ITOT 45 Spring 2024 · THCS · A (Advanced)

Twelve students have pairwise different strengths. The coach divides them into two teams of six on ten occasions, with a different division each time. Could all ten tug-of-war games be draws, meaning that the total strengths of the two teams are equal each time?

Prove that among the vertices of any convex nonagon there are three forming an obtuse triangle, none of whose sides is a side of the nonagon.

Two players take stones from a pile of100. On move1 the first player takes1; on move2 the second takes1 or2; on move3 the first takes1,2 or3; and so on, with at most the move number taken each time. Whoever takes the last stone wins. Which player can force a win?

Tom chooses a positive reduced fraction$x=m/n$. You may choose a positive rational$y\lt 1$, and Tom reports only the numerator of$x+y$ after reduction. How can you determine$x$ with certainty using two such queries?

Nine vertical pillars stand in a row, with horizontal sticks between neighboring pillars at pairwise different heights. A beetle climbs upward; whenever it meets a stick it crosses to the neighboring pillar and continues upward. Starting at the bottom of pillar1, it finishes at the top of pillar9. Can one always remove a single stick so that a beetle starting at pillar1 finishes at pillar5?

In the source example, the original route is solid; removing its third encountered stick gives the dotted route.

On the circumcircle of triangle$ABC$, let$M,N,P,Q$ be the midpoints of arcs$BAC,CBA,BC,AC$, respectively; the two-letter arcs are the arcs not containing the third vertex. The$A$-excircle touches$BC$ at$A_1$ and the extensions of$AC,AB$. The$B$-excircle touches$AC$ at$B_1$ and the extensions of$BA,BC$. Suppose$A_1$ lies on segment$NP$. Prove that$B_1$ lies on segment$MQ$.

Distinct real numbers are written on99 cards, and their total sum is irrational. An ordering is called unfortunate if, for every$k=1,\ldots,99$, the sum of its first$k$ cards is irrational. What is the smallest possible number of unfortunate orderings?