Inequality Proof Problems [51-100]
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P51. (IneqMath) 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) $<$
\[\begin{aligned} a^2 + b^2 + c^2 &\ge ab + bc + ca \quad \text{(by the Rearrangement Inequality)} \\[4pt] \therefore \quad (a+b+c)^2 &= 1 \\ &= a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \\ &\ge 3ab + 3bc + 3ca \\[4pt] \therefore \quad 7(ab+bc+ca) &< \frac{7}{3} < 3 \le 3 + 9abc \end{aligned}\]P52. (IneqMath) 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) $<$
Solved by UVW principle.
P53. (IneqMath) 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) $<$
Solved by UVW principle.
P54. (IneqMath) 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) $\geq$
Solved by UVW principle.
P55. (IneqMath) Let $a, b, c > 0$ be positive real numbers such that $ab + bc + ca = 1$. Consider the following inequality:
\[abc\left(a+\sqrt{a^2+1}\right)\left(b+\sqrt{b^2+1}\right)\left(c+\sqrt{c^2+1}\right) \quad () \quad 2.\]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) $<$
Solved by UVW princple.
P56. (IneqMath) 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) $<$
When $x_i + x_j = k$, it can be shown that $\frac{x_i+n}{1+x_i^2}+\frac{x_j+n}{1+x_j^2}$ is maximized when $x_i = x_j$. Therefore, for any $(x_1, x_2, \ldots, x_n)$, if we repeatedly select two elements at random and replace them by their average, then in the limit we can obtain the maximum value of the left-hand side, $n^2$, with equality attained when
\[x_i = x_j \quad \text{for all } i,j.\]P57. (IneqMath) 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) $>$
Solved by UVW principle.
P58. (IneqMath) 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) $\geq$
Solved by UVW princple.
P59. (IneqMath) 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. (IneqMath) 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. (IneqMath) 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) $>$
Solved by UVW princple.
P62. (IneqMath) 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. (IneqMath) 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) $\geq$
Solved by UVW principle.
P64. (IneqMath) 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) $>$
Solved by UVW princple.
P65. (IneqMath) 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. (IneqMath) 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) $>$
Solved by UVW principle.
P67. (IneqMath) 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. (IneqMath) 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. (IneqMath) 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. (IneqMath) 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) $\geq$
Solved by UVW princple.
P71. (IneqMath) 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) $\leq$
Solved by UVW principle.
P72. (IneqMath) 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) $\leq$
Solved by UVW principle.
P73. (IneqMath) 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) $\leq$
Solved by UVW principle.
P74. (IneqMath) 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. (IneqMath) 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) $\leq$
Solved by UVW principle.
| P76. (IneqMath) 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: |
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. (IneqMath) 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) $\leq$
Solved by UVW principle.
P78. (IneqMath) 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. (IneqMath) 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) $\leq$
Solved by UVW princple.
P80. (IneqMath) 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. (IneqMath) 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. (IneqMath) 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. (IneqMath) Let $ABCD$ be a cyclic quadrilateral. Determine the smallest constant $C$ such that the following inequality holds if and only if $AB \cdot BC = 2AD \cdot DC$:
\[BD^2 \leq C \cdot AC^2\]P84. (IneqMath) 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. (IneqMath) 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) $>$
By setting $b = 1-a$, the left-hand side becomes a function of a single variable, which makes the problem easier to solve.
P86. (IneqMath) 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. (IneqMath) 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) $>$
Solved by UVW princple.
P88. Will be updated later.
P89. (IneqMath) 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) $>$
Solved by UVW principle.
P90. (IneqMath) 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. (IneqMath) 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 $a, b, c, d$:
\[(a + 1)(d + 1)-\frac {3}{4d^3} \geq 2C.\]P92. (IneqMath) 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. (IneqMath) 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. (IneqMath) Let $a, b, c > 0$ such that $a+b+c=1$. Determine the maximal constant $C$ such that the following inequality holds for all $a, b, c$:
\[\frac{1+a+b}{2+c}+\frac{1+b+c}{2+a}+\frac{1+c+a}{2+b} \geq C.\]S94. $C = 15/7$
Solved by UVW princple.
P95. (IneqMath) 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. (IneqMath) Let $a, b, c \geq 0$ and $a+b+c=1$. Determine the minimal constant $C$ such that the following inequality holds for all $a, b, c$:
\[\frac{a}{\sqrt{b^2+3 c}}+\frac{b}{\sqrt{c^2+3 a}}+\frac{c}{\sqrt{a^2+3 b}} \geq \frac{1}{\sqrt{1+2C a b c}}.\]S96. $C = 3/2$
Solved by UVW principle.
P97. (IneqMath) 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. (IneqMath) Let $0 < a < b$ and $x_i \in [a, b]$. Determine the largest constant $C$ such that the following inequality holds for all $x_i$:
\[\left(x_1+x_2+\ldots+x_n\right)\left(\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n}\right) \leq \frac{n^2(a+b)^2}{Cab}.\]P99. (IneqMath) 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 $a, b, c$:
\[\sum \frac{1}{a^3+2} \leq C.\]S99. $C=1/3$
Solved by UVW principle.
P100. (IneqMath) 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