Difference between revisions of "2013 TSTST Problems"
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===Problem 2=== | ===Problem 2=== | ||
− | A finite sequence of integers <math>a_1, a_2, \dots, a_n</math> is called regular if there exists a real number <math>x</math> satisfying<cmath> \left\lfloor kx \right\rfloor = a_k \quad \text{for } 1 \le k \le n. </cmath>Given a regular sequence <math>a_1, a_2, \dots, a_n</math>, for <math>1 \le k \le n</math> we say that the term <math>a_k</math> is forced if the following condition is satisfied: the sequence<cmath> a_1, a_2, \dots, a_{k-1}, b </cmath>is regular if and only if <math>b = a_k</math>. Find the maximum possible number of forced terms in a regular sequence with <math>1000</math> terms. | + | A finite sequence of integers <math>a_1, a_2, \dots, a_n</math> is called regular if there exists a real number <math>x</math> satisfying<cmath> \left\lfloor kx \right\rfloor = a_k \quad \text{for } 1 \le k \le n. </cmath>Given a regular sequence <math>a_1, a_2, \dots, a_n</math>, for <math>1 \le k \le n</math> we say that the term <math>a_k</math> is forced if the following condition is satisfied: the sequence<cmath> a_1, a_2, \dots, a_{k-1}, b </cmath>is regular if and only if <math>b = a_k</math>. Find the maximum possible number of forced terms in a regular sequence with <math>1000</math> terms. |
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[[2013 TSTST Problems/Problem 2|Solution]] | [[2013 TSTST Problems/Problem 2|Solution]] | ||
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Divide the plane into an infinite square grid by drawing all the lines <math>x=m</math> and <math>y=n</math> for <math>m,n \in \mathbb Z</math>. Next, if a square's upper-right corner has both coordinates even, color it black; otherwise, color it white (in this way, exactly <math>1/4</math> of the squares are black and no two black squares are adjacent). Let <math>r</math> and <math>s</math> be odd integers, and let <math>(x,y)</math> be a point in the interior of any white square such that <math>rx-sy</math> is irrational. Shoot a laser out of this point with slope <math>r/s</math>; lasers pass through white squares and reflect off black squares. Prove that the path of this laser will form a closed loop. | Divide the plane into an infinite square grid by drawing all the lines <math>x=m</math> and <math>y=n</math> for <math>m,n \in \mathbb Z</math>. Next, if a square's upper-right corner has both coordinates even, color it black; otherwise, color it white (in this way, exactly <math>1/4</math> of the squares are black and no two black squares are adjacent). Let <math>r</math> and <math>s</math> be odd integers, and let <math>(x,y)</math> be a point in the interior of any white square such that <math>rx-sy</math> is irrational. Shoot a laser out of this point with slope <math>r/s</math>; lasers pass through white squares and reflect off black squares. Prove that the path of this laser will form a closed loop. | ||
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[[2013 TSTST Problems/Problem 3|Solution]] | [[2013 TSTST Problems/Problem 3|Solution]] | ||
==Day 2== | ==Day 2== | ||
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for all <math>a,b,c \ge 2</math>. | for all <math>a,b,c \ge 2</math>. | ||
(Here <math>f^1(n) = f(n)</math> and <math>f^k(n) = f(f^{k-1}(n))</math> for every integer <math>k</math> greater than <math>1</math>.) | (Here <math>f^1(n) = f(n)</math> and <math>f^k(n) = f(f^{k-1}(n))</math> for every integer <math>k</math> greater than <math>1</math>.) | ||
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[[2013 TSTST Problems/Problem 6|Solution]] | [[2013 TSTST Problems/Problem 6|Solution]] | ||
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(a) For all odd <math>n</math>, prove that <math>T_n</math> is divisible by <math>n</math>. | (a) For all odd <math>n</math>, prove that <math>T_n</math> is divisible by <math>n</math>. | ||
(b) For all even <math>n</math>, prove that <math>T_n</math> is divisible by <math>n/2</math>. | (b) For all even <math>n</math>, prove that <math>T_n</math> is divisible by <math>n/2</math>. | ||
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[[2013 TSTST Problems/Problem 7|Solution]] | [[2013 TSTST Problems/Problem 7|Solution]] | ||
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Define a function <math>f: \mathbb N \to \mathbb N</math> by <math>f(1) = 1</math>, <math>f(n+1) = f(n) + 2^{f(n)}</math> for every positive integer <math>n</math>. Prove that <math>f(1), f(2), \dots, f(3^{2013})</math> leave distinct remainders when divided by <math>3^{2013}</math>. | Define a function <math>f: \mathbb N \to \mathbb N</math> by <math>f(1) = 1</math>, <math>f(n+1) = f(n) + 2^{f(n)}</math> for every positive integer <math>n</math>. Prove that <math>f(1), f(2), \dots, f(3^{2013})</math> leave distinct remainders when divided by <math>3^{2013}</math>. | ||
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[[2013 TSTST Problems/Problem 8|Solution]] | [[2013 TSTST Problems/Problem 8|Solution]] | ||
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Let <math>r</math> be a rational number in the interval <math>[-1,1]</math> and let <math>\theta = \cos^{-1} r</math>. Call a subset <math>S</math> of the plane good if <math>S</math> is unchanged upon rotation by <math>\theta</math> around any point of <math>S</math> (in both clockwise and counterclockwise directions). Determine all values of <math>r</math> satisfying the following property: The midpoint of any two points in a good set also lies in the set. | Let <math>r</math> be a rational number in the interval <math>[-1,1]</math> and let <math>\theta = \cos^{-1} r</math>. Call a subset <math>S</math> of the plane good if <math>S</math> is unchanged upon rotation by <math>\theta</math> around any point of <math>S</math> (in both clockwise and counterclockwise directions). Determine all values of <math>r</math> satisfying the following property: The midpoint of any two points in a good set also lies in the set. | ||
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[[2013 TSTST Problems/Problem 9|Solution]] | [[2013 TSTST Problems/Problem 9|Solution]] | ||
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+ | {{MAA Notice}} |
Latest revision as of 07:45, 24 February 2023
Contents
[hide]Day 1
Problem 1
Let be a triangle and , , be the midpoints of arcs , , on the circumcircle. Line passes through the feet of the perpendiculars from to and . Line passes through the feet of the perpendiculars from to and . Let denote the intersection of lines and . Define points and similarly. Prove that triangle and are similar to each other.
Problem 2
A finite sequence of integers is called regular if there exists a real number satisfyingGiven a regular sequence , for we say that the term is forced if the following condition is satisfied: the sequenceis regular if and only if . Find the maximum possible number of forced terms in a regular sequence with terms.
Problem 3
Divide the plane into an infinite square grid by drawing all the lines and for . Next, if a square's upper-right corner has both coordinates even, color it black; otherwise, color it white (in this way, exactly of the squares are black and no two black squares are adjacent). Let and be odd integers, and let be a point in the interior of any white square such that is irrational. Shoot a laser out of this point with slope ; lasers pass through white squares and reflect off black squares. Prove that the path of this laser will form a closed loop.
Day 2
Problem 4
Circle , centered at , is internally tangent to circle , centered at , at . Let and be variable points on and , respectively, such that line is tangent to (at ). Determine the locus of -- the circumcenter of triangle .
Problem 5
Let be a prime. Prove that any complete graph with vertices, whose edges are labelled with integers, has a cycle whose sum of labels is divisible by .
Problem 6
Let be the set of positive integers. Find all functions that satisfy the equation for all . (Here and for every integer greater than .)
Problem 7
A country has cities, labelled . It wants to build exactly roads between certain pairs of cities so that every city is reachable from every other city via some sequence of roads. However, it is not permitted to put roads between pairs of cities that have labels differing by exactly , and it is also not permitted to put a road between cities and . Let be the total number of possible ways to build these roads. (a) For all odd , prove that is divisible by . (b) For all even , prove that is divisible by .
Problem 8
Define a function by , for every positive integer . Prove that leave distinct remainders when divided by .
Problem 9
Let be a rational number in the interval and let . Call a subset of the plane good if is unchanged upon rotation by around any point of (in both clockwise and counterclockwise directions). Determine all values of satisfying the following property: The midpoint of any two points in a good set also lies in the set.
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.