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Inequality Proof Problems [51-100]

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Restructured the IneqMath training data.

P51. Let $a, b, c \geq 0$ be non-negative real numbers such that $a + b + c = 1$. Consider the following inequality: \(7(ab + bc + ca) \quad () \quad 3 + 9abc.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

S51. (D) $<$

P52. Let $a, b, c \in (0,1)$. Consider the following inequality: \(\sqrt{a b c} + \sqrt{(1-a)(1-b)(1-c)} \quad () \quad 1.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S52. (D) $<$

P53. Let $a, b, c > 0$ be positive real numbers such that $a b c = 1$. Consider the following inequality: \(\frac{a b}{a^2+b^2+\sqrt{c}}+\frac{b c}{b^2+c^2+\sqrt{a}}+\frac{c a}{c^2+a^2+\sqrt{b}} \quad () \quad \frac{3}{2}.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S53. (D) $<$

P54. Let $a, b, c > 0$ be positive real numbers such that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \sqrt{abc}$. Consider the following inequality: \(abc \quad () \quad \sqrt{3(a+b+c)}.\) Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S54. (B) $\geq$

P55. Let $a, b, c > 0$ be positive real numbers such that $ab + bc + ca = 1$. Consider the following inequality: \(a b c\left(a+\sqrt{a^2+1}\right)\left(b+\sqrt{b^2+1}\right)\left(c+\sqrt{c^2+1}\right) \quad () \quad 2.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S55. (D) $<$

P56. Given $x_1, x_2, \ldots, x_n > 0$ such that $\sum_{i=1}^n x_i = 1$, consider the following inequality: \(\sum_{i=1}^n \frac{x_i+n}{1+x_i^2} \quad () \quad n^2 + n.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S56. (D) $<$

P57. Given $a, b, c \geq 0$, consider the following inequality: \(\sum_{cyc} \sqrt{\frac{a+b}{b^2+4bc+c^2}} \quad () \quad \frac{2}{\sqrt{a+2b+4c}}.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S57. (E) $>$

P58. Let $a, b, c > 0$ such that $a b c = 1$. Consider the following inequality: \(\frac{a^{10}}{b+c}+\frac{b^{10}}{c+a}+\frac{c^{10}}{a+b} \quad () \quad \frac{a^7}{b^7+c^7}+\frac{b^7}{c^7+a^7}+\frac{c^7}{a^7+b^7}.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S58. (B) $\geq$

P59. Let $x, y, z \in \mathbb{R}$ such that $x^2 + 2y^2 + z^2 = \frac{2}{5} a^2$, where $a > 0$. Consider the following expression: \(|x - y + z| \quad () \quad a.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S59. (A) $\leq$

P60. Let $x, y, z$ be positive real numbers. Consider the following inequality: \(\left(\frac{2x}{y}+3\right)\left(\frac{3y}{z}+1\right)\left(\frac{z}{x}+1\right) \quad () \quad 2+\frac{2(x+y+z)}{\sqrt[3]{x y z}}.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S60. (E) $>$

P61. Let $x, y, z > 0$ be positive real numbers such that $xy + yz + zx = 1$. Consider the following inequality: \(\frac{1}{x+x^3} + \frac{1}{y+y^3} + \frac{1}{z+z^3} \quad () \quad \frac{9 \sqrt{2}}{4}.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S61. (E) $>$

P62. Let $a, b, c \geq 0$ such that $a^4 + b^4 + c^4 \leq 2(a^2 b^2 + b^2 c^2 + c^2 a^2)$. Consider the following inequality: \(a^2 + b^2 + c^2 \quad () \quad 2(2ab + 3bc + ca).\) Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S62. (A) $\leq$

P63. Let $x, y, z \in \mathbb{R}^{+}$ such that $\sqrt{x} + \sqrt{y} + \sqrt{z} = 1$. Consider the following inequality: \((1-x)^2(1-y)^2(1-z)^2 \quad () \quad 2^{15} x y z (x+y)(y+z)(z+x).\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S63. (B) $\geq$

P64. Let $a, b, c > 0$ be positive real numbers such that \(\frac{1}{1+a} + \frac{1}{1+b} + \frac{1}{1+c} = 2.\)

Consider the following inequality: \(2(2ab + 3bc + ca) \quad () \quad \sqrt{ab} + \sqrt{bc} + \sqrt{ca}.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S64. (E) $>$

P65. Let $a, b, c, d \in \mathbb{R}$ such that $a^2 + b^2 + c^2 + d^2 = 4$. Consider the following inequality: \(\frac{1}{5-a} + \frac{1}{5-b} + \frac{1}{5-c} + \frac{1}{5-d} \quad () \quad \frac{5}{4}.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S65. (D) $<$

P66. Let $a, b, c \in \mathbb{R}^{+}$ be positive real numbers such that $abc = 1$. Consider the following expressions: \(a^3 + b^3 + 2c^3 \quad () \quad a^2 + b^2 + c^2.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S66. (E) $>$

P67. Let $a, b, c$ be positive numbers. Consider the following inequality: \(\frac{a^2}{b} + \frac{b^2}{c} + \frac{4c^2}{a} \quad () \quad -3a + b + 7c.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S67. (E) $>$

P68. Let $a, b, c$ be positive real numbers. Consider the following inequality: \(\sum_{cyc} \frac{b+c}{\sqrt{(a+b)(a+c)}} \quad () \quad 2 \sum_{cyc} \frac{2a}{2b+c}.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S68. (D) $<$

P69. Let $a_1, a_2, \ldots, a_n$ be positive real numbers such that $a_1 + a_2 + \ldots + a_n = 1$. Consider the following inequality: \(\left(a_1 a_2 + a_2 a_3 + \ldots + a_n a_1\right)\left(\frac{a_1}{a_2 + a_2^2} + \frac{a_2}{a_3 + a_3^2} + \ldots + \frac{a_n}{a_1 + a_1^2}\right) \quad () \quad \frac{n}{n+1}.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S69. (B) $\geq$

P70. Let $a, b, c$ be positive numbers such that $a^2 + b^2 + c^2 = 3$. Consider the following inequality: \(\frac{b+c}{a^2} + \frac{c+a}{b^2} + \frac{a+b}{c^2} \quad () \quad 3\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} - a b c\right).\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S70. (B) $\geq$

P71. Let $a, b, c$ be positive real numbers. Consider the following inequality: \(\frac{a b}{3 a+4 b+2 c}+\frac{b c}{3 b+4 c+2 a}+\frac{c a}{3 c+4 a+2 b} \quad () \quad \frac{a+b+c}{9}.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S71. (A) $\leq$

P72. Let $a, b, c$ be positive real numbers such that $abc = 1$. Consider the following inequality: \(\sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c} \quad () \quad \sqrt[3]{3(3 + a + b + c + ab + bc + ca)}.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S72. (A) $\leq$

P73. Let $a, b, c > 0$ be positive real numbers. Consider the following inequality: \(\frac{a^2}{a^2+bc} + \frac{b^2}{b^2+ca} + \frac{c^2}{c^2+ba} \quad () \quad \frac{(a+b+c)^2}{2(ab+bc+ca)}.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S73. (A) $\leq$

P74. Let $a, b, c, d$ be non-negative numbers. Consider the following inequality: \(a^4 +2 b^4 + 3c^4 + d^4 + 3abcd \quad () \quad \frac{1}{2} \sum_{cyc} a^3(b+c+d).\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S74. (B) $\geq$

P75. Let $a, b, c \geq 0$ be real numbers such that $a^2 + b^2 + c^2 = 3$. Consider the following inequality: \(\sum_{cyc} \frac{a}{a+2} \quad () \quad 1.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S75. (A) $\leq$

P76. Let $a_1, a_2, \ldots, a_n$ be real numbers such that $a_1 + a_2 + \ldots + a_n = 0$ and $\max {|a_i - a_j| \mid 1 \leq i, j \leq n} \leq 1$. Consider the following expression: \(a_1^2 + a_2^2 + \ldots + a_n^2 \quad () \quad \frac{1}{n}\left[\frac{n}{2}\right]\left[\frac{n+1}{2}\right]\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S76. (A) $\leq$

P77. Let $a, b, c > 0$ be positive real numbers such that $a + b + c = 1$. Consider the following inequality: \(5(a^2 + b^2 + c^2) \quad () \quad 6(a^3 + b^3 + c^3) + 1.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S77. (A) $\leq$

P78. Let $a, b, c, d$ be positive real numbers such that \(c^2 + d^2 = (a^2 + b^2)^3.\)

Consider the following expression: \(\frac{a^3}{c} + \frac{b^3}{d} \quad () \quad 1.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S78. (B) $\geq$

P79. Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. Consider the following inequality: \(a \sqrt[3]{1+b-c} + b \sqrt[3]{1+c-a} + c \sqrt[3]{1+a-b} \quad () \quad 1.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S79. (A) $\leq$

P80. Let $a, b, c \in \mathbb{R}^{+}$ be positive real numbers. Consider the following inequality: \(\frac{a}{\sqrt{a^2+b^2}} + \frac{b}{\sqrt{2b^2+c^2}} + \frac{c}{\sqrt{3c^2+a^2}} \quad () \quad \frac{3 \sqrt{2}}{2}.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S80. (D) $<$

P81. Let $x, y, z \in \mathbb{R}$. Consider the following expression: \(\frac{x^2-y^2}{2x^2+1} + \frac{y^2-z^2}{2y^2+1} + \frac{z^2-x^2}{2z^2+1} \quad () \quad 0.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S81. (A) $\leq$

P82. Let $a, b, c > 0$ be positive real numbers such that $a + b + c = 3$. Consider the following inequality: \(a^2 + 4b^2 + 4b + 8c \quad () \quad 7 + 2\sqrt{2(b^2 + c^2)} + 4\sqrt{bc}.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S82. (B) $\geq$

P83. In an acute triangle $ABC$, consider the following inequality: \(\frac{\cos A}{\cos B \cos C} + \frac{\cos B}{\cos C \cos A} + \frac{\cos C}{\cos A \cos B} \quad () \quad 3\left(\frac{1}{1+\cos A} + \frac{1}{1+\cos B} + \frac{1}{1+\cos C}\right).\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S83. (B) $\geq$

P84. Let $a, b > 0$ be positive real numbers such that $a + b \leq \frac{3}{2}$. Consider the following inequality: \(\frac{a}{2a^2+1} + \frac{4b}{3b^2+4} \quad () \quad \frac{7}{5}.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S84. (D) $<$

P85. Let $a, b \geq 0$ be non-negative real numbers such that $a + b = 1$. Consider the following inequality: \(a^5 + 16b^5 \quad () \quad \frac{1}{6}.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S85. (E) $>$

P86. Let $a, b > 0$ be positive real numbers such that $a + b + \sqrt{a^2 + b^2} \leq 10$. Consider the following inequality: \(\frac{1}{a} + \frac{2}{b} \quad () \quad 1.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S86. (B) $\geq$

P87. Let $a, b > 0$ be positive real numbers such that $a^2 + b^2 + 3 = 4ab$. Consider the following inequality: \(2\sqrt{2ab} - \frac{1}{2}a - b \quad () \quad \frac{1}{2}.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S87. (E) $>$

P88. Let $x, y \in \mathbb{R}$ such that $(x+1)(y+2) = 8$. Consider the following expression: \(|xy - 10| \quad () \quad 7.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S88. (E) $>$

P89. Let $a, b, c$ be positive real numbers such that $a b c = 1$. Consider the following inequality: \(\frac{1}{a} + \frac{2}{b} + \frac{1}{c} \quad () \quad 2.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S89. (E) $>$

P90. Let $a, b, c > 0$. Consider the following inequality: \(\sqrt{\frac{2 a b\left(a^2-a b+b^2\right)}{2a^4+b^4}} + \sqrt{\frac{2 b c\left(b^2-b c+c^2\right)}{b^4+3c^4}} + \sqrt{\frac{2 c a\left(c^2-c a+a^2\right)}{2c^4+a^4}} \quad () \quad \frac{2(\sqrt{a}+\sqrt{b}+\sqrt{c})^2}{a+b+c}.\)

Determine the correct inequality relation to fill in the blank.

Options:

(A) $\leq$

(B) $\geq$

(C) $=$

(D) $<$

(E) $>$

(F) None of the above

S90. (D) $<$

P91. Let $a, b, c, d > 0$ such that $a \leq b \leq c \leq d$ and $abcd = 1$. Determine the minimal constant $C$ such that the following inequality holds for all $a, b, c, d$:

P92. Let $a, b, c$ be positive real numbers. Find the largest positive constant $C$ such that the inequality below holds for all $a, b, c \in \mathbb{R}^{+}$: \(\frac{a}{(b+c)^2}+\frac{b}{(c+a)^2}+\frac{c}{(a+b)^2} \geq \frac{C}{a+b+c}\)

P93. Let $a, b, c$ be real numbers such that $0 < a \leq b \leq c$. Find the largest constant $C$ such that the following inequality holds for all $a, b, c$ satisfying the given constraints: \((a+b)(c+a)^2 \geq C a b c\)

P94. Let $a, b, c > 0$ such that $a+b+c=1$. Determine the maximal constant $C$ such that the following inequality holds for all $a, b, c$:

P95. Let $a, b, c$ be the lengths of the sides of a triangle. Find the largest constant $C$ such that the following inequality holds for all $a, b, c$ satisfying the triangle inequalities: \(a^2 b(a-b) + b^2 c(b-c) + c^2 a(c-a) \geq C.\)

P96. Let $a, b, c \geq 0$ and $a+b+c=1$. Determine the minimal constant $C$ such that the following inequality holds for all $a, b, c$:

P97. Let $a$ and $b$ be non-negative real numbers such that $a \geq b$. Determine the largest constant $C$ such that the following inequality holds for all $a \geq b \geq 0$: \(a + \frac{1}{b(a-b)} \geq C.\)

P98. Let $0 < a < b$ and $x_i \in [a, b]$. Determine the largest constant $C$ such that the following inequality holds for all $x_i$:

P99. Let $a, b, c > 0$ satisfy the condition $\sum \frac{1}{a^2+1} = \frac{1}{2}$. Find the smallest constant $C$ such that the following inequality holds for all $a, b, c$: \(\sum \frac{1}{a^3+2} \leq C.\)

P100. Let $a, b, c$ be positive real numbers. Find the smallest constant $C$ such that the following inequality holds for all $a, b, c \in \mathbb{R}^{+}$: \(\frac{(2 a+b+c)^2}{2 a^2+(b+c)^2}+\frac{(2 b+a+c)^2}{2 b^2+(a+c)^2}+\frac{(2 c+a+b)^2}{2 c^2+(a+b)^2} \leq C.\)

Input: 2025.12.08 15:51