投稿日: 投稿者: babalabo

Your Daily Epsilon of Math 2023

  1. 2023-01-01 整方程式の定数項は根の約数：Let p(y) be a polynomial with integer coefficients. p(0)=104729 x, x∈Z such that p(x) = x and 0<x<104729
  2. 2023-01-02 部分分数分解による無限級数の和：(lim{n→∞}1/(2n-1)(2n+1))^-1
  3. 2023-01-03 数学パズル：111/(1+1+1) + 222/(2+2+2) + …+ 999/(9+9+9)
  4. 2023-01-04 因数定理，整係数方程式の有理数解 -x^4+16x^3-96x^2+256x-256=0
  5. 2023-01-05 ピタゴラスの定理：The area of the square ABCD is 20.25. Find EC. (Round)
  6. 2023-01-06 合成関数（三角関数）の微分：Find the value of the derivative of f(x)=sin(x^3+6x) at x=0
  7. 2023-01-07 直線で囲まれた矩形面積：What is the area bounded by y=3-4x/9, x=3 and the x and y axes?
  8. 2023-01-08 三角関数，逆三角関数：cot^2(π/6)+csc(5π/6)+3tan^2(π/6)+sec(11π/6)cot(7π/6)
  9. 2023-01-09 階乗：4!÷4+(4-4÷4)
  10. 2023-01-10 三角関数の積分：∫{0~4π}(20x^2 + 30sin~2(x))dx / (x^2cos^2(x) + cos^2(x)) (Round)
  11. 2023-01-11 階乗：(x!)^2/(x-1)!(x+1)! = 11/22
  12. 2023-01-12 方角・距離：Anirban runs to a tree 2 m north of his starting location, then runs to a spot 4 m south and 8 m west of his starting location. How far did he run?
  13. 2023-01-13 タイリング問題：How many ways are there of tiling a 2×6 space with 2×1 tiles?
  14. 2023-01-14 正七角形，二面体群，対称群：What is the order of the symmetry group of a regular heptagon.
  15. 2023-01-15 指数計算：225^1/200*225^3/200*225^5/200*…*225^19/200
  16. 2023-01-16 複素数の乗除算，極座標表示：(i-1)^8
  17. 2023-01-17 接線の傾き，合成関数の微分：Find the slope of the line tangent to the curve y=2^x at x=4.6 (Round)
  18. 2023-01-18 連立整方程式：For a, b, c, d, x positive integers : abcd = 8!, ab+a+b=524, bc+b+c=146, cd+c+d=104, a-c=x
  19. 2023-01-19 素因数分解：Find the second smallest prime factor of 1,000,027
  20. 2023-01-20 四面体数・三角数：What is the fourth tetrahedral number?
  21. 2023-01-21 直線の傾き：Find the slope of the line that passes through (-10, –21) and (-7, 42)
  22. 2023-01-22 １次方程式：(8-x)/(1-x) = 2/3
  23. 2023-01-23 交差図形の面積，定積分による求積：Four circles inscribed by a big circle. An uncovered area = 23. What is the area x of a crossing area?
  24. 2023-01-24 待ち行列，ｎ次対称群（ｎ次置換群）：How many ways can Alfonso, Bianca, Catriona, and Domingo queue up?
  25.  2023-01-25 地図の縮尺：If two mountain peaks are 10 cm apart on a map with scale 4 cm to 10 km, how many kilometers separate the peaks?
  26. 2023-01-26 無限級数の和，区分求積法，置換積分，マクローリン展開：13e^(1-1/2+1/3-1/4+…)
  27. 2023-01-27 三角形の面積，三角形の相似：An equilateral triangle of area=9 is inscribed by an equilateral triangle. Find the area of the outer triangle.
  28. 2023-01-28 MMMMDCCLVI÷CXVI-XIII
  29. 2023-01-29 平面的最短距離，展開図：What is the shortest distance from A to B on the outside of the box?
  30. 2023-01-30 余弦定理・ヘロンの公式：In the diagram a square is placed on each side of a right triangle (5, 12, 13) . How much the area sandwiched  by the square 5 and square 13?
  31. 2023-01-31 二次方程式：Find the smaller root of x^2-103x+2232=0
  32. 2023-02-01 開被覆，開集合，閉包：For any set X ∈ [0, 1], let E(X) be the largest value such that for any open cover of X, the sum of its lengths ≧E(X). If Z={Q∩[0, 1]}, what is E([0, 1])-E(Z)?
  33. 2023-02-02 対数計算：1/log4(10) + 2/log5(10)
  34. 2023-02-03 置換積分法（合成関数の積分）：π/∫{0→1}1/sqrt(x)sqrt(4-x) dx
  35. 2023-02-04 フィボナッチ数列，ピタゴラス数：Starting with x+1, every second Fibonacci number is the largest number in a Pythagorean triple.
  36. 2023-02-05 方べきの定理：The circumference/inscribed angle theorem
  37. 2023-02-06 因数定理，整係数方程式の有理解，定数項の約数：Find the integer x that (x-3)^3+(x-2)^3+(x-1)^3=x^3
  38. 2023-02-07 因子木：All letters in this factor tree represent distinct digits. Primes are circled.
  39. 2023-02-08 三角形の面積：Find the triangular area from given value of sides of the squares.
  40. 2023-02-09 両替問題：How many ways can you make change for a twenty collar bill with one, five, and ten dollar bills.
  41. 2023-02-10 関数の最小値，導関数=0，合成関数の微分：Find the minimum value of sqrt(x^2+1+2/x+1/x^2) + sqrt(x^2-12x+85-14/x+1/x^2)
  42. 2023-02-11 素因数分解：Find the number of odd numbers that divide 3^12-1 but do not divide any of 3^11-1, 3^10-1, …, 3^1-1
  43. 2023-02-12 倍角・半角の公式，加法定理：tan(θ)=(sqrt(145) – 1)/12  Find tan(2θ)
  44. 2023-02-13 排他的論理和：10110B ⊕ 11011B
  45. 2023-02-14 複素数：(3-i)(2+i) + (3+i)(2-i)
  46. 2023-02-15 合同式，フェルマーの小定理，２項定理：15^113 mod 113
  47. 2023-02-16 複素数累乗根：Find the sum of the absolute value of the real parts of the roots of  z^3 – 512
  48. 2023-02-17 三角形の面積：area(⊿ABC)＝136 The degrees of ∠ABC is optional and the ratio of the sides AB : BC is free, hence the length of AC is arbitrary. Then how on earth the assigned area can be determined?
  49. 2023-02-18 切頂４面体：How many edges does a truncated tetrahedron have?
  50. 2023-02-19 整数解：A and B are whole numbers such that A/3 + B/37 = 91/111  Find A+B
  51. 2023-02-20 合成関数（有理関数）の微分：Find the derivative of  f(x) = (x^2-10)/(2x+1) at x = 0
  52. 2023-02-21 完全平方数：What is the smallest integer n such that 4725/n is a perfect square?
  53. 2023-02-22 長方形の面積：Find the third area from given two areas and two sides
  54. 2023-02-23 交代数列の累和：–1 + 2 – 3 + 4 – 5 + 6 – … –45 + 46
  55. 2023-02-24 順列・組み合わせ：Four cards are drawn at random from a deck numbered 1 to 100. What is the inverse of the probability that the cards are drawn in increasing order?
  56. 2023-02-25 指数関数の極限公式，ロピタルの定理：lim{x→0}(5x/(e^0.2x – 1))
  57. 2023-02-26 二次方程式：x^2-(x-2)^2=100
  58. 2023-02-27 ウィルソンの定理：(16!)^2 mod 37
  59. 2023-02-28 １次方程式：(x + 7) / 5 = x / 4
  60. 2023-03-01 級数の部分分数分解 Σ{1→∞}1/{n^2+n}
  61. 2023-03-02 行列の基本変形，ベクトルの合成 How many dimensions does the subspace spanned by vectors (2, 3, 4), (1, –1, 1), (5, 5, 9) have?
  62. 2023-03-03 連立方程式，因数分解 Find the number of non-negative integers n such that 16n+9 and 9n+15 are both perfect squares.
  63. 2023-03-04 逆三角関数 csc(π/6)sec(π/3)
  64. 2023-03-05 対数の微分公式 Find the derivative of y=ln(x) at x=0.2
  65. 2023-03-06 ヘロンの公式 正三角形に内接する円に内接する正方形の面積
  66. 2023-03-07 虚数の対数 {14i/π}ln({1-i}/{1+i})
  67. 2023-03-08 置換積分法 積分を指数とするネピア数の値 e^∫{0→π}{-3le(1+cos(x))}/π dx
  68. 2023-03-09 ２つの円の界斜に似た接線の長さ Two circles with radius 3 and 6 that touch each other
  69. 2023-03-10  連立方程式 Allana buys pens and pencils at $4 and 50 ¢ respectively. If she buys 100 items for $85  total, how many pens did she buy?
  70. 2023-03-11 サイコロの出目の確率 Two six-sided dice are rolled until a sum of 7 or 8 appears.
  71. 2023-03-12 平方差の絶対値，偶奇 |x^2-y^2|=52 0<x<y x, y ∈Z
  72. 2023-03-13 方程式の変形 Find the radius of the circle x^2-4x+y^2+2Y=164
  73. 2023-03-14 ８角形内の四辺形の面積 Nested squares in a regular octagon
  74. 2023-03-15 素因数分解 Find the smallest integer x>0 such that x! ends in 000.
  75. 2023-03-16 円に内接する三角形に内接する円の半径 Find the diameter of the larger circle.
  76. 2023-03-17 方程式の解と係数の関係 Find the sum of the cubes of the roots of x^3-2x^2+2x-7=0
  77. 2023-03-18 線分の長さ Given triangle with height 30 and bottom 15
  78. 2023-03-19 置換積分法 積分を指数とするネピア数の値 e^∫{0→1}{x^56-x^2}/ln(x) dx
  79. 2023-03-20 八角形の対角線数 How many diagonals does an octagon have?
  80. 2023-03-21 負数・分数の合同式 (-2023)^-1 mod 43
  81. 2023-03-22 コンウェイの円 Find AB for this Conway circle
  82. 2023-03-23 調和級数の部分和 H(n)=Σ{1→n}1/i Find {Σ{1→22}H(k)} / {H(23) – 1}
  83. 2023-03-24 等比級数 Let a_i be a geometric sequence such that a_8-a_6 = 1944 and a_8/a_6=9 What is a_4-a_2?
  84. 2023-03-25 倍角公式 cos(θ/2)=√13/5 Find sec(θ)
  85. 2023-03-26 順列・組み合わせ Find the numbers of permutations (a1, a2, a3, a4, a5) of (1, 2, 3, 4, 5) such that there is exactly one index with a_i>a_(i+1)
  86. 2023-03-27 底の変換公式 2log_3(x)+2log_9(x)=9
  87. 2023-03-28 正接の倍角半角公式 Find the sum of the square of 5 tangents
  88. 2023-03-29 正整数を定義域/値域とする関数 f(x):Z+→Z+ is such that f(x+1)>f(x) and f(f(x))=3x if f(n)=56, what is n?
  89. 2023-03-30 三角形の面積 Find the area of a triangle inside of the square with sides of length 10
  90. 2023-03-31 多角形の頂角の大きさ If the sum of all but one angle of a convex polygon is 1949,
  91. 2023-04-01 テイラー展開の公式　Let A be the coefficient of x^5 in the Taylor expansion of e^x Find 5!*A
  92. 2023-04-02 多項式の因数分解 係数の連立方程式 　Find the integer a such that x^2-x+a is a factor of x^13+x+90
  93. 2023-04-03 合同式 べき乗剰余の周期 久留島・オイラーの公式 ガロア群　3^2023^2023 mod 7
  94. 2023-04-04 指数法則　3√y^18 / y^2 = y^x for x, y ∈ Z
  95. 2023-04-05 期待値　expected number of diamond cards in a random 20 card hand?
  96. 2023-04-06 分数べき方程式の２階微分　Find the second derivative of f(x)=x^{3/2} at x=1/64
  97. 2023-04-07 暗算問題　For w=2, y=5, z=7, evaluate z^2-y^3+5w^2+9z
  98. 2023-04-08 因数定理 組み立て除法　Find the largest zero of y^3-7y^2-10y+16
  99. 2023-04-09 ２次関数の定積分　Find the area between the x-axis and y=3x/2-x^2/4
  100. 2023-04-10 交代数列　1-2+3-4+….-14+15-16+17-18+19
  101. 2023-04-11 暗算 連立１次方程式 平均速度　A boat travels 18 miles upstream in 2 hours and travels 39 miles downstream in 3 hours
  102. 2023-04-12 暗算 ２次方程式　Find the distance between the roots of y^2-6y-27
  103. 2023-04-13 置換積分　∫{0→√π/2}26y*cos(y^2)dy
  104. 2023-04-14 完全平方数 素因数分解　What is the smallest positive integer n such that 126n is a perfect square?
  105. 2023-04-15 中心角 弦の長さ　Find the length of the chord of a cycle with radius 5√3
  106. 2023-04-16 グラフのラベリング　How many differently labeld trees can one make on 4 vertices?
  107. 2023-04-17 マクローリン展開　f(z)=e^z-1 if z≠0, 1 if z=0 if g(z) is the sixteenth differential of f(z), what is 1/g(0)?
  108. 2023-04-18 ２変数１次方程式の整数解　non-negative solutions (y, z) to 7y+8z=1000
  109. 2023-04-19 和算数能極形術　Find the area of the circle inscribed in a semicircle
  110. 2023-04-20 等差数列・等比数列の和の公式 等比級数の部分和の積分 メルカトール級数とライプニッツ級数　5π/{Σ{k=0→∞}(-1)^k/(2k+1)}
  111. 2023-04-21 等差数列の和 平方数は２つの三角数の和　6^2-Σ{n=1→5}n
  112. 2023-04-22 オイラーの多面体定理 包絡線図との関係　What is the maximum number of regions formed by 6 lines in a plane?
  113. 2023-04-23 暗算問題　11/.3-37/6-15/2
  114. 2023-04-24 約数の個数の公式　How many factors does 78645 have?
  115. 2023-04-25 ２倍角の公式 置換積分　∫{0→200π}{1/π}sin^2(t)cos^2(t)dt
  116. 2023-04-26 平方の和 剰余ゼロ条件　x^2+y^2≡0 mod 0 where 0<x<20 and 10<y<100
  117. 2023-04-27 余角　Two complementary angles have measure x+3 and 2x+6
  118. 2023-04-28 約数条件付き２変数２次方程式 x^2+y^2/(x-y) is an integer that divides 1995 and x+y > 4
  119. 2023-04-29 和と差の積　Π(1-1/x^2)=15/29
  120. 2023-04-30 階乗　(3!)!/4!

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