Geometry

May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies

jlacosta

May 1, 2025

0 replies

orthocenter on sus circle

DVDTSB 2

N 10 minutes ago by Diamond-jumper76

Source: Romania TST 2025 Day 2 P1

Let $ABC$ be an acute triangle with $AB < AC$ , and let $O$ be the center of its circumcircle. Let $A'$ be the reflection of $A$ with respect to $BC$ . The line through $O$ parallel to $BC$ intersects $AC$ at $F$ , and the tangent at $F$ to the circle $\odot(BFC)$ intersects the line through $A'$ parallel to $BC$ at point $M$ . Let $K$ be a point on the ray $AB$ , starting at $A$ , such that $AK = 4AB$ .
Show that the orthocenter of triangle $ABC$ lies on the circle with diameter $KM$ .

Proposed by Radu Lecoiu

2 replies

DVDTSB

Yesterday at 12:18 PM

Diamond-jumper76

10 minutes ago

problem 5

termas 74

N 22 minutes ago by maromex

Source: IMO 2016

The equation
$(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)$ is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?

74 replies

termas

Jul 12, 2016

maromex

22 minutes ago

Inspired by nhathhuyyp5c

sqing 2

N 25 minutes ago by lbh_qys

Source: Own

Let $x,y>0, x+2y- 3xy\leq 4.$ Prove that
$\frac{1}{x^2} + 2y^2 + 3x + \frac{4}{y} \geq 3\left(2+\sqrt[3]{\frac{9}{4} }\right)$

2 replies

sqing

an hour ago

lbh_qys

25 minutes ago

I think I know why this problem was rejected by IMO PSC several times...

mshtand1 1

N 42 minutes ago by sarjinius

Source: Ukrainian Mathematical Olympiad 2025. Day 2, Problem 11.8

Exactly $102$ country leaders arrived at the IMO. At the final session, the IMO chairperson wants to introduce some changes to the regulations, which the leaders must approve. To pass the changes, the chairperson must gather at least $\frac{2}{3}$ of the votes "FOR" out of the total number of leaders. Some leaders do not attend such meetings, and it is known that there will be exactly $81$ leaders present. The chairperson must seat them in a square-shaped conference hall of size $9 \times 9$ , where each leader will be seated in a designated $1 \times 1$ cell. It is known that exactly $28$ of these $81$ leaders will surely support the chairperson, i.e., they will always vote "FOR." All others will vote as follows: At the last second of voting, they will look at how their neighbors voted up to that moment — neighbors are defined as leaders seated in adjacent cells $1 \times 1$ (sharing a side). If the majority of neighbors voted "FOR," they will also vote "FOR." If there is no such majority, they will vote "AGAINST." For example, a leader seated in a corner of the hall has exactly $2$ neighbors and will vote "FOR" only if both of their neighbors voted "FOR."

(a) Can the IMO chairperson arrange their $28$ supporters so that they vote "FOR" in the first second of voting and thereby secure a "FOR" vote from at least $\frac{2}{3}$ of all $102$ leaders?

(b) What is the maximum number of "FOR" votes the chairperson can obtain by seating their 28 supporters appropriately?

Proposed by Bogdan Rublov

1 reply

mshtand1

Mar 14, 2025

sarjinius

42 minutes ago

No more topics!

Geometry

VicKmath7 3

N Mar 28, 2020 by sunken rock

Source: 12th Russian MO "Leonhard Euler", final round

Given is a triangle ABC with 2<B=180°+<A. Point K is in the interior of the triangle, so that <ACK=2<BCK and L is on AB, so that BK=KL. Prove that CK+AL=AC

3 replies

VicKmath7

Mar 25, 2020

sunken rock

Mar 28, 2020

Geometry

G H J

Reveal topic tags

geometry

G H BBookmark kLocked kLocked NReply

Source: 12th Russian MO "Leonhard Euler", final round

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VicKmath7

1390 posts

VicKmath7

#1 h Mar 25, 2020, 7:44 PM

Y by

Given is a triangle ABC with 2<B=180°+<A. Point K is in the interior of the triangle, so that <ACK=2<BCK and L is on AB, so that BK=KL. Prove that CK+AL=AC

Z K Y

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