The 2014-15 USAMTS Round 3 set has been released, and unsurprisingly some problems from it have already been spotted on the main site.

This is another request for all users to be on the lookout for these problems, and to flag them for moderator attention when you see them so that we can lock them as per our policy on ongoing contest problems. The current problem set can be found here, and again I transcribe the problems below:

  1. Fill in each blank unshaded cell with a positive integer less than $100$, such that every consecutive group of unshaded cells within a row or column is an arithmetic sequence.

    2014-15 USAMTS round 3, problem 1 image

    You do not need to prove that your answer is the only one possible; you merely need to find and answer that satisfies the constraints above.

  2. Let $A_1A_2A_3A_4A_5$ be a regular pentagon with side length $1$. The sides of the pentagon are extended to form a 10-sided polygon shown in bold at right. Find the ratio of the area of the quadrilateral $A_2A_5B_2B_5$ (shaded in the picture to the right) to the area of the entire 10-sided polygon.

    2014-15 USAMTS round 3, problem 2 image

  3. Let $a_1, a_2, a_3 , \ldots$ be a sequence of positive real numbers such that

    1. For all positive integers $m$ and $n$ we have $a_{mn} = a_ma_n$, and
    2. There exists a positive real number $B$ such that for all positive integers $m$ and $n$ with $m < n$ we have $a_m < B a_n$.

    Find all possible values of $\log_{2015} ( a_{2015} ) - \log_{2014} ( a_{2014} )$.

  4. Nine distinct positive integers are arranged in a circle such that the product of any two non-adjacent numbers in the circle is a multiple of $n$ and the product of any two adjacent numbers in the circle is not a multiple of $n$, where $n$ is a fixed positive integer. Find the smallest possible value for $n$.

  5. A set $S$ of unit squares is chosen out of a large finite grid of unit squares. The squares of $S$ are tiled with isosceles right triangles of hypotenuse $2$ so that the triangles do not overlap each other, do not extend past $S$, and all of $S$ is fully covered by the triangles. Additionally, the hypotenuse of each triangle lies along a grid line, and the vertices of the triangles lie at the corners of the squares. Show that the number of triangles must be a multiple of $4$.

  • 5
    $\begingroup$ Question 1 has already been answered here: puzzling.stackexchange.com/q/6150/5095 $\endgroup$ – Joel Reyes Noche Dec 15 '14 at 5:17
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    $\begingroup$ @Integrator So... do you realize that you let through 3 of the above questions yesterday on the First Posts queue? While I don't expect you to have known that they were from USAMTS, you should have voted to close them for lack of context. $\endgroup$ – epimorphic Dec 15 '14 at 6:11
  • 1
    $\begingroup$ @epimorphic I'd prefer not to say anything. $\endgroup$ – Aditya Hase Dec 15 '14 at 6:24
  • $\begingroup$ OT: 2. is pretty simple for a contest problem lol $\endgroup$ – AlexR Dec 18 '14 at 23:33
  • $\begingroup$ Edited the OP to reflect the recent change in Problem 5. $\endgroup$ – Ahaan S. Rungta Dec 20 '14 at 16:11
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    $\begingroup$ This guy was clever enough to change the notation and the numbers in problem 3 before posting it. math.stackexchange.com/questions/1078201 $\endgroup$ – JimmyK4542 Dec 22 '14 at 23:28

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