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• 03:32, 8 October 2014 (diff | hist) . . (-15)‎ . . m 1970 Canadian MO Problems/Problem 2 ‎
• 03:32, 8 October 2014 (diff | hist) . . (+537)‎ . . 1970 Canadian MO Problems/Problem 2 ‎ (→‎Problem)
• 03:32, 8 October 2014 (diff | hist) . . (+324)‎ . . 1970 Canadian MO Problems ‎ (→‎Problem 2) (current)
• 03:22, 8 October 2014 (diff | hist) . . (+284)‎ . . 1970 Canadian MO Problems ‎ (→‎Problem 2)
• 03:04, 8 October 2014 (diff | hist) . . (+396)‎ . . N 1970 Canadian MO Problems/Problem 10 ‎ (Created page with "== Problem == Given the polynomial <math>f(x)=x^n+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-1}x+a_n</math> with integer coefficients <math>a_1,a_2,\ldots,a_n</math>, and given also ...")
• 03:03, 8 October 2014 (diff | hist) . . (+237)‎ . . N 1970 Canadian MO Problems/Problem 9 ‎ (Created page with "== Problem == Let f(n) be the sum of the first n terms of the sequence 0, 1,1, 2,2, 3,3, 4,4, 5,5, 6,6, \ldots\, . a) Give a formula for f(n). b) Prove that f(s+t)-f(s-t)=st wh...")
• 03:02, 8 October 2014 (diff | hist) . . (+219)‎ . . N 1970 Canadian MO Problems/Problem 8 ‎ (Created page with "== Problem == Consider all line segments of length 4 with one end-point on the line y=x and the other end-point on the line y=2x. Find the equation of the locus of the midpoints...")
• 03:02, 8 October 2014 (diff | hist) . . (+179)‎ . . N 1970 Canadian MO Problems/Problem 7 ‎ (Created page with "== Problem == Show that from any five integers, not necessarily distinct, one can always choose three of these integers whose sum is divisible by <math>3</math>. == Solution ==")
• 03:01, 8 October 2014 (diff | hist) . . (+342)‎ . . N 1970 Canadian MO Problems/Problem 4 ‎ (Created page with "== Problem == a) Find all positive integers with initial digit <math>6</math> such that the integer formed by deleting <math>6</math> is <math>1/25</math> of the original intege...")
• 03:00, 8 October 2014 (diff | hist) . . (+325)‎ . . N 1970 Canadian MO Problems/Problem 3 ‎ (Created page with "== Problem == A set of balls is given. Each ball is coloured red or blue, and there is at least one of each colour. Each ball weighs either <math>1</math> pound or <math>2</math...")
• 03:00, 8 October 2014 (diff | hist) . . (+371)‎ . . N 1970 Canadian MO Problems/Problem 2 ‎ (Created page with "== Problem == Given a triangle <math>ABC</math> with angle <math>A</math> obtuse and with altitudes of length <math>h</math> and <math>k</math> as shown in the diagram, prove th...")
• 02:56, 8 October 2014 (diff | hist) . . (+667)‎ . . N 2007 iTest Problems/Problem 60 ‎ (Created page with "== Problem == Let <math>T=\text{TNFTPP}</math>. Triangle <math>ABC</math> has <math>AB=6T-3</math> and <math>AC=7T+1</math>. Point <math>D</math> is on <math>BC</math> so that <...")
• 02:55, 8 October 2014 (diff | hist) . . (+240)‎ . . N 2007 iTest Problems/Problem 57 ‎ (Created page with "== Problem == Let <math>T=\text{TNFTPP}</math>. How many positive integers are within <math>T</math> of exactly <math>\lfloor \sqrt T\rfloor</math> perfect squares? (Note: <math>...")
• 02:55, 8 October 2014 (diff | hist) . . (+234)‎ . . N 2007 iTest Problems/Problem 56 ‎ (Created page with "== Problem == Let <math>T=\text{TNFTPP}</math>. In the binary expansion of <math>\dfrac{2^{2007}-1}{2^T-1}</math>, how many of the first <math>10,000</math> digits to the right o...")
• 02:54, 8 October 2014 (diff | hist) . . (+233)‎ . . N 2007 iTest Problems/Problem 55 ‎ (Created page with "== Problem == Let <math>T=\text{TNFTPP}</math>, and let <math>R=T-914</math>. Let <math>x</math> be the smallest real solution of <math>3x^2+Rx+R=90x\sqrt{x+1}</math>. Find the v...")
• 02:53, 8 October 2014 (diff | hist) . . (+816)‎ . . N 2007 iTest Problems/Problem 54 ‎ (Created page with "== Problem == Let <math>T=\text{TNFTPP}</math>. Consider the sequence <math>(1, 2007)</math>. Inserting the difference between <math>1</math> and <math>2007</math> between them,...")
• 02:53, 8 October 2014 (diff | hist) . . (+307)‎ . . N 2007 iTest Problems/Problem 53 ‎ (Created page with "== Problem == Let <math>T=\text{TNFTPP}</math>. Three distinct positive Fibonacci numbers, all greater than <math>T</math>, are in arithmetic progression. Let <math>N</math> be t...")
• 02:52, 8 October 2014 (diff | hist) . . (+790)‎ . . N 2007 iTest Problems/Problem 59 ‎ (Created page with "== Problem == Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where ...")
• 02:52, 8 October 2014 (diff | hist) . . (+371)‎ . . N 2007 iTest Problems/Problem 58 ‎ (Created page with "== Problem == Let <math>T=\text{TNFTPP}</math>. For natural numbers <math>k,n\geq 2</math>, we define <cmath>S(k,n)=\left\lfloor\frac{2^{n+1}+1}{2^{n-1}+1}\right\rfloor+\left\lfl...")
• 02:50, 8 October 2014 (diff | hist) . . (-1)‎ . . m 2007 iTest Problems ‎ (→‎Problem 58)
• 02:49, 8 October 2014 (diff | hist) . . (+3,403)‎ . . m 2007 iTest Problems ‎
• 22:03, 7 October 2014 (diff | hist) . . (+510)‎ . . N 2007 iTest Problems/Problem 50 ‎ (Created page with "== Problem == A block <math>Z</math> is formed by gluing one face of a solid cube with side length <math>6</math> onto one of the circular faces of a right circular cylinder wit...")
• 22:03, 7 October 2014 (diff | hist) . . (+13)‎ . . 2007 iTest Problems ‎
• 22:03, 7 October 2014 (diff | hist) . . (+181)‎ . . N 2007 iTest Problems/Problem 49 ‎ (Created page with "== Problem == How many <math>7</math>-element subsets of <math>\{1, 2, 3,\ldots , 14\}</math> are there, the sum of whose elements is divisible by <math>14</math>? == Solution ==")
• 22:02, 7 October 2014 (diff | hist) . . (+369)‎ . . N 2007 iTest Problems/Problem 48 ‎ (Created page with "== Problem == Let a and b be relatively prime positive integers such that <math>a/b</math> is the maximum possible value of <math>\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2...")
• 22:02, 7 October 2014 (diff | hist) . . (+619)‎ . . N 2007 iTest Problems/Problem 47 ‎ (Created page with "== Problem == Let <math>\{X_n\}</math> and <math>\{Y_n\}</math> be sequences defined as follows: <math>X_0=Y_0=X_1=Y_1=1</math>, <cmath>\begin{align*}X_{n+1}&=X_n+2X_{n-1}\qqu...")
• 22:01, 7 October 2014 (diff | hist) . . (+382)‎ . . N 2007 iTest Problems/Problem 46 ‎ (Created page with "== Problem == Let <math>(x,y,z)</math> be an ordered triplet of real numbers that satisfies the following system of equations: <cmath>\begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,...")
• 22:01, 7 October 2014 (diff | hist) . . (+221)‎ . . N 2007 iTest Problems/Problem 45 ‎ (Created page with "== Problem == Find the sum of all positive integers <math>B</math> such that <math>(111)_B=(aabbcc)_6</math>, where <math>a,b,c</math> represent distinct base <math>6</math> dig...")
• 22:01, 7 October 2014 (diff | hist) . . (+357)‎ . . N 2007 iTest Problems/Problem 44 ‎ (Created page with "== Problem == A positive integer <math>n</math> between <math>1</math> and <math>N=2007^{2007}</math> inclusive is selected at random. If <math>a</math> and <math>b</math> are n...")
• 22:00, 7 October 2014 (diff | hist) . . (+638)‎ . . N 2007 iTest Problems/Problem 43 ‎ (Created page with "== Problem == Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spre...")
• 22:00, 7 October 2014 (diff | hist) . . (+752)‎ . . N 2007 iTest Problems/Problem 42 ‎ (Created page with "== Problem == During a movie shoot, a stuntman jumps out of a plane and parachutes to safety within a <math>100</math> foot by <math>100</math> foot square field, which is entir...")
• 21:59, 7 October 2014 (diff | hist) . . (+538)‎ . . N 2007 iTest Problems/Problem 41 ‎ (Created page with "== Problem == The sequence of digits <math>123456789101112131415161718192021\ldots</math> is obtained by writing the positive integers in order. If the <math>10^{nth}</math> dig...")
• 21:58, 7 October 2014 (diff | hist) . . (+194)‎ . . N 2007 iTest Problems/Problem 40 ‎ (Created page with "== Problem == Let <math>S</math> be the sum of all <math>x</math> such that <math>1\leq x\leq 99</math> and <math>\{x^2\}=\{x\}^2</math>. Compute <math>\lfloor S\rfloor</math>. ...")
• 21:58, 7 October 2014 (diff | hist) . . (+240)‎ . . N 2007 iTest Problems/Problem 39 ‎ (Created page with "== Problem == Let a and b be relatively prime positive integers such that a/b is the sum of the real solutions to the equation <math>\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+...")
• 21:58, 7 October 2014 (diff | hist) . . (+117)‎ . . N 2007 iTest Problems/Problem 38 ‎ (Created page with "== Problem == Find the largest positive integer that is equal to the cube of the sum of its digits. == Solution ==")
• 21:57, 7 October 2014 (diff | hist) . . (+872)‎ . . N 2007 iTest Problems/Problem 37 ‎ (Created page with "== Problem == Rob is helping to build the set for a school play. For one scene, he needs to build a multi-colored tetrahedron out of cloth and bamboo. He begins by fitting three...")
• 21:57, 7 October 2014 (diff | hist) . . (+358)‎ . . N 2007 iTest Problems/Problem 36 ‎ (Created page with "== Problem == Let b be a real number randomly selected from the interval <math>[-17,17]</math>. Then, m and n are two relatively prime positive integers such that m/n is the pro...")
• 21:56, 7 October 2014 (diff | hist) . . (+206)‎ . . N 2007 iTest Problems/Problem 35 ‎ (Created page with "== Problem == Find the greatest natural number possessing the property that each of its digits except the first and last one is less than the arithmetic mean of the two neighbor...")
• 21:56, 7 October 2014 (diff | hist) . . (+258)‎ . . N 2007 iTest Problems/Problem 34 ‎ (Created page with "== Problem == Let <math>a/b</math> be the probability that a randomly selected divisor of <math>2007</math> is a multiple of <math>3</math>. If <math>a</math> and <math>b</math>...")
• 21:55, 7 October 2014 (diff | hist) . . (+177)‎ . . N 2007 iTest Problems/Problem 33 ‎ (Created page with "== Problem == How many <math>\textit{odd}</math> four-digit integers have the property that their digits, read left to right, are in strictly decreasing order? == Solution ==")
• 21:54, 7 October 2014 (diff | hist) . . (+832)‎ . . N 2007 iTest Problems/Problem 32 ‎ (Created page with "== Problem == When a rectangle frames a parabola such that a side of the rectangle is parallel to the parabola's axis of symmetry, the parabola divides the rectangle into region...")
• 21:53, 7 October 2014 (diff | hist) . . (+245)‎ . . N 2007 iTest Problems/Problem 31 ‎ (Created page with "== Problem == Let <math>x</math> be the length of one side of a triangle and let y be the height to that side. If <math>x+y=418</math>, find the maximum possible <math>\textit{i...")
• 21:53, 7 October 2014 (diff | hist) . . (+819)‎ . . N 2007 iTest Problems/Problem 30 ‎ (Created page with " == Problem == While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of deat...")
• 21:53, 7 October 2014 (diff | hist) . . (+174)‎ . . N 2007 iTest Problems/Problem 29 ‎ (Created page with "== Problem == Let <math>S</math> be equal to the sum <math>1+2+3+\cdots+2007</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. == Solution ==")
• 21:52, 7 October 2014 (diff | hist) . . (+151)‎ . . N 2007 iTest Problems/Problem 28 ‎ (Created page with "== Problem == The space diagonal (interior diagonal) of a cube has length 6. Find the <math>\textit{surface area}</math> of the cube. == Solution ==")
• 21:51, 7 October 2014 (diff | hist) . . (+186)‎ . . N 2007 iTest Problems/Problem 27 ‎ (Created page with "== Problem == The face diagonal of a cube has length <math>4</math>. Find the value of n given that <math>n\sqrt2</math> is the <math>\textit{volume}</math> of the cube. == Sol...")
• 21:51, 7 October 2014 (diff | hist) . . (+563)‎ . . N 2007 iTest Problems/Problem 26 ‎ (Created page with "== Problem == Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of &#0...")
• 21:50, 7 October 2014 (diff | hist) . . (+100)‎ . . m 2007 iTest Problems ‎
• 21:46, 7 October 2014 (diff | hist) . . (+6,367)‎ . . m 2007 iTest Problems ‎
• 21:36, 7 October 2014 (diff | hist) . . (+2,753)‎ . . 2007 iTest Problems ‎

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