36th United States of America Mathematical Olympiad : USAMO 2007

1. Let n be a positive integer. Define a sequence by setting a₁ = n and, for each k > 1, letting a_k be the unique integer in the range 0 ≤ a_k ≤ k – 1 for which a₁ + a₂ + … + a_k is divisible by k. For instance, when n = 9 the obtained sequence is 9, 1, 2, 0, 3, 3, 3, … .

Prove that for any n the sequence a₁, a₂, a₃, …. eventually becomes constant.

2. A square grid on the Euclidean plane consists of all points (m, n), where m and n are integers. Is it possible to cover all grid points by an infinite family of discs with non-overlapping interiors if each disc in the family has radius at least 5?

3. Let S be a set containing n² + n – 1 elements, for some positive integer n. Suppose that the n-element subsets of S are partitioned into two classes. Prove that there are at least n pairwise disjoint sets in the same class.

4. An animal with n cells is a connected figure consisting of n equal-sized square cells. The figure below shows an 8-cell animal.

A dinosaur is an animal with at least 2007 cells. It is said to be primitive if its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.

5. Prove that for every nonnegative integer n, the number 7^{7^n} + 1 is the product of at least 2n + 3 (not necessarily distinct) primes.

6. Let ABC be an acute triangle with ω, Ω and R being its incircle, circumcircle, and cir- cumradius, respectively. Circle ω_A is tangent internally to Ω at A and tangent externally to ω. Circle Ω_A is tangent internally to Ω at A and tangent internally to ω. Let P_A and Q_A denote the centers of ω_A and Ω_A, respectively. Define points P_B, Q_B, P_C, Q_C analogously.

Prove that

8P_AQ_A . P_BQ_B . P_CQ_C ≤ R³

with equality if and only if triangle ABC is equilateral.

Faktor yang Mempengaruhi Hasil Belajar

Hasil belajar siswa dipengaruhi oleh dua faktor utama yaitu faktor dari dalam diri siswa dan faktor yang datang dari luar diri siswa atau faktor lingkungan. Menurut Slameto (2003:54-72), faktor-faktor yang mempengaruhi belajar adalah :
1. Faktor-faktor Internal
– Jasmaniah (kesehatan, cacat tubuh)
– Psikologis (intelegensi, perhatian, minat, bakat, motif, kematangan, kesiapan)
– Kelelahan
2. Faktor-faktor Eksternal