Ahora tenemos uno de combinatoria del Putnam. En el examen no me salió, pero el problema es muy bonito.
Un subconjunto de {1,2,3,...,n} se llama "mediocre" si para cualesquiera dos elementos a,b del subconjunto,si su promedio es entero se tiene que también esta en el subconjunto. Sea A(n) el número de subconjuntos mediocres de {1,2,...,n}. (Por ejemplo, todos los subconjuntos de {1,2,3} excepto {1,3} son mediocres, así que A(3) = 7). Encuentra todos los números naturales n tales que A(n+2) -2A(n+1) + A(n) = 1.
lunes, 28 de diciembre de 2009
lunes, 14 de diciembre de 2009
Problema del Día
Ahora yo quiero poner un problema.
Agarré el libro de desigualdades y seleccioné un problema al azar, no se si esta difícil o fácil ya que aun no lo hago.
(Lista Corta Iberoamericana, 2003)
Sean a,b,c números reales positivos muestre que:
(a^3)/(b^2 - bc + c^2) + (b^3)/(c^2 - ca + a^2) + (c^3)/(a^2 - ab + b^2) >= a + b + c
Agarré el libro de desigualdades y seleccioné un problema al azar, no se si esta difícil o fácil ya que aun no lo hago.
(Lista Corta Iberoamericana, 2003)
Sean a,b,c números reales positivos muestre que:
(a^3)/(b^2 - bc + c^2) + (b^3)/(c^2 - ca + a^2) + (c^3)/(a^2 - ab + b^2) >= a + b + c
viernes, 11 de diciembre de 2009
Problema del Día
Sigamos con otro problema fácil del Putnam reciente (aunque admito que a mi no me salió).
Putnam A1:
Sea f una función del plano a los números reales, es decir, para cada punto P en el plano, f(P) es un número real. Supón que para cualquier cuadrado ABCD se cumple que f(A) + f(B) + f(C) + f(D) = 0. ¿Será cierto que f(P) = 0 para todo punto P en el plano?
Putnam A1:
Sea f una función del plano a los números reales, es decir, para cada punto P en el plano, f(P) es un número real. Supón que para cualquier cuadrado ABCD se cumple que f(A) + f(B) + f(C) + f(D) = 0. ¿Será cierto que f(P) = 0 para todo punto P en el plano?
miércoles, 9 de diciembre de 2009
Problema Del Día
Se tardaron un buen rato en poner soluciones para el de la cuadrícula de n x n por eso me tarde en poner otro. Ya puse uno de combinatoria y uno de geometría así que hoy toca de números.
Este el el problema B1 del Putnam 2009 (examen fue el 5 de diciembre):
Demuestra que puedes escribir todo número racional como la división de productos de facoriales de primos. Ejemplo: 10/9 = (2! 5!) / (3! 3! 3!).
Este el el problema B1 del Putnam 2009 (examen fue el 5 de diciembre):
Demuestra que puedes escribir todo número racional como la división de productos de facoriales de primos. Ejemplo: 10/9 = (2! 5!) / (3! 3! 3!).
Las matemáticas, ¿se crean o se descubren?
Analizando las palabras que vienen al pie del block de hojas que se nos entrego en una competencia del ITCJ, la cual parafrasea a Galileo con "Las matemáticas son el alfabeto con el que Dios escribió el universo", me entró la curiosidad por ver o entender la opinion de aquellos que tienen o no tienen referencia respecto a (algun) Dios.
Es dificil, o por lo menos abstracto, el hecho de imaginarse el inicio del universo, y retomando las ideas creacionistas en las que la religión tiene la principal aportación, lo es aún más pensarlo de manera en que una CIENCIA como lo son las matemáticas, pueda interferir en la perfección y armonía del equilibrio universal.
Volviendo al tema, y sin adentrarme tanto al ámbito religioso, que al fin y al cabo no viene a ser relevante para la discusión de esta ponencia, solicito (si me lo permiten) a la comunidad olimpica, departir al respecto.
Investigando más a fondo sobre si las matematicas son parte de una naturaleza tal cual la conocemos o si solo es un "invento" que el hombre ha creado como satisfactor de necesidades; encontré reflexiones muy interesantes que exponen filosofos, tales como Platón, quien se plantea de cierta forma y con mis palabras: ¿qué es realmente lo imaginario, e imaginariamente qué es lo real?, y no encontró algo que fuera totalmente imaginario, puesto que la verdad, la única verdad, deberá ser universal e inmutable, pero esto viene a contradecir el famosísimo planteamiento: "Nada es absoluto... todo es relativo", misma que a un tiempo se contradice a si. ¿Porqué se afirma que nada es absoluto?, ¿es acaso esa afirmación la única razón NO relativa?, ¿porqué la verdad que Platón nos muestra es 100% INMUTABLE?.
Como sea, las matemáticas, como las conocemos, son y han sido parte de la vida en todos los tiempos, con objetivos que van desde la necesidad de contar (he aqui que son un SATISFACTOR), hasta la solución de problemas complejos propios de la vida cotidiana.
Las matemáticas por sí solas son la base de un todo (mi interpretación de la frase de Galileo, siendo ésta la que comparto); sabemos que no llegó algún espíritu y le mostró a las diferentes civilizaciones cómo contar, pero también es de nuestro conocimiento que sin matemáticas simplemente la vida como la conocemos, no seria posible (ah?). Es aquí donde viene muy a do el planteamiento de qué entendemos por descubrimiento y creación. Un ejemplo, el cual no se si es el mas correcto utilizar en este caso, es el oxígeno. ¿Se descubrió o se inventó?, la respuesta más lógica es POR SUPUESTO QUE SIEMPRE HA EXISTIDO (considerese despreciable la polisemia del termino -siempre-), ENTONCES SE DESCUBRIÓ. Pero... y antes de que supiera que existía, ya se utilizaba ¿o no?, y a lo que voy es a lo siguiente: antes de que el hombre tuviera noción de que podía contar, ya utilizaba no métodos, sino comparaciones de qué lado tiene más o si faltaba mucho camino por recorrer, si era más rápido ir directamente a un lugar que rodear y luego encontrarlo... Desde ahí ya existían (por naturaleza), como ya se dijo LAS COMPARACIONES (con la definición más sencilla que se le pueda asignar) mismas que son ampliamente utilizadas matemáticamente (y más aún nosotros como olímpicos y exolímipcos), y no el término MATEMÁTICAS, el cual se fue construyendo con el paso del tiempo.
Conocemos sobre economía, algoritmos informáticos, aritmética, logaritmos, geometría (todas las que existen), ecuaciones, combinatoria, (...) en fin, muchas ramas que ya desarrolladas son las que entendemos dentro del termino MATEMÁTICAS.
Entonces he aquí la pregunta: ¿se crearon o se descubrieron? Parece broma que después de todo el rollo que te acabo de hacer leer se siga con la misma pregunta. Aunque en el contenido de este post no hice mas que merodear y en el peor de los casos confundir, si bien no se llegó a una respuesta no digamos absoluta, sino convincente, pudimos interesarnos sobre el tema y la resolución del mismo.
Creo que, sin mofarme de la Ley de la Conservación de la materia, establecida por Antoine de Lavoisier, LAS MATEMÁTICAS NO SE CREAN NI SE DESCUBREN, SÓLO SE TRANSFORMAN, es decir, TODO (nuevamente despreciese la polisemia) ya existe, el hombre sólo le da nuevas interpretaciones, las cuales hacen más favorable el entendimiento del mundo natural.
Propiamente considero que todo esto no es más que un juego de palabras que torna todo el tema en una discusión de interpretaciones distanciadas en su congruencia y en muchos de los casos asimiladas como un problema matemático cuya solución se dará reduciendo al absurdo.
O.. ¿Ustedes qué opinan?
Daniel.
martes, 8 de diciembre de 2009
El Nacional according to Quique
Se me hizo suave lo que Quique publico en su blog sobre la Olimpiada Nacional así que aquí les hago un copy/paste:
Saturday November 7, 2009:
Because of lack of flights to Campeche, we had to leave on Saturday instead of Sunday. Four of the students and me left on Saturday. We were in good spirits. When I arrived, the hotel wasn't expecting me a day early so they gave me another room which was nicer than the usual and I had a view to the ocean. I was very happy specially because I thought I would have to be in a room with all four students trying to save money, but luckily they assigned us rooms without having to pay much more. One thing that was funny that happened was this guy at the gas station that wanted to get a tax receipt for his 2 dollar purchase. I thought it was ridiculous how he made us all wait for it. The guy ended up being one of the Gold medalists in the competition.
Sunday:
I slept in late and relaxed. I walked a bit with the students around the "malecón" (street next to the ocean). At night, after registering the team (with David and Héctor) we ate a really crappy supper. Throughout the week, the service in the hotel wasn't very good.
Monday:
We rode a bus to the Nautical Club where the exam would be. Because we are super competitive, David, Héctor and I have our own little competition to see who can first finish the problems. Although this time, David wanted clearer rules and said that it would end at 11am. I didn't understand that part and that it was about speed, so when I finished at 10am I left. David got upset but he still managed to finish before 11am. There was some controversy however as Hector and I heard a hint which we thought David heard to, so we told him the hint at 10am when we found out he hadn't heard it. I made a hint-free solution anyway to make my victory certain. However David kept insisted that it was a tie and not my victory.
The students had a bad day, although not terrible. Manuel had a perfect first day and the rest were between 9 and 13 points (out of 21). It seems okay, but this exam was very easy (note how I finished in less than 75 minutes and they had 4.5 hours). Many students got a perfect exam, so those scores were pretty low.
The rest of the day was meetings about the grading scheme and a proposal to change the selection of the Mexican team for the IMO. Besides that, it just consisted of me having verbal fights with David about our silly competition.
Later at night I went out walking with my mom who had come to Campeche to visit. It was a very pleasant walk.
Tuesday:
Day 2 of the competition. This time I was okay with David's rules. I solved the first two problems in 30 minutes and had plenty of time for the third (just under two hours). However, the problem was very hard. David managed to solve it at around 10:59 right at the buzzer and I would have needed a few more minutes to finish. I had the idea of the solution but didn't finish writing it out, I predict I would have needed between 5 and 15 more minutes. David clearly won day 2 and he claims he should get the sum of both days, but I argue that my hint-free solution should compensate. We had this discussion for the whole week and Héctor seems to side with him. I was very happy with this exam and with the fact that I was able to solve it all. The second day exam was a beautiful exam and problem 6 is one of my favorite problems I've seen in the Mexican Mathematical Olympiad.
The students didn't do so well. Manuel came out saying he had them all but not being sure of the last one. Karina came out happy (the first day she was very sad). Daniel came out extremely sad (he would only have 2 points). The rest were neither very happy or sad. They did so so.
Right after eating, we played beach volleyball and we beat every one. I didn't know David played a lot of volleyball in high school. One memorable thing was how a few times the ball went into the sea and since the place was surrounded by rocks we couldn't go for the ball, however the sea would push the ball further and further north until it would reach the sand part of the beach and where we could pick it up. It was really cool.
At night we graded.
Wednesday
During the day we graded and defended problems 1,2,3 and 4. With problems 2 and 4 we didn't have much trouble. In problem 2 we had a bit of a discussion about one student to get one point and they yielded to my argument. I did a very good job with that one and I think the table was very just with the students. In problem 4 we essentially just explained what Pepe had done. Pepe doesn't write very nice so we would always explain what he did to every table. Once the explanation was said we got the points we thought he deserved (3 instead of 1). In problem 3 we argued for 4 points for Daniel and they wanted 2 at first but quickly changed to 3. We had a very long discussion (over an hour) which ended with we signing the 3 although we still felt it was a 4. This was the only point we wouldn't get that we thought was deserved.
Problem 1 deserves its own paragraph because it had many anecdotes. We mainly had issues with two students. They were giving Luis 1 and Pepe 4 when we thought they had 5 and 7 (we asked for 6 and 7). I argued for a while and they were okay with giving 4 instead of 1 to Luis but after those thirty minutes we didn't have time to discuss Pepe's exam. Later I came back and argued for the seven (full credit) showing them where the proof was in the paper. They nodded to everything, asked about why we wrote 90-a in an angle that was inside a right triangle opposite the a angle. After deliberating they said the student had only 4 points. I couldn't believe that now that they understand the solution they still want to give 4 points instead of 7. We keep arguing and at some point in my incredulity I yell at them "WHAT IS WRONG WITH YOU?" At that point David decided to go argue another problem elsewhere. After the table had discussed the problem with the other table (each problem is graded by two tables each with two people) we came back and they decided that Luis had 2 instead of 4 (we wanted 5) and Pepe remained at 4. Once I heard the reversal of the 4 to a 2 I lost my calm again. However we discussed Pepe's exam first. I again showed the solution and we kept explaining to them why they shouldn't get so technical on this problem, as in wanting the student to explain why that angle is 90-a. They remained stubborn so we asked the problem to be read by the head coordinator without their input to see what he would give them. While he read this we discussed Luis' exam. I explained the alternative solution to them and what was lacking from that in his exam and equated it to a grading scheme they had to argue for 5 points. They would then ask ridiculous things and I was so upset that I would answer in incredibly sarcastic tone "Oh my God, what did the student do here, what oh what? Yes, he substracted" (showing proof in the picture that the student had that theta + p = 45 implies p = 45 - theta). We then had to leave the room and when we came back they offered 5 for Luis (success) and 6 for Pepe. After we explain to the head coordinator where the seventh point is, he agrees and we got what we deserved. This was a very infuriating discussion and we were saved by having a great head coordinator. I would also point out that David was also using sarcasm in this discussion, I wasn't the only one losing my temper.
At night it was time to grade more and then rest for a long day the next day.
Thursday:
I woke up early to keep grading. Problem 6 was a difficult one to grade as Manuel's solution was a mess and there seemed to be a good idea in Karina's exam. Héctor finished grading problem 5 so we went downstairs to defend it. In problem 5 the table had very good graders and essentially we only noticed one point they didn't notice. They did a superb job. Héctor also did a great job finding that hard to find point (which was a point between the lines in the grading scheme too) in the student's exam.
After that we came back to grade more. I wrote a clean version of the proof to explain to the judges. I also tried to use an idea of Karina to finish the proof. I discussed it with Héctor and he came out with the proof of my Moonwalker conjecture which would then help make Karina's idea a strong one. After that we finished (incorrectly) the proof of Karina's idea. We didn't know how much to ask for her because when we had to go to the judges we didn't have a clean version of the alternative proof so we asked for 4 (they had 1). They also had 1 point for Manuel. The table was very far behind on time, it was 1pm when we passed and all states should have finished by 1pm. I showed them Manuel's proof (my explanation was flawed) and they were convinced that it was much more than 1 but they wanted to read it to make sure it was 7 points. After dinner we came back to discuss. We got to the table at 5:20pm. They told us what wasn't convincing to them about Manuel's proof and I explained everything. I convinced one of them but not the other, so they asked the other table to read the exam. In the meantime we discussed Karina's exam. During my explanation I made a flawed statement, but then realized it wasn't needed in the proof and showed them the alternative solution (corrected). I then asked for 4 points and they agreed liking the alternative solution very much (I am very proud of this, because I worked on grading this exam for more than 3 hours where at least 2 hours were used creating the proof with Héctor). At about 6:20pm Carlitos (a grader from the other table) told us his reservations with the proof and saying he would give 5 points. I realized his objection was correct but tried finding in the exam something that would fix it. With help from Héctor and a reinterpretation of a 5 to a 3, I think I found the final piece of the puzzle and Carlos was utterly convinced that the student had 7 points (I must note that David thinks Carlos got convinced by another argument Héctor was using, but I think the last page of Manuel's exam was the main ingredient in the change). Having Carlos convinced wasn't enough because two people out of the four felt the student didn't deserve seven points. Hence we had to go to the head coordinator again. He had a tough decision (at 7:45pm after the super long discussion we had had) and decided to trust in Carlos and gave the student the 7 which settled Manuel to have a perfect exam.
I was extremely happy after this and we thought that this was one of our best performances in terms of defending points for our students. I am very proud of getting the alternative solution to Karina's exam and David thinks the defense of Manuel's 7 is the finest moment.
At night, I went to get drinks with Pedroza, Rogelio and David and we had a very interesting conversation. I kept wanting to go dance to the dancefloor as they were playing salsa, but I ended up not going. When we were walking back to the hotel we saw a large crowd coming out of it. They had just competed in the Geometrense (a competition consisting of 9 geometry problems for ex-Olympians, I didn't compete because I am not a big fan of geometry) and they were heading out to party. We went back to the bar and talked for hours. I had a pleasant conversation with Pablín (el oro), who had invented problem 6 of the exam (my favorite problem this year). It was also very funny to see David's pajonez shine in the room with his awkward questions. A very fun time. However we went to bed at 5am.
Friday:
Didn't get much sleep because we had to go to a 4 hour meeting. After the meeting I went to ear with my mom and then I went to the prize ceremony. There I met the governor. I was very happy that the governor came to the ceremony, this is the first time I see this in the 9 Olympiads I've attended. The ceremony was a bit long but nice. I had to be standing up all the time because there weren't enough seats. My mom seemed to have a good time too.
After the ceremony I went to the closing party which was okay. We were a bit sad because the team had performed poorly getting a seventh place (I thought we had a chance to win) and very far from the first place. We did well in geometry (doing better than Morelos), but we got only 145 points while first place got 191. The students seemed sad, however they did some funny things like Pepe as part of a punishment went to a table took a bite from somebody else's food then drank from another person's glass. The reactions were very funny.
When I came back from the party I went salsa dancing with my mom. I had a great time dancing with her. I think my dancing has improved a lot. I also danced with her and a group of her friends the Friday before flying to Campeche and that was a lot of fun. It was also nice meeting her tango friends.
After dancing salsa with my mom, I went to my bedroom and outside my bedroom were a couple of students and Niño talking. Niño is a very good friend of mine so I joined the conversation. I had a lot of laughs talking to María, Irune, Nacho and Niño.
Saturday:
After relaxing all day, we went to eat dinner with my mom and then caught the airplane to head back home. When we arrived to Juárez, Pepe's family received us with big poster that said "Campeones" (champions). It was very nice to have such a warm reception. It was the first time a team was all from one city (Juárez) and so this was the first time the whole team arrived at the same time to the same city. It was great.
I had a really wonderful time and there were a lot of positive things. I was happy to solve all the problems. I was happy to play volleyball. I was happy to defend points for my students (and do such a good job at it) and I was happy to spend a lot of time with my great friends David and Héctor. Even though the team didn't do as well as predicted I am very proud of them and of the hard work I did in helping in their preparation. I love the Mathematical Olympiad.
Because of lack of flights to Campeche, we had to leave on Saturday instead of Sunday. Four of the students and me left on Saturday. We were in good spirits. When I arrived, the hotel wasn't expecting me a day early so they gave me another room which was nicer than the usual and I had a view to the ocean. I was very happy specially because I thought I would have to be in a room with all four students trying to save money, but luckily they assigned us rooms without having to pay much more. One thing that was funny that happened was this guy at the gas station that wanted to get a tax receipt for his 2 dollar purchase. I thought it was ridiculous how he made us all wait for it. The guy ended up being one of the Gold medalists in the competition.
Sunday:
I slept in late and relaxed. I walked a bit with the students around the "malecón" (street next to the ocean). At night, after registering the team (with David and Héctor) we ate a really crappy supper. Throughout the week, the service in the hotel wasn't very good.
Monday:
We rode a bus to the Nautical Club where the exam would be. Because we are super competitive, David, Héctor and I have our own little competition to see who can first finish the problems. Although this time, David wanted clearer rules and said that it would end at 11am. I didn't understand that part and that it was about speed, so when I finished at 10am I left. David got upset but he still managed to finish before 11am. There was some controversy however as Hector and I heard a hint which we thought David heard to, so we told him the hint at 10am when we found out he hadn't heard it. I made a hint-free solution anyway to make my victory certain. However David kept insisted that it was a tie and not my victory.
The students had a bad day, although not terrible. Manuel had a perfect first day and the rest were between 9 and 13 points (out of 21). It seems okay, but this exam was very easy (note how I finished in less than 75 minutes and they had 4.5 hours). Many students got a perfect exam, so those scores were pretty low.
The rest of the day was meetings about the grading scheme and a proposal to change the selection of the Mexican team for the IMO. Besides that, it just consisted of me having verbal fights with David about our silly competition.
Later at night I went out walking with my mom who had come to Campeche to visit. It was a very pleasant walk.
Tuesday:
Day 2 of the competition. This time I was okay with David's rules. I solved the first two problems in 30 minutes and had plenty of time for the third (just under two hours). However, the problem was very hard. David managed to solve it at around 10:59 right at the buzzer and I would have needed a few more minutes to finish. I had the idea of the solution but didn't finish writing it out, I predict I would have needed between 5 and 15 more minutes. David clearly won day 2 and he claims he should get the sum of both days, but I argue that my hint-free solution should compensate. We had this discussion for the whole week and Héctor seems to side with him. I was very happy with this exam and with the fact that I was able to solve it all. The second day exam was a beautiful exam and problem 6 is one of my favorite problems I've seen in the Mexican Mathematical Olympiad.
The students didn't do so well. Manuel came out saying he had them all but not being sure of the last one. Karina came out happy (the first day she was very sad). Daniel came out extremely sad (he would only have 2 points). The rest were neither very happy or sad. They did so so.
Right after eating, we played beach volleyball and we beat every one. I didn't know David played a lot of volleyball in high school. One memorable thing was how a few times the ball went into the sea and since the place was surrounded by rocks we couldn't go for the ball, however the sea would push the ball further and further north until it would reach the sand part of the beach and where we could pick it up. It was really cool.
At night we graded.
Wednesday
During the day we graded and defended problems 1,2,3 and 4. With problems 2 and 4 we didn't have much trouble. In problem 2 we had a bit of a discussion about one student to get one point and they yielded to my argument. I did a very good job with that one and I think the table was very just with the students. In problem 4 we essentially just explained what Pepe had done. Pepe doesn't write very nice so we would always explain what he did to every table. Once the explanation was said we got the points we thought he deserved (3 instead of 1). In problem 3 we argued for 4 points for Daniel and they wanted 2 at first but quickly changed to 3. We had a very long discussion (over an hour) which ended with we signing the 3 although we still felt it was a 4. This was the only point we wouldn't get that we thought was deserved.
Problem 1 deserves its own paragraph because it had many anecdotes. We mainly had issues with two students. They were giving Luis 1 and Pepe 4 when we thought they had 5 and 7 (we asked for 6 and 7). I argued for a while and they were okay with giving 4 instead of 1 to Luis but after those thirty minutes we didn't have time to discuss Pepe's exam. Later I came back and argued for the seven (full credit) showing them where the proof was in the paper. They nodded to everything, asked about why we wrote 90-a in an angle that was inside a right triangle opposite the a angle. After deliberating they said the student had only 4 points. I couldn't believe that now that they understand the solution they still want to give 4 points instead of 7. We keep arguing and at some point in my incredulity I yell at them "WHAT IS WRONG WITH YOU?" At that point David decided to go argue another problem elsewhere. After the table had discussed the problem with the other table (each problem is graded by two tables each with two people) we came back and they decided that Luis had 2 instead of 4 (we wanted 5) and Pepe remained at 4. Once I heard the reversal of the 4 to a 2 I lost my calm again. However we discussed Pepe's exam first. I again showed the solution and we kept explaining to them why they shouldn't get so technical on this problem, as in wanting the student to explain why that angle is 90-a. They remained stubborn so we asked the problem to be read by the head coordinator without their input to see what he would give them. While he read this we discussed Luis' exam. I explained the alternative solution to them and what was lacking from that in his exam and equated it to a grading scheme they had to argue for 5 points. They would then ask ridiculous things and I was so upset that I would answer in incredibly sarcastic tone "Oh my God, what did the student do here, what oh what? Yes, he substracted" (showing proof in the picture that the student had that theta + p = 45 implies p = 45 - theta). We then had to leave the room and when we came back they offered 5 for Luis (success) and 6 for Pepe. After we explain to the head coordinator where the seventh point is, he agrees and we got what we deserved. This was a very infuriating discussion and we were saved by having a great head coordinator. I would also point out that David was also using sarcasm in this discussion, I wasn't the only one losing my temper.
At night it was time to grade more and then rest for a long day the next day.
Thursday:
I woke up early to keep grading. Problem 6 was a difficult one to grade as Manuel's solution was a mess and there seemed to be a good idea in Karina's exam. Héctor finished grading problem 5 so we went downstairs to defend it. In problem 5 the table had very good graders and essentially we only noticed one point they didn't notice. They did a superb job. Héctor also did a great job finding that hard to find point (which was a point between the lines in the grading scheme too) in the student's exam.
After that we came back to grade more. I wrote a clean version of the proof to explain to the judges. I also tried to use an idea of Karina to finish the proof. I discussed it with Héctor and he came out with the proof of my Moonwalker conjecture which would then help make Karina's idea a strong one. After that we finished (incorrectly) the proof of Karina's idea. We didn't know how much to ask for her because when we had to go to the judges we didn't have a clean version of the alternative proof so we asked for 4 (they had 1). They also had 1 point for Manuel. The table was very far behind on time, it was 1pm when we passed and all states should have finished by 1pm. I showed them Manuel's proof (my explanation was flawed) and they were convinced that it was much more than 1 but they wanted to read it to make sure it was 7 points. After dinner we came back to discuss. We got to the table at 5:20pm. They told us what wasn't convincing to them about Manuel's proof and I explained everything. I convinced one of them but not the other, so they asked the other table to read the exam. In the meantime we discussed Karina's exam. During my explanation I made a flawed statement, but then realized it wasn't needed in the proof and showed them the alternative solution (corrected). I then asked for 4 points and they agreed liking the alternative solution very much (I am very proud of this, because I worked on grading this exam for more than 3 hours where at least 2 hours were used creating the proof with Héctor). At about 6:20pm Carlitos (a grader from the other table) told us his reservations with the proof and saying he would give 5 points. I realized his objection was correct but tried finding in the exam something that would fix it. With help from Héctor and a reinterpretation of a 5 to a 3, I think I found the final piece of the puzzle and Carlos was utterly convinced that the student had 7 points (I must note that David thinks Carlos got convinced by another argument Héctor was using, but I think the last page of Manuel's exam was the main ingredient in the change). Having Carlos convinced wasn't enough because two people out of the four felt the student didn't deserve seven points. Hence we had to go to the head coordinator again. He had a tough decision (at 7:45pm after the super long discussion we had had) and decided to trust in Carlos and gave the student the 7 which settled Manuel to have a perfect exam.
I was extremely happy after this and we thought that this was one of our best performances in terms of defending points for our students. I am very proud of getting the alternative solution to Karina's exam and David thinks the defense of Manuel's 7 is the finest moment.
At night, I went to get drinks with Pedroza, Rogelio and David and we had a very interesting conversation. I kept wanting to go dance to the dancefloor as they were playing salsa, but I ended up not going. When we were walking back to the hotel we saw a large crowd coming out of it. They had just competed in the Geometrense (a competition consisting of 9 geometry problems for ex-Olympians, I didn't compete because I am not a big fan of geometry) and they were heading out to party. We went back to the bar and talked for hours. I had a pleasant conversation with Pablín (el oro), who had invented problem 6 of the exam (my favorite problem this year). It was also very funny to see David's pajonez shine in the room with his awkward questions. A very fun time. However we went to bed at 5am.
Friday:
Didn't get much sleep because we had to go to a 4 hour meeting. After the meeting I went to ear with my mom and then I went to the prize ceremony. There I met the governor. I was very happy that the governor came to the ceremony, this is the first time I see this in the 9 Olympiads I've attended. The ceremony was a bit long but nice. I had to be standing up all the time because there weren't enough seats. My mom seemed to have a good time too.
After the ceremony I went to the closing party which was okay. We were a bit sad because the team had performed poorly getting a seventh place (I thought we had a chance to win) and very far from the first place. We did well in geometry (doing better than Morelos), but we got only 145 points while first place got 191. The students seemed sad, however they did some funny things like Pepe as part of a punishment went to a table took a bite from somebody else's food then drank from another person's glass. The reactions were very funny.
When I came back from the party I went salsa dancing with my mom. I had a great time dancing with her. I think my dancing has improved a lot. I also danced with her and a group of her friends the Friday before flying to Campeche and that was a lot of fun. It was also nice meeting her tango friends.
After dancing salsa with my mom, I went to my bedroom and outside my bedroom were a couple of students and Niño talking. Niño is a very good friend of mine so I joined the conversation. I had a lot of laughs talking to María, Irune, Nacho and Niño.
Saturday:
After relaxing all day, we went to eat dinner with my mom and then caught the airplane to head back home. When we arrived to Juárez, Pepe's family received us with big poster that said "Campeones" (champions). It was very nice to have such a warm reception. It was the first time a team was all from one city (Juárez) and so this was the first time the whole team arrived at the same time to the same city. It was great.
I had a really wonderful time and there were a lot of positive things. I was happy to solve all the problems. I was happy to play volleyball. I was happy to defend points for my students (and do such a good job at it) and I was happy to spend a lot of time with my great friends David and Héctor. Even though the team didn't do as well as predicted I am very proud of them and of the hard work I did in helping in their preparation. I love the Mathematical Olympiad.
Problemas con el Sidebar
Mientras encuentro como rayos arregarlo, digan si les gusta la plantilla.
Saludos.
Isaí
EDIT:
YA ESTA! xD
sábado, 5 de diciembre de 2009
Problema del Día
Trataré de empezar a poner problemas que me gustan en el blog. Me gustaría que los intentaran y comentaran sobre el problema.
Ya que hace poco puse el problema 6 del nacional de Morelia, ahora pondré el 5 de ese mismo nacional. En lo personal me parece casi de la misma dificultad y me dio gusto intentarlo hace 3 días. No lo había revisitado desde mi nacional y me dio gusto que ahora si me salió. Además siento que el problema en sí es muy bonito.
Allí les va:
Se tiene un tablero de n x n pintado como tablero de ajedrez. Está permitido efectuar la siguiente operación en el tablero:
Escoger un rectángulo en la cuadrícula tal que las longitudes de sus lados sean ambas pares o ambas impares, pero que no sean las dos iguales a 1 al mismo tiempo, e invertir los colores de los cuadritos de ese rectángulo (es decir, los cuadritos del rectángulo que eran negros se convierten en blancos y los que eran blancos, se convierten en negros).
Encuentra para que n's es posible lograr que todos los cuadritos queden de un mismo color después de haber efectuado la operación el número de veces que sea necesario.
Nota: Las dimensiones de los rectángulos que se escogen pueden ir cambiando.
Ya que hace poco puse el problema 6 del nacional de Morelia, ahora pondré el 5 de ese mismo nacional. En lo personal me parece casi de la misma dificultad y me dio gusto intentarlo hace 3 días. No lo había revisitado desde mi nacional y me dio gusto que ahora si me salió. Además siento que el problema en sí es muy bonito.
Allí les va:
Se tiene un tablero de n x n pintado como tablero de ajedrez. Está permitido efectuar la siguiente operación en el tablero:
Escoger un rectángulo en la cuadrícula tal que las longitudes de sus lados sean ambas pares o ambas impares, pero que no sean las dos iguales a 1 al mismo tiempo, e invertir los colores de los cuadritos de ese rectángulo (es decir, los cuadritos del rectángulo que eran negros se convierten en blancos y los que eran blancos, se convierten en negros).
Encuentra para que n's es posible lograr que todos los cuadritos queden de un mismo color después de haber efectuado la operación el número de veces que sea necesario.
Nota: Las dimensiones de los rectángulos que se escogen pueden ir cambiando.
miércoles, 2 de diciembre de 2009
¿Cuándo son los entrenamientos Nacionales?
Manuel y Karina, ¿cuándo son los entrenamientos nacionales? Estaría padre si nos mandan los problemas que les encargaron de tarea (en caso de que les hayan encargado tarea). Me gustaría ver los problemas y creo que sería bueno que los intentaran también Luis y Alberto.
Si no tienen problemas de tarea les recomiendo que intenten al menos un problema nivel nacional diario. Si se les hace difícil encontrar problemas, les recomiendo tratar nacionales de años pasados. Esta semana me he puesto a resolver problemas de nacionales de hace rato.
Les escribo un problema para los amantes de geometría. Este apareció en el XIV nacional Morelia, 2000:
Sea ABC un triángulo en el que el ángulo en B es mayor a 90 grados y en el que un punto H sobre AC tiene la propiedad de que AH = BH y BH es perpendicular a BC. Sean D y E los puntos medios de AB y BC, respectivamente. Por H traza una paralela a AB que corta a DE en F. Prueba que el ángulo BCF es igual al ángulo ACD.
Saludos,
Enrique
Si no tienen problemas de tarea les recomiendo que intenten al menos un problema nivel nacional diario. Si se les hace difícil encontrar problemas, les recomiendo tratar nacionales de años pasados. Esta semana me he puesto a resolver problemas de nacionales de hace rato.
Les escribo un problema para los amantes de geometría. Este apareció en el XIV nacional Morelia, 2000:
Sea ABC un triángulo en el que el ángulo en B es mayor a 90 grados y en el que un punto H sobre AC tiene la propiedad de que AH = BH y BH es perpendicular a BC. Sean D y E los puntos medios de AB y BC, respectivamente. Por H traza una paralela a AB que corta a DE en F. Prueba que el ángulo BCF es igual al ángulo ACD.
Saludos,
Enrique
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