29. There are 10 birds on the ground pecking at food, and among any 5 birds, at least 4 are on the same circle. How many birds are on the circle with the most birds, at a minimum?
(1991 6th CMO Problem)
29. First, prove that the circle with the most birds has at least 5 birds. Since each set of 4 birds on a circle is contained in $C_{6}^{1}=6$ sets of 5 birds, let the total number of sets of 4 birds on a circle be $X$, then these sets are contained in $6X$ sets of 5 birds (some sets of 5 birds may be counted multiple times). These $6X$ sets of 5 birds include all possible sets of 5 birds (since any 5 birds must have 4 birds on a circle), so $6X \geqslant C_{10}^{5}=252$, thus $X \geqslant 42$. Next, consider the sets of 3 birds. 10 birds can form $C_{10}^{3}=120$ sets of 3 birds, but 42 sets of 4 birds on a circle can form $42 C_{4}^{3}=168$ sets of 3 birds, which means at least 2 sets of 4 birds on a circle contain the same set of 3 birds, hence at least 5 birds are on the same circle. 

Next, let $S$ be the circle with the most birds, then $S$ has at least 5 birds $A, B, C, D, E$. Suppose there are two birds $U$ and $V$ not on $S$, then among $A, B, C$, there are 2 birds (say $A$ and $B$) with $U$ and $V$ on the same circle $S_{1}$, and among $C, D, E$, there are 2 birds (say $C, D$) with $U$ and $V$ on the same circle $S_{2}$. Now consider the 5 birds $A, C, E, U, V$, among them, there must be 4 birds on the same circle, but $U, V$ are not on the circle $S$ determined by $A, C, E$, so among $A, C, E$, there must be 2 birds with $U, V$ on the same circle; if $A, C, U, V$ are on the same circle $S_{3}$, then $S_{1}$ and $S_{2}$ both coincide with $S_{3}$, thus $S_{1}, S_{2}, S_{3}$ all coincide with $S$, leading to $U, V$ both on $S$, a contradiction; if $A$ (or $C$), $E, U, V$ are on the same circle $S_{4}$, then $S$ and $S_{1}$ (or $S_{2}$) both coincide with $S_{4}$, thus $S_{1}$ (or $S_{2}$) coincides with $S$, also leading to $U, V$ both on $S$, a contradiction. The above contradiction shows: there is at most 1 bird not on $S$. 

In summary, we conclude: the circle with the most birds has at least 9 birds.