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| − | Going to put practice problems here. | + | Going to put practice problems here: |
| − | ==2013 AMC 10A Problem 7== | + | == Problem 1== |
| | + | As shown in the figure, [[triangle]] <math>ABC</math> is divided into six smaller triangles by [[line]]s drawn from the [[vertex | vertices]] through a common interior point. The [[area]]s of four of these triangles are as indicated. Find the area of triangle <math>ABC</math>. |
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| − | A student must choose a program of four courses from a menu of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen?
| + | [[Image:AIME 1985 Problem 6.png]] |
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| − | <math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 16 </math>
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| − | [[2013 AMC 10A Problems/Problem 7|Solution]]
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| − | ==2013 AMC 10A Problem 10==
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| − | A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?
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| − | <math> \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 40 \qquad\textbf{(D)}\ 60 \qquad\textbf{(E)}\ 70 </math>
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| − | [[2013 AMC 10A Problems/Problem 10|Solution]] | |
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| − | ==2013 AMC 10A Problem 13==
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| − | How many three-digit numbers are not divisible by <math>5</math>, have digits that sum to less than <math>20</math>, and have the first digit equal to the third digit?
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| − | <math> \textbf{(A)}\ 52 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 66 \qquad\textbf{(D)}\ 68 \qquad\textbf{(E)}\ 70 </math>
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| − | [[2013 AMC 10A Problems/Problem 13|Solution]]
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| − | ==2013 AMC 10A Problem 16==
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| − | A triangle with vertices <math>(6, 5)</math>, <math>(8, -3)</math>, and <math>(9, 1)</math> is reflected about the line <math>x=8</math> to create a second triangle. What is the area of the union of the two triangles?
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| − | <math> \textbf{(A)}\ 9 \qquad\textbf{(B)}\ \frac{28}{3} \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ \frac{31}{3} \qquad\textbf{(E)}\ \frac{32}{3} </math>
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| − | [[2013 AMC 10A Problems/Problem 16|Solution]]
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| − | ==2013 AMC 10A Problem 17==
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| − | Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next 365-day period will exactly two friends visit her?
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| − | <math> \textbf{(A)}\ 48\qquad\textbf{(B)}\ 54\qquad\textbf{(C)}\ 60\qquad\textbf{(D)}\ 66\qquad\textbf{(E)}\ 72 </math>
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| − | ==2013 AMC 10A Problem 18==
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| − | Let points <math>A = (0, 0)</math>, <math>B = (1, 2)</math>, <math>C=(3, 3)</math>, and <math>D = (4, 0)</math>. Quadrilateral <math>ABCD</math> is cut into equal area pieces by a line passing through <math>A</math>. This line intersects <math>\overline{CD}</math> at point <math>\bigg(\frac{p}{q}, \frac{r}{s}\bigg)</math>, where these fractions are in lowest terms. What is <math>p+q+r+s</math>?
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| − | <math> \textbf{(A)}\ 54\qquad\textbf{(B)}\ 58\qquad\textbf{(C)}\ 62\qquad\textbf{(D)}\ 70\qquad\textbf{(E)}\ 75 </math>
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| − | [[2013 AMC 10A Problems/Problem 18|Solution]]
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| − | ==2013 AMC 10A Problem 20==
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| − | A unit square is rotated <math>45^\circ</math> about its center. What is the area of the region swept out by the interior of the square?
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| − | <math> \textbf{(A)}\ 1 - \frac{\sqrt2}{2} + \frac{\pi}{4}\qquad\textbf{(B)}\ \frac{1}{2} + \frac{\pi}{4} \qquad\textbf{(C)}\ 2 - \sqrt2 + \frac{\pi}{4}</math>
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| − | <math>\textbf{(D)}\ \frac{\sqrt2}{2} + \frac{\pi}{4} \qquad\textbf{(E)}\ 1 + \frac{\sqrt2}{4} + \frac{\pi}{8} </math>
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| − | [[2013 AMC 10A Problems/Problem 20|Solution]]
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| − | ==2013 AMC 10A Problem 22==
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| − | Six spheres of radius <math>1</math> are positioned so that their centers are at the vertices of a regular hexagon of side length <math>2</math>. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?
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| − | <math> \textbf{(A)}\ \sqrt2\qquad\textbf{(B)}\ \frac{3}{2}\qquad\textbf{(C)}\ \frac{5}{3}\qquad\textbf{(D)}\ \sqrt3\qquad\textbf{(E)}\ 2 </math>
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| − | [[2013 AMC 10A Problems/Problem 22|Solution]]
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| − | ==2013 AMC 10A Problem 23==
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| − | In <math>\triangle ABC</math>, <math>AB = 86</math>, and <math>AC=97</math>. A circle with center <math>A</math> and radius <math>AB</math> intersects <math>\overline{BC}</math> at points <math>B</math> and <math>X</math>. Moreover <math>\overline{BX}</math> and <math>\overline{CX}</math> have integer lengths. What is <math>BC</math>?
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| − | <math> \textbf{(A)}\ 11\qquad\textbf{(B)}\ 28\qquad\textbf{(C)}\ 33\qquad\textbf{(D)}\ 61\qquad\textbf{(E)}\ 72 </math>
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| − | [[2013 AMC 10A Problems/Problem 23|Solution]]
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| − | ==2013 AMC 10A Problem 24==
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| − | Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled?
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| − | <math> \textbf{(A)}\ 540\qquad\textbf{(B)}\ 600\qquad\textbf{(C)}\ 720\qquad\textbf{(D)}\ 810\qquad\textbf{(E)}\ 900</math>
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| − | [[2013 AMC 10A Problems/Problem 24|Solution]]
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| − | ==2013 AMC 10A Problem 25==
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| − | All 20 diagonals are drawn in a regular octagon. At how many distinct points in the interior
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| − | of the octagon (not on the boundary) do two or more diagonals intersect?
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| − | <math> \textbf{(A)}\ 49\qquad\textbf{(B)}\ 65\qquad\textbf{(C)}\ 70\qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 128 </math>
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| − | ==2014 AMC 10A Problem 22==
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| − | In rectangle <math>ABCD</math>, <math>AB=20</math> and <math>BC=10</math>. Let <math>E</math> be a point on <math>\overline{CD}</math> such that <math>\angle CBE=15^\circ</math>. What is <math>AE</math>?
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| − | <math> \textbf{(A)}\ \dfrac{20\sqrt3}3\qquad\textbf{(B)}\ 10\sqrt3\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 11\sqrt3\qquad\textbf{(E)}\ 20 </math>
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| − | [[2014 AMC 10A Problems/Problem 22|Solution]]
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| − | ==2014 AMC 10A Problem 23==
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| − | A rectangular piece of paper whose length is <math>\sqrt3</math> times the width has area <math>A</math>. The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area <math>B</math>. What is the ratio <math>B:A</math>?
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| − | <asy>
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| − | import graph;
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| − | size(6cm);
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| − | real L = 0.05;
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| − | pair A = (0,0);
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| − | pair B = (sqrt(3),0);
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| − | pair C = (sqrt(3),1);
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| − | pair D = (0,1);
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| − | pair X1 = (sqrt(3)/3,0);
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| − | pair X2= (2*sqrt(3)/3,0);
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| − | pair Y1 = (2*sqrt(3)/3,1);
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| − | pair Y2 = (sqrt(3)/3,1);
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| − | dot(X1);
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| − | dot(Y1);
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| − | draw(A--B--C--D--cycle, linewidth(2));
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| − | draw(X1--Y1,dashed);
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| − | draw(X2--(2*sqrt(3)/3,L));
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| − | draw(Y2--(sqrt(3)/3,1-L));
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| − | </asy>
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| − | <math> \textbf{(A)}\ 1:2\qquad\textbf{(B)}\ 3:5\qquad\textbf{(C)}\ 2:3\qquad\textbf{(D)}\ 3:4\qquad\textbf{(E)}\ 4:5 </math>
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| − | [[2014 AMC 10A Problems/Problem 23|Solution]]
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| − | ==2014 AMC 10A Problem 24==
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| − | A sequence of natural numbers is constructed by listing the first <math>4</math>, then skipping one, listing the next <math>5</math>, skipping <math>2</math>, listing <math>6</math>, skipping <math>3</math>, and, on the <math>n</math>th iteration, listing <math>n+3</math> and skipping <math>n</math>. The sequence begins <math>1,2,3,4,6,7,8,9,10,13</math>. What is the <math>500,000</math>th number in the sequence?
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| − | <math> \textbf{(A)}\ 996,506\qquad\textbf{(B)}\ 996,507\qquad\textbf{(C)}\ 996,508\qquad\textbf{(D)}\ 996,509\qquad\textbf{(E)}\ 996,510 </math>
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| − | [[2014 AMC 10A Problems/Problem 24|Solution]]
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| − | ==2014 AMC 10A Problem 25==
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| − | The number <math>5^{867}</math> is between <math>2^{2013}</math> and <math>2^{2014}</math>. How many pairs of integers <math>(m,n)</math> are there such that <math>1\leq m\leq 2012</math> and <cmath>5^n<2^m<2^{m+2}<5^{n+1}?</cmath>
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| − | <math>\textbf{(A) }278\qquad\textbf{(B) }279\qquad\textbf{(C) }280\qquad\textbf{(D) }281\qquad\textbf{(E) }282\qquad</math>
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| − | [[2014 AMC 10A Problems/Problem 25|Solution]]
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is divided into six smaller triangles by 
