Hi, does anyone happen to have MathCounts National round test problems from 2018-2024? I found 2000-2017 online but can't find anything past that year.

Alex is training to make . Currently he will score a on . He can expend units of effort to gain points on the , units of effort to gain points on the , and units of effort to gain point on the . He will need to get at least points on and combined and get at least points on to make . What is the minimum amount of effort he can expend to make ?

Let be an acute scalene triangle. Denote by the circle with diameter , and let be the contact points of the tangents from to , chosen so that and lie on opposite sides of and and lie on opposite sides of . Similarly, let be the circle with diameter , with tangents from touching at , and the circle with diameter , with tangents from touching at .
Prove that the lines are concurrent.

Friday May 9th

I spent my evening after school, packing for the trip, using the checklist given by my coach.
I didn’t do much preparation, as I was mostly chilling out for the upcoming days.

I also played basketball with my cousin, Kevin, who met Gotham Chess and stayed at his home!


Saturday, May 10th

I woke up at 5:30 AM, ate a light breakfast, and headed out to the airport with my luggage.

I met my teacher, but was surprised that Archishman split up with his own family.
Waiting for the TSA was pretty boring, but we soon got through, and after I found our gate.

A couple of minutes pass by, as I review an AOPS mock where I meet Archischman;
Afterward, we chill out, watch the rube goldberg machine in the airport, and wait to board the plane.

During the plane ride, I played games; however, during our descent, I heard a loud crack, and our plane started wobbling, and we heard cracking sounds in the seats. Fortunately, we were able to land and were able to attend the competition the next day.

After this, heading out, we went to the shuttle; however, we had 35 minutes. We tried to solve the Jane Street card puzzle but failed, and ended up socializing.

After we arrived at the hotel, we received a MASSIVE amount of stuff, like calculators, shirts, coupons, plaques, stickers, etc.
I also saw and got a signature from Richard Rusczyk, which was really cool.

Then, we went to a restaurant named “Chinatown Garden”, with the worst food I’ve ever had.

We then chilled in our rooms, studied for a bit, and started organizing plans for pin trading.

Our goal was to scam as many people as possible by doing 2:1 trades, as we had a “limited”
amount of pins. (We even got 5:1 and 10:1 trades)
A Virginia kid scammed me with a STEM pin, so I chased him down and got our pin back.

We got through around half the states organizing in and out of what pins we had.

Finally, we got some food from the buffet (which was surprisingly decent) and had a good time trading some more.

We ended the day with a short and brief CDR, where we had some fun, and then we went to sleep to anticipate the next day.

At night, I showered and sang karaoke with Archi.

Sunday, May 11th

Getting ready, I found out that a mock (outside the box) was recently released and took it through breakfast.

Then, once we got there at 8:30, there was a mob of parents taking pictures, and music played.

Then every team did introductions/attendance and their chants, most of which were really cringe.

I took the test; however was too slow on the sprint round and got a predicted 16.

On the target round, I was able to get through and got a 12, despite barely not solving p8 to my frustration.

Team round we did decently, scoring a 14/20, which was one of the best scores around us, that even orz states like Texas and Washington didn’t beat.

Predicted Scores:
Caleb and I got a similar score (around 28), and Henry got around 35, and Archi sold on the target and got a 24.

After this, we teamed up with North Carolina (chill af) and went to a pho shop (54 Restaurant), which tasted amazing. (A far contrast from Chinatown Garden)

Then, we went to an aerospace museum, where we played Brawl Stars and went around. Eventually, we saw models of blackholes and air vacuums, and played a flight simulator.

Then we went to our hotel, chilled, and watched basketball games.

After, we went to an Indian restaurant named “Himilayan Doko” which was really delicious!

Then we raided different rooms, from NC, HAWAII, Idaho, Virgin Islands, and accidentally a random dudes room who was ticked at us.

Finally, we chilled and went to sleep, though I tried to get Henry and Archi to sleep since they were being annoying.

Monday, May 12th

We start the day forming my pin badge, and then we went to get some breakfast.

After that, we met in the breakfast area with 2 teams for table, and I actually got a 10:1 pin trade which was pretty cool.

After that, we lined up and got our thunderstick/clapping machines, and ran through the entrance of the CDR.

Sadly, we didn’t win anything, but it was cool seeing the results.

Then, we started to watch the CDR, which was really exciting.
It got really interesting when everyone saw Nathan Liu cook his opponent in half a second.

In the semifinals, it was insane, and Advait and Nathan, buzzed every question that was around mid-sprint level.

Then, it finished with Nathan beating Brandon with a 2-second solve, absolute insanity.

Finally, we went back to our rooms and got lunch in the hotel.
A few hours later, we received our scores, and I had bombed, scoring a 26 with 7 sillies. (ouch)

Unfortunately, my teammates Henry and Archishman sillied a bunch of questions.

After, we played Brawl Stars, and went to explore the hotel, where we went up a random staircase and got stuck. We went to the roof, but got scared and yelled out for help on the gym floor. Thankfully, we got back, and I went and reviewed the test.

After we reviewed the test, and went to the Mathcounts Party.

The food was mid, but the games were pretty fun.

We met a bunch of people, played air hockey, foosball, and basketball, while listening to the not so great music in the background.

Then we went back to our rooms at 8PM, to put our pins on, and I got 38/56!

Finally, we met up in a room with a bunch of Cali, and NC kids, and talked about the test, the people, and played Brawl Stars. Even Josh Frost came up to us and asked us how the trip was.

Tuesday, May 13th

I started the day waking up at 6:20, and packed up and ate breakfast. After that, Henry was late, so we packed food for him and went to the bus shuttle.

Eventually, we arrived at the airport, went through security (which was suspiciously fast), and played Brawl Stars. We also ate five guys fries, which was pretty good. Eventually, we had to part our ways with Henry and headed out to our flights, which marked the end of the trip.

Conclusion:

Although we didn’t do amazingly well in the contest, going to DC was an amazing experience. I got to meet people who were passionate about math, and hang out with them, goofing around.

This was the best math contest experience that I’ll likely ever have, and I’m glad I went through it.

What's the goofiest thing you've seen happen at a math competition? :starwars:

The numbers and are prime and satisfy

for some positive integer . Find all possible values of .

Luxembourg (Pierre Haas)

Let Prove that

Given a with orthocenter and a point lying on the Euler line of Prove that the midpoint of lies on the Thomson cubic of the pedal triangle of WRT

Could anyone post some problems which are similar to the below problem:

Find the real solution of:
Sol(outline)

Let . Prove that
W here

Let be an acute triangle with , and let be the center of its circumcircle. Let be the reflection of with respect to . The line through parallel to intersects at , and the tangent at to the circle intersects the line through parallel to at point . Let be a point on the ray , starting at , such that .
Show that the orthocenter of triangle lies on the circle with diameter .

Proposed by Radu Lecoiu

The equation
is written on the board, with linear factors on each side. What is the least possible value of for which it is possible to erase exactly of these linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?

Exactly 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 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 leaders present. The chairperson must seat them in a square-shaped conference hall of size , where each leader will be seated in a designated cell. It is known that exactly of these 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 (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 neighbors and will vote "FOR" only if both of their neighbors voted "FOR."

(a) Can the IMO chairperson arrange their supporters so that they vote "FOR" in the first second of voting and thereby secure a "FOR" vote from at least of all leaders?

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

Proposed by Bogdan Rublov

Let be a quadrilateral with . Diagonals and meet at . Let and denote the circumcircle and the circumcenter of triangle . Let and denote the circumcircle and circumcenter of triangle . Segment meets and again at and (other than and ), respectively. Let and be the midpoints of minor arcs (not including ) and (not including ). Prove that .