\sqrt{a+h_B}+\sqrt{b+h_C}+\sqrt{c+h_A}=\sqrt{a+h_C}+\sqrt{b+h_A}+\sqrt{c+h_B}$ is isosceles.

Combinatorics game

by VicKmath7, Mar 4, 2020, 6:52 PM

Players A and B play a game. They are given a box with $n=>1$ candies. A starts first. On a move, if in the box there are $k$ candies, the player chooses positive integer $l$ so that $l<=k$ and $(l, k) =1$ , and eats $l$ candies from the box. The player who eats the last candy wins. Who has winning strategy, in terms of $n$ .

This post has been edited 2 times. Last edited by VicKmath7, Mar 11, 2020, 11:19 AM

combinatorics

TST

four points lie on a circle

by pohoatza, Jun 28, 2007, 6:32 PM

Let $ABCD$ be a trapezoid with parallel sides $AB > CD$ . Points $K$ and $L$ lie on the line segments $AB$ and $CD$ , respectively, so that $AK/KB=DL/LC$ . Suppose that there are points $P$ and $Q$ on the line segment $KL$ satisfying $\angle{APB} = \angle{BCD}\qquad\text{and}\qquad \angle{CQD} = \angle{ABC}.$ Prove that the points $P$ , $Q$ , $B$ and $C$ are concyclic.

Proposed by Vyacheslev Yasinskiy, Ukraine

This post has been edited 2 times. Last edited by wu2481632, Dec 10, 2018, 2:24 AM