Inequality Proof Problems [101-150]
Recommended posting: 【Algebra】 Algebra Index
Restructured the IneqMath training data.
P101. Let $a, b, c > 0$ such that $abc = 8$. Determine the largest constant $C$ such that the following inequality holds for all $a, b, c$:
\[\frac{a^2}{\sqrt{\left(1+a^3\right)\left(1+b^3\right)}}+\frac{b^2}{\sqrt{\left(1+b^3\right)\left(1+c^3\right)}}+\frac{c^2}{\sqrt{\left(1+c^3\right)\left(1+a^3\right)}} \geq C.\]P102. Let $a_1, a_2, \ldots, a_n > 0$ such that $a_1 + a_2 + \ldots + a_n < 1$. Determine the minimal constant $C$ such that the following inequality holds for all $a_1, a_2, \ldots, a_n$:
\[\frac{a_1 \cdot a_2 \ldots a_n \left(1 - a_1 - a_2 - \ldots - a_n\right)}{\left(a_1 + a_2 + \ldots + a_n\right)\left(1-a_1\right)\left(1-a_2\right) \ldots \left(1-a_n\right)} \leq C\frac{3}{n^{n-1}}.\]P103. Let $a, b, c > 0$ such that $a+b+c=1$. Determine the maximal constant $C$ such that the following inequality holds for all $a, b, c$:
\[\frac{a^2+a b}{1-a^2}+\frac{b^2+b c}{1-b^2}+\frac{c^2+c a}{1-c^2} \geq C.\]P104. Let $a, b, c, x, y, z$ be positive real numbers such that $a+x \geq b+y \geq c+z$ and $a+b+c = x+y+z$. Find the largest constant $C$ such that the following inequality holds for all $a, b, c, x, y, z$ satisfying the given conditions: \(a y + b x \geq 3C(a c + x z).\)
P105. For arbitrary real numbers $a, b, c$, find the largest constant $C$ such that the following inequality holds for all $a, b, c \in \mathbb{R}$: \(\sqrt{a^2+(1-b)^2}+\sqrt{b^2+(1-c)^2}+\sqrt{c^2+(1-a)^2} \geq C.\)
P106. Let $x, y, z$ be positive real numbers. Find the largest constant $C$ such that the following inequality holds for all $x, y, z \in \mathbb{R}^{+}$: \(\left(x^2-y z\right)^2 \geq C\left(x z-y^2\right)\left(x y-z^2\right)\)
P107. Let $a, b, c$ be positive real numbers. Find the largest constant $C$ such that the following inequality holds for all $a, b, c \in \mathbb{R}^{+}$: \(\frac{a^3}{b^2}+\frac{b^3}{c^2}+\frac{c^3}{a^2} \geq C\left(\frac{a^2}{4b}+\frac{b^2}{4c}+\frac{c^2}{4a}\right)\)
P108. Let $a, b, c \in \mathbb{R}^{+}$ such that $a + b + c = 1$. Determine the minimal constant $C$ such that the following inequality holds for all $a, b, c$:
\[\frac{a}{a+bc} + \frac{b}{b+ac} + \frac{\sqrt{abc}}{c+ab} \leq C.\]P109. Let $m, n$ be natural numbers. Find the smallest constant $C$ such that for all real numbers $x$ and $y$, the following inequality holds: \(\sin^{2m} x \cdot \cos^{2n} y \leq C\frac{3m^n n^n}{(m+n)^{m+n}}.\)
P110. For $a, b, c > 0$, find the largest constant $C$ such that the following inequality holds for all $a, b, c$:
\[\left\{\left(a^2-a b+b^2\right)^2+\left(b^2-b c+c^2\right)^2+\left(c^2-c a+a^2\right)^2+3\right\}\left(\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c a}\right) \geq C\]P111. For $a, b > 0$, find the minimal constant $C$ such that the following inequality holds for all $a, b$: \(\frac{1}{2}(a+b)^2 + C(a+b) \geq a \sqrt{b} + b \sqrt{a}.\)
P112. Let $x, y, z > 0$ such that $x + y + z = xyz$. Find the smallest constant $C$ such that the following inequality holds for all $x, y, z$ satisfying the given constraint: \((x-1)(y-1)(z-1) \leq C\)
P113. Let $a, b, c > 0$. Find the smallest constant $C$ such that the following inequality holds for all positive $a, b, c$: \(4Cabc(a b+b c+c a) \geq\left(a^2+b^2+c^2\right)(a+b-c)(b+c-a)(c+a-b)\)
P114. For all $a, b, c > 0$ satisfying $a+b+c=2$, determine the largest constant $C$ such that the following inequality holds: \(\frac{a}{1-a} \cdot \frac{b}{1-b} \cdot \frac{c}{1-c} \geq C.\)
P115. Let $a, b, c \geq 0$ satisfy $ab + bc + ca = 1$. Determine the minimal constant $C$ such that the following inequality holds for all $a, b, c$:
\[\sum \frac{ab}{(a+b)(c^2+1)} \leq \frac{C}{a+b+c}.\]P116. Find the largest constant $C$ such that for all real numbers $x, y, z$, the following inequality holds: \(\frac{x^2-y^2}{2 x^2+1}+\frac{y^2-z^2}{2 y^2+1}+\frac{z^2-x^2}{2 z^2+1} \geq C\)
P117. Let $a_1, a_2, a_3, \ldots, a_n$ be $n$ real numbers all greater than 1, such that $\left|a_k-a_{k+1}\right| \leq 1$ for $1 \leq k \leq n-1$. Find the smallest constant $C$ such that the following inequality holds for all such sequences: \(\frac{a_1}{a_2}+\frac{a_2}{a_3}+\cdots+\frac{a_n}{a_1} \leq Cn-1.\)
P118. Let $a_1, a_2, \ldots, a_n$ be real numbers and $S$ be a non-empty subset of ${1,2, \ldots, n}$. Find the largest constant $C$ such that the following inequality holds for all $a_1, a_2, \ldots, a_n$ and $S$: \(2C \left(\sum_{i \in S} a_i\right)^2 \leq \sum_{1 \leq i \leq j \leq n}\left(a_i+\cdots+a_j\right)^2\)
P119. Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Determine the minimal constant $C$ such that the following inequality holds for all $a_1, a_2, \ldots, a_n$:
\[\left(a_1+\ldots+a_n\right)^2 \leq C\left(1^2 a_1^2+2^2 a_2^2+\ldots+n^2 a_n^2\right)\]P120. Let $x, y, z > 0$ such that $x+y+z=1$. Determine the minimal constant $C$ such that the following inequality holds for all $x, y, z$:
\[x y(y+4 z)+y z(z+4 x)+z x(x+4 y) \leq C.\]P121. Given $a, b, c > 0$ such that $ab + bc + ca = abc$, find the smallest constant $C$ such that the following inequality holds for all $a, b, c$: \(\frac{1}{a+3b+2c} + \frac{1}{b+3c+2a} + \frac{1}{c+3a+2b} \leq C.\)
P122. Let $ABCD$ be a cyclic quadrilateral. Determine the smallest constant $C$ such that the following inequality holds if and only if $AB \cdot BC = 2AD \cdot DC$: \(BD^2 \leq C \cdot AC^2\)
P123. Let $n$ be a natural number such that $n \geq 2$. Find the largest constant $C$ such that the following inequality holds for all $n$: \(\frac{1}{n+1}\left(1+\frac{1}{3}+\cdots+\frac{1}{2n-1}\right) \geq C \cdot \frac{1}{n}\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right).\)
P124. Let $a, b, c$ be positive real numbers. Determine the minimal constant $C$ such that the following inequality holds for all $a, b, c \in \mathbb{R}^{+}$: \(a+b+c+\sqrt{a^2+b^2+c^2} \leq C \cdot \left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \cdot \left(a^2+b^2+c^2\right)\)
P125. Let $a, b, c$ be the side lengths of a triangle. Determine the largest constant $C$ such that the following inequality holds for all $a, b, c$ satisfying the triangle inequality: \(\frac{b+c}{a}+\frac{c+a}{b}+\frac{a+b}{c}+\frac{(b+c-a)(c+a-b)(a+b-c)}{a b c} \geq C.\)
P126. Let $P$ be a point inside a triangle $ABC$, and let $d_a, d_b, d_c$ be the distances from point $P$ to the triangle’s sides. Determine the largest constant $C$ such that the following inequality holds for all points $P$ inside the triangle:
\[d_a \cdot h_a^2 + d_b \cdot h_b^2 + d_c \cdot h_c^2 \geq C \cdot (d_a + d_b + d_c)^3\]where $h_a, h_b, h_c$ are the altitudes of the triangle.
P127. Let $a, b, c$ be distinct real numbers. Find the largest constant $C$ such that the following inequality holds for all distinct $a, b, c \in \mathbb{R}$:
\[\begin{aligned} & \left|\frac{b-c}{c-a}\right|(|a-b|+|b-c|)\left|+\left|\frac{c-a}{a-b}\right|(|b-c|+|c-a|)\right|+\left|\frac{a-b}{b-c}\right|(|c-a|+|a-b|) \\ & +|a-b\|b-c\| c-a|+1 \geq C \sqrt{|a-b\|b-c\| c-a|} \end{aligned}\]P128. Let $a, b, c > 0$ such that $a + b + c = abc$. Find the largest constant $C$ such that the following inequality holds for all $a, b, c$ satisfying the given constraint: \(\frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2} \geq 2C \sqrt{a^2 + b^2 + c^2}\)
P129. Find the largest constant $C$ such that for all real numbers $a, b, c$, the following inequality holds: \(a(b-c)^4+b(a-c)^4+c(a-b)^4 \geq C\)
P130. Let $x, y, z \in \mathbb{R}$ satisfy the following constraints: \(\left\{\begin{array}{r} x^2+x y+y^2=16 ; \\ y^2+y z+z^2=3 ; \end{array}\right.\)
Find the smallest constant $C$ such that the following inequality holds for all $x, y, z$ satisfying the constraints: \(x y+y z+z x \leq C.\)
P131. Let $x, y, z$ be three non-negative reals such that $xyz = x + y + z$. Find the largest constant $C$ such that the following inequality holds for all $x, y, z$ satisfying the given constraint: \(xy + yz + zx \geq C + \sqrt{x^2+1} + \sqrt{y^2+1} + \sqrt{z^2+1}\)
P132. Let $\left(a_n\right)_{n \geq 0}$ be a sequence of real numbers such that for all $n \geq 0$,
\[a_{n+1} \geq a_n^2 + \frac{1}{5}\]Determine the maximal constant $C$ such that the following inequality holds for all $n \geq 5$:
\[\sqrt{a_{n+5}} \geq a_{n-5}^2 + C.\]P133. Let $x, y, z$ be real numbers such that $0 \leq x, y, z \leq 1$. Find the smallest constant $C$ such that the following inequality holds for all $x, y, z$ in the given domain: \(\frac{x}{y+z+1}+\frac{y}{z+x+1}+\frac{z}{x+y+1} \leq 3C -(1-x)(1-y)(1-z).\)
P134. Let $a, b, c$ be real numbers such that $a+b+c=1$. Find the smallest constant $C$ such that the following inequality holds for all $a, b, c$ satisfying the given constraint: \(\frac{11 a-2}{1+a^2}+\frac{11 b-2}{1+b^2}+\frac{11 c-2}{1+c^2} \leq C.\)
P135. Given positive reals $a, b, c$, find the largest constant $C$ such that the following inequality holds for all $a, b, c \in \mathbb{R}^{+}$: \(\frac{(a+b+c)^2}{ab+bc+ca} \geq C \cdot \frac{(a+b+c)^2}{ab+bc+ca} \cdot \left(\frac{b}{a+b}+\frac{c}{b+c}+\frac{a}{c+a}\right) + 1.\)
P136. Let $a, b, c$ be positive real numbers greater than 1. Find the smallest constant $C$ such that the following inequality holds for all $a, b, c > 1$: \(\sqrt{a-1} + \sqrt{b-1} + \sqrt{c-1} \leq 3C \sqrt{c(ab+1)}\)
P137. Let $a, b, c, d, e, f \in \mathbb{R}$ with $ab+bc+cd+de+ef=1$. Find the largest constant $C$ such that the following inequality holds for all $a, b, c, d, e, f$ satisfying the given constraint: \(a^2+b^2+c^2+d^2+e^2+f^2 \geq C.\)
P138. Let $a, b, c$ be positive numbers. Determine the largest constant $C$ such that the following inequality holds for all $a, b, c \in \mathbb{R}^{+}$: \(\frac{a b c(a b+a c+b c)^2}{\left(a^2+b^2+c^2\right)^2} \geq 2C(a+b-c)(a+c-b)(b+c-a).\)
P139. Let $a, b, c \in \mathbb{R}_{+}$. Find the largest constant $C$ such that the following inequality holds for all positive $a, b, c$: \(\sum a \frac{b^2+c^2}{a^2+bc} \geq C \sum 3a\)
P140. Let $a, b, c \in \mathbb{R}$ such that $a + 2b + 3c = 2$ and $2ab + 3ac + 6bc = 1$. Determine the minimal constants $C$ such that the following inequalities hold for all $a$ satisfying the given constraints: \(a \leq C\)
P141. Find the smallest constant $C$ such that for all non-negative real numbers $a, b, c$, the following inequality holds: \(\sqrt{\frac{a b+b c+c a}{3}} \leq C \cdot \sqrt[3]{\frac{(a+b)(b+c)(c+a)}{2}}\)
P142. Let $a_1, a_2, \ldots, a_n$ be real numbers such that $a_1 + a_2 + \cdots + a_n \geq n$ and $a_1^2 + a_2^2 + \cdots + a_n^2 \geq n^2$. Find the largest constant $C$ such that the following inequality holds: \(\max(a_1, a_2, \ldots, a_n) \geq C\) for all $a_1, a_2, \ldots, a_n$ satisfying the given constraints.
P143. Let $x_1, x_2, \cdots, x_n$ be real numbers in the interval $[-1, 1]$ such that $\sum_{i=1}^{n} x_i^3 = 0$. Find the smallest constant $C$ such that the following inequality holds: \(\sum_{i=1}^{n} x_i \leq Cn.\)
P144. Let $a, b, c$ be the side lengths of a triangle. Find the smallest constant $C$ such that the following inequality holds for all $a, b, c$:
\[\left|\sqrt{\frac{a}{b}}-\sqrt{\frac{b}{a}}+\sqrt{\frac{b}{c}}-\sqrt{\frac{c}{b}}+\sqrt{\frac{c}{a}}-\sqrt{\frac{a}{c}}\right| \leq C.\]P145. Let $a, b, c > 0$ satisfy $a + b + c = 3$. Find the largest constant $C$ such that the following inequality holds for all $a, b, c$: \(\sqrt[3]{a^2 + ab + bc} + \sqrt[3]{b^2 + bc + ca} + \sqrt[3]{c^2 + ca + ab} \geq C(ab + bc + ca).\)
P146. For positive reals $x, y, z$, determine the largest constant $C$ such that the following inequality holds for all $x, y, z \in \mathbb{R}^{+}$:
\[\sum_{cyc} \frac{x(y+z)^2}{2x+y+z} + \frac{9}{2}\left(\frac{x+y+z}{xy+yz+zx}\right)^2 \geq C + \frac{9}{2(xy+yz+zx)}\]P147. Find the smallest constant $C$ such that for all real numbers $a, b, c$, the following inequality holds: \(a^4+b^4+c^4+2a+2b+2c+C \geq a^2+b^2+c^2+2a^2c+2b^2a+2c^2b\)
P148. Let $a, b, c$ be positive real numbers such that:
\[\left\{\begin{array}{l} a+b \leq c+1 \\ b+c \leq a+1 \\ c+a \leq b+1 \end{array}\right.\]Find the minimal constant $C$ such that the following inequality holds for all $a, b, c$ satisfying the given constraints:
\[a^2+b^2+c^2 \leq 2 (a b c + C).\]P149. Let $P$ be a point in $\triangle ABC$, and denote $\angle PAB = \alpha$, $\angle PBC = \beta$, $\angle PCA = \gamma$. Find the largest constant $C$ such that the following inequality holds for all points $P$ within the triangle: \(\cot \alpha + \cot \beta + \cot \gamma \geq C + \cot A + \cot B + \cot C\)
P150. Let $x_1, x_2, \ldots, x_n$ be positive numbers satisfying: \(\frac{1}{x_1+1998}+\frac{1}{x_2+1998}+\cdots+\frac{1}{x_n+1998}=\frac{1}{1998}.\) Determine the largest constant $C$ such that the following inequality holds for all $x_1, x_2, \ldots, x_n$: \(\frac{\sqrt[n]{x_1 x_2 \cdots x_n}}{n-1} \geq C.\)
Input: 2025.12.08 15:51