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AIME2018试题

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闲来无事,测试一下爪机QQ的OCR。

将楼主的几个图发给任意QQ好友,点开再点菜单中的“提取图中文字”,提取到如下东东:
  1.   2018 AIME II Problems
  2.   1. PointsA. B, and C lie in that order along a straight path where the distance
  3.   from A toC is 1800 meters. Inaruns twice as fast as Eve, and Paul runs twiceas fast as Ina. The three runners start running at the same time with Ina starting
  4.   &vo  atAand running toward C, Paul starting at B and running toward C, and Eve
  5.   starting at C and running toward A. When Paúi meets Eve, he turns around andruns toward A. Paul and Ina both arrive at B at the same time. Find the numberof meters from A to B.
  6.   2. Letao =2, a1 =5,anda2= 8,andforn >2define an recursively
  7.   to be the remainder when 4(an-1 + an2+ an-3) is divided by 11.  Find
  8.   7-  a2oY8 .a2d20 a22
  9.   7 È. Find the sum of all positive integersb < 1000 such that the base-b integer 36b
  10.   is a perfect square and the base-b integer 27b is a perfect cube.
  11.   4. In equiangular octagon CAROLINE CA =RO =LI= NE =V2 and
  12.   AR= OLeINe EC =1 The self-intersecting octagon CORNELIA
  13.   4  encloses six non-overlapping triangular regions. Let K be the area enclosed byCORNELIA, that is, the total area of the six triangular regions. ThenK= g,
  14.   where a and b are relatively prime positive integers. Finda + b.

  15.   5Suppose that x, ys and sare complex numbers such thatxy =80  320i,
  16.   yz =l60, andzx =96 424i, wherei = VT. Then there are real numbers
  17.   a and b such thatx+ y+z =a+bi. Finda? + b2.  Juui  x6xit8232
  18.   Otbt2ob1  1ecvi  zwotH9s6 Areal numbera is chosen randomly and uniformly from the interval [20.18].  4-i
  19.   The probability that the roots of the polynomial
  20.   x+2ax +(2a2)x2 +(4a + 3)x2z (-D(e2( X+ (>a-l)x+1
  21.   are all real can be written in the form n. where m and n are relatively prime
  22.   positive integers. Find m+ n.
  23.   7. Triangle ABC has side lengthsAB =9. BC=5V3, and AC= 12. Points
  24.   A=Po.Pi,Pa..... P2450 = B are on segment AB with Pk between P._ IandPk+i fork=12...2449, and pointsA= Qo. Qi. Q..... Q2450=C are on segment AC with Qk betweenQk I and Qk+I fork = 1, 2.....2449.q8 Furthermore, each segment P.Qr.k = 1. 2....2449, is parallelto BC. The
  25.   segments cut the triangle into 2450 regions, consisting of 2449 trapezoids and1 triangle. Each of the 2450 regions has the same area. Find the number ofsegments PQk,k=l.2.....2450 that have rational length.

  26.   8. Afrog is positioned at the origin in the coordinate plane. From the point(x. y)
  27.   the frog can jump to any of the points (x + l.,y). (x+2,y), (x.y+ 1), or
  28.   96.  (x,y+ 2). Find the number of distinct sequences of jumps in which the frog
  29.   begins at (0.0) and ends at (4, 4).  qot6+?o

  30. 2018 AIME II Problems
  31. 9. Octagon ABCDEFGH with side lengthsAB = CD= EF =GH= 10
  32. and BC =DE= FG= HA = 11 is formed by removing four 6 8- 10
  33. triangles from the cornersofa23x 27 rectangle with side AH on a short sideof
  34. 28o
  35. the rectangle, as shown. Let J be the midpoint of HÃ, and partition the octagon
  36. into 7 triangles by drawing segments JB, JC. JD, JE.JF, and JG. Find the
  37. area of the convex polygon whose vertices are the centroids of these 7 triangles
  38. H
  39. E
  40. Y
  41. ase 63wz2
  42. .
  43. =[?4p
  44. 9S
  45. B
  46. tgo3 qNA 6ee.


  47.   10. Find the number of functions f(x) from ¡1,2,3,45 to [1.23.45 that
  48.   vo  satisfy ff(x)= f(f(f(x) forallxin23,45
  49.   ltXt20
  50.   ANFind the number of permutationsof 1,2,3,4,5,6 such that for each k with
  51.   1 <k 5,atleast one of the first k terms of the permutation is greater than k.
  52.   12. Let ABCD be a convex quadrilateral with AB = CD= 10, BC= 14, and
  53.   AD = 2V65. Assume that the diagonals of ABCD intersect at point P, andthat the sum of the areas of. AAPB and ACPD equals the sum of the areas ofABPC and AAPD. Find the area of quadrilateral ABCD.
  54.   d3X Misha rolls. a standard, fair six-sided die until she rolls 1-2-3 in that order on
  55.   three consecutive rolls. The probability that she will roll the die an odd numberof times is m, where m and n are relatively prime positive integers. Findm + n.
  56.   14. The incircle w of AABC is tangenttoBCatX LetY#Xbethe other
  57.   intersection of AX and o. Points P and Q lie on AB and AC , respectively, sothat PQ is tangent tooat Y. Assume thatAP= 3, PB=4AC = 8, andAQ = m, where m and n are relatively prime positive integers. Findm +n.
  58.   15. Find the number of functions f from [01.2.3,4, 5.6) to the integers such that
  59.   f(0)=0, f(6)= 12, and
复制代码
乱七八zao的东东并不算很多,公式当然识别得不好,文字基本正确,偶尔有些空格没了,其他的主要是有些用笔写的东东也识别进去了,早知道就有画图工具先擦一下。

大概地稍微整理一下,码一些有必要码的公式,就成如下的:
2018 AIME II Problems

1. Points A, B, and C lie in that order along a straight path where the distance
from A to C is 1800 meters. Ina runs twice as fast as Eve, and Paul runs twice
as fast as Ina. The three runners start running at the same time with Ina starting
at A and running toward C, Paul starting at B and running toward C, and Eve
starting at C and running toward A. When Paúi meets Eve, he turns around and
runs toward A. Paul and Ina both arrive at B at the same time. Find the number
of meters from A to B.

2. Let `a_0 =2, a_1 =5`, and `a_2= 8`, and for `n >2` define an recursively
to be the remainder when `4(a_{n-1} + a_{n-2}+ a_{n-3})` is divided by 11.  Find
`a_{2018}a_{2020}a_{2022}`.

3. Find the sum of all positive integers `b < 1000` such that the base-`b` integer `36_b`
is a perfect square and the base-`b` integer `27_b` is a perfect cube.

4. In equiangular octagon CAROLINE `CA =RO =LI= NE =\sqrt2` and
`AR= OL=IN= EC =1`. The self-intersecting octagon CORNELIA
encloses six non-overlapping triangular regions. Let `K` be the area enclosed by
CORNELIA, that is, the total area of the six triangular regions. Then `K= a/b`,
where `a` and `b` are relatively prime positive integers. Find `a + b`.

5. Suppose that `x, y` and `z` are complex numbers such that `xy =-80 - 320i`,
`yz =60`, and `zx =-96 +24i`, where `i = \sqrt{-1}`. Then there are real numbers
`a` and `b` such that `x+ y+z =a+bi`. Find `a^2 + b^2`.

6. A real number `a` is chosen randomly and uniformly from the interval `[-20,18]`.
The probability that the roots of the polynomial
\[x^4+2ax^3+(2a-2)x^2 +(-4a + 3)x-2\] are all real can be written in the form `m/n`. where `m` and `n` are relatively prime
positive integers. Find `m+ n`.

7. Triangle `ABC` has side lengths `AB =9`. `BC=5\sqrt3`, and `AC= 12`. Points
`A=P_0,P_1,...,P_{2450} = B` are on segment `AB` with `P_k` between `P_{k-1}`
and `P_{k+1}` for `k=1,2,...,2449`, and points `A= Q_0, Q_1,...,Q_{2450}=C`
are on segment `AC` with `Q_k` between `Q_{k-1}` and `Q_{k+1}` for `k = 1, 2,...,2449`.
Furthermore, each segment `P_kQ_k, k= 1, 2,...,2449`, is parallel to `BC`. The
segments cut the triangle into 2450 regions, consisting of 2449 trapezoids and
1 triangle. Each of the 2450 regions has the same area. Find the number of
segments `P_kQ_k,k=l,2,...,2450` that have rational length.

8. A frog is positioned at the origin in the coordinate plane. From the point `(x, y)`
the frog can jump to any of the points `(x + 1,y), (x+2,y), (x,y+ 1)`, or
`(x,y+ 2)`. Find the number of distinct sequences of jumps in which the frog
begins at (0.0) and ends at (4, 4).

9. Octagon `ABCDEFGH` with side lengths `AB = CD= EF =GH= 10`
and `BC =DE= FG= HA = 11` is formed by removing four 6- 8- 10
triangles from the corners of a 23x27 rectangle with side `AH` on a short side of
the rectangle, as shown. Let `J` be the midpoint of `HA`, and partition the octagon
into 7 triangles by drawing segments `JB, JC, JD, JE, JF`, and `JG`. Find the
area of the convex polygon whose vertices are the centroids of these 7 triangles.
QQ截图20180523021350.jpg
2018-5-23 02:14

10. Find the number of functions `f(x)` from {1,2,3,4,5} to {1,2,3,4,5} that
satisfy `f(f(x))= f(f(f(x)))` for all `x` in {1,2,3,4,5}.

11. Find the number of permutations of 1,2,3,4,5,6 such that for each `k` with
`1 \le k\le5`, at least one of the first `k` terms of the permutation is greater than `k`.

12. Let `ABCD` be a convex quadrilateral with `AB = CD= 10, BC= 14`, and
`AD = 2\sqrt{65}`. Assume that the diagonals of `ABCD` intersect at point `P`, and
that the sum of the areas of. `\triangle APB` and `\triangle CPD` equals the sum of the areas of
`\triangle BPC` and `\triangle APD`. Find the area of quadrilateral `ABCD`.

13. Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on
three consecutive rolls. The probability that she will roll the die an odd number
of times is `m/n`, where `m` and `n` are relatively prime positive integers. Find `m + n`.

14. The incircle `w` of `\triangle ABC` is tangent to `BC` at `X` Let `Y\ne X` be the other
intersection of `AX` and `w`. Points `P` and `Q` lie on `AB` and `AC` , respectively, so
that `PQ` is tangent to `w` at `Y`. Assume that `AP= 3, PB=4,AC = 8`, and
`AQ = m/n`, where `m` and `n` are relatively prime positive integers. Find `m +n`.

15. Find the number of functions f from {0,1,2,3,4,5,6} to the integers such that
`f(0)=0, f(6)= 12`, and
\[|x-y|\le|f(x)-f(y)|\le3|x-y|\]for all `x` and `y` in {0,1,2,3,4,5,6}.

这样,一楼的图就可以扔掉了。
$\href{https://kuingggg.github.io/}{\text{About Me}}$

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第8题,题意大概就是一步可以沿 x 或 y 轴的正向跳 1 或 2 个单位,问从原点到 (4,4) 的跳法数。
按照前天这帖 http://kuing.orzweb.net/viewthread.php?tid=5363 从13楼起的那个方法,只需展开
\[\sum_{k=4}^8(x + x^2 + y + y^2)^k\]
看看 `x^4y^4` 的系数即可。
在MMC中输入:
Coefficient[Sum[(x + x^2 + y + y^2)^k, {k, 4, 8}], x^4 y^4]
输出结果为 556

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本帖最后由 isee 于 2018-5-23 16:49 编辑

回复 2# kuing
图还是保留的好,当有疑议时,图片能排除文字疑义

PS:OCR对文字的识别一很高,特别是纯文字。图文混排,或者公式很多时,OCR就很难看了。

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