Inequality Proof Problems [01-50]
Recommended posting: 【Algebra】 Algebra Index
Restructured the IneqMath training data.
P1. Let $a, b, c > 0$. Determine the largest constant $C$ such that the following inequality holds for all $a, b, c$:
\[\frac{a+b+c}{\sqrt[3]{a b c}}+\frac{8 a b c}{(a+b)(b+c)(c+a)} \geq C.\]S1. $C = 4$
P2. For $a, b, c > 0$, find the largest constant $C$ such that the following inequality holds for all $a, b, c$: \(\frac{a^3}{a^3 + abc + b^3} + \frac{b^3}{b^3 + abc + c^3} + \frac{c^3}{c^3 + abc + a^3} \geq C.\)
S2. $C = 1$
P3. Find the smallest constant $C$ such that for all real numbers $x$ and $y$, the following inequality holds: \(x^2 + x + y^2 + y + C \geq x y\)
S3. $C = 1$
P4. Let $a, b, c \neq 0$ such that $a^2+b^2+c^2=2(ab+bc+ca)$. Find the largest constant $C$ such that the following inequality holds for all $a, b, c$ satisfying the given condition: \((a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \geq C\)
S4. $C = \frac{27}{2}$
P5. Let $a, b, c > 0$ such that $a + b + c = abc$. Determine the largest constant $C$ such that the following inequality holds for all $a, b, c$: \(\frac{a}{b^3} + \frac{b}{c^3} + \frac{c}{a^3} \geq C.\)
S5. $C = 1$
P6. Let $x, y, z$ be positive real numbers such that $x+y+z=1$. Determine the largest constant $C$ such that the following inequality holds for all $x, y, z$:
\[(x+\frac{1}{x})(y+\frac{1}{y})(z+\frac{1}{z}) \geq C.\]S6. $C = \frac{1000}{27}$
P7. Let $a, b, c > 0$. Determine the largest constant $C$ such that the following inequality holds for all $a, b, c$: \(\left(\frac{a}{b+c}\right)^2 + \frac{b}{c+a} + \left(\frac{c}{a+b}\right)^2 \geq C.\)
S7. $C = 1$
P8. Let $a, b, c > 0$. Determine the largest constant $C$ such that the following inequality holds for all positive $a, b, c$: \(\frac{c^2+a b}{a+b}+\frac{a^2+b c}{b+c}+\frac{b^2+c a}{c+a} \geq C(a+b+c).\)
S8. $C = 1$
P9. Let $a, b, c > 0$ such that $a+b+c=2$ and $a^2+b^2+c^2=2$. Determine the minimal constant $C$ such that the following inequality holds for all $a, b, c$ satisfying the given constraints: \(a b c \leq C.\)
S9. $C = \frac{4}{27}$
P10. Let $a, b, c$ be positive real numbers such that $a \geq b+c$. Determine the largest constant $C$ such that the following inequality holds for all $a, b, c$:
\[\frac{a^3+2 b^3+2 c^3}{a b c} \geq C.\]S10. $C = 6$
P11. Let $a, b, c > 0$ and $k \in \mathbb{N}^{+}$. Determine the largest constant $C$ such that the following inequality holds for all $a, b, c$:
\[\left(a^2+\frac{2(k+1)^2}{b+k}\right)\left(b^2+\frac{2(k+1)^2}{c+k}\right)\left(c^2+\frac{2(k+1)^2}{a+k}\right) \geq (Ck+3)^3.\]S11. $C = 2$
P12. Let $a, b, c > 0$. Find the largest constant $C$ such that the following inequality holds for all $a, b, c \in \mathbb{R}^{+}$: \(\frac{1}{b(a+b)}+\frac{1}{c(b+c)}+\frac{1}{a(c+a)} \geq \frac{C}{(a+b+c)^2}\)
S12. $C = \frac{27}{2}$
P13. Let $a_1, a_2, a_3, \ldots, a_n$ be real numbers, where $n > 1$. Find the largest constant $C$ such that the following inequality holds for all $a_1, a_2, \ldots, a_n$: \(\sqrt{a_1^2+\left(1-a_2\right)^2}+\sqrt{a_2^2+\left(1-a_3\right)^2}+\ldots +\sqrt{a_n^2+\left(1-a_1\right)^2} \geq Cn\)
S13. $C = \frac{1}{\sqrt{2}}$
P14. Let $x$ and $y$ be two positive real numbers such that $x + y = 1$. Determine the largest constant $C$ such that the following inequality holds for all $x, y$:
\[\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right) \geq C.\]S14. $C = 9$
P15. Let $a, b, c \geq 0$ such that $a^2+b^2+c^2+abc=4$. Find the smallest constant $C$ such that the following inequality holds for all $a, b, c$ satisfying the given constraint: \(a+b+c+\sqrt{\frac{a^2+b^2+c^2}{3}} \leq C\)
S15. $C = 4$
P16. In an acute triangle $ABC$, find the largest constant $C$ such that the following inequality holds for all angles $A, B, C$: \(\frac{\cos A}{\cos B \cos C}+\frac{\cos B}{\cos C \cos A}+\frac{\cos C}{\cos A \cos B} \geq C\left(\frac{1}{1+\cos A}+\frac{1}{1+\cos B}+\frac{1}{1+\cos C}\right)\)
S16. $C = 3$
P17. Let $a, b, c$ be three non-negative real numbers. Find the largest constant $C$ such that the following inequality holds for all $a, b, c \geq 0$: \(\left(a^2+1\right)\left(b^2+1\right)\left(c^2+1\right) \geq (ab+bc+ca-abc)^2 + C \cdot abc\)
S17. $C = 4$
P18. Let $a, b, c$ be nonnegative real numbers such that $ab+bc+ca=4$. Determine the largest constant $C$ such that the following inequality holds for all $a, b, c$:
\[\left(a^2+b^2+c^2+1\right)\left(\frac{1}{a^2+b^2}+\frac{1}{b^2+c^2}+\frac{1}{c^2+a^2}\right) \geq C.\]S18. $C = \frac{45}{8}$
P19. Let $a, b, c, m, n$ be positive real numbers. Find the largest constant $C$ such that the following inequality holds for all $a, b, c, m, n \in \mathbb{R}^{+}$: \(\frac{a^2}{b(m a+n b)}+\frac{b^2}{c(m b+n c)}+\frac{c^2}{a(m c+n a)} \geq \frac{C}{m+n}.\)
S19. $C = 3$
P20. Given positive real numbers $a, b, c$ satisfying $ab + bc + ca = 3$, find the largest constant $C$ such that the following inequality holds for all $a, b, c$: \(\frac{1}{\sqrt{a+b}} + \frac{1}{\sqrt{c+b}} + \frac{1}{\sqrt{a+c}} \geq \sqrt{\frac{C}{a+b+c+3}}.\)
S20. $C = 27$
P21. Given $a, b, c \geq 0$ satisfying $a b+b c+c a>0$, find the largest constant $C$ such that the following inequality holds for all $a, b, c$:
\[\sqrt{\frac{a}{b^2+b c+c^2}}+\sqrt{\frac{b}{c^2+c a+a^2}}+\sqrt{\frac{c}{a^2+a b+b^2}} \geq C \sqrt{\frac{a b+b c+c a}{(a+b)(b+c)(c+a)}}\]S21. $C = 2\sqrt{2}$
P22. Let $x, y, z \in [0,1]$. Determine the minimal constant $C$ such that the following inequality holds for all $x, y, z$: \(x(x-y)(z-x) \leq C.\)
S22. $C = \frac{4}{27}$
P23. Let $a, b \geq 0$ and $(a+b)(1+4ab)=2$. Find the smallest constant $C$ such that the following inequality holds for all $a, b$ satisfying the given constraint: \(ab(a+b+ab) \leq C\)
S23. $C = \frac{5}{16}$
P24. Let $a, b, c$ be positive real numbers such that $a+b+c=3$. Determine the maximal constant $C$ such that the following inequality holds for all $a, b, c$:
\[\frac{a}{1+2 b^3}+\frac{b}{1+2 c^3}+\frac{c}{1+2 a^3} \geq C.\]S24. $C = 1$
P25. Let $a, b > 0$ and $a + b \leq \frac{3}{2}$. Find the smallest constant $C$ such that the following inequality holds for all $a, b$ satisfying the given constraint: \(\frac{a}{a^2+1} + \frac{4b}{b^2+4} \leq C.\)
S25. $C = \frac{6}{5}$
P26. Let $a, b, c, d \in \mathbb{R}^{+}$. Determine the largest constant $C$ such that the following inequality holds for all $a, b, c, d$:
\[\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d} \geq \frac{C}{a+b+c+d}\]S26. $C = 64$
P27. Given non-negative real numbers $a, b, c$ satisfying $ab + bc + ca = 1$, find the largest constant $C$ such that the following inequality holds for all $a, b, c$:
\[\frac{c}{\sqrt{a+b+3c}} + \frac{a}{\sqrt{c+b+3a}} + \frac{b}{\sqrt{a+c+3b}} \geq C.\]S27. $C = 1$
P28. Let $x, y$ be positive real numbers such that $x + y = 2$. Determine the smallest constant $C$ such that the following inequality holds for all $x, y$:
\[x^3 y^3 (x^3 + y^3) \leq C.\]S28. $C = 2$
P29. Let $a, b, c, d \in \mathbb{R}$ such that $a^2 + b^2 + c^2 + d^2 = 4$. Find the minimal constant $C$ such that the following inequality holds for all $a, b, c, d$ satisfying the given constraint: \(a^3 + b^3 + c^3 + d^3 \leq C.\)
S29. $C = 8$
P30. Find the largest constant $C$ such that for all real numbers $a$ and $b$, the following inequality holds: \(\sqrt{a^2+b^2+\sqrt{2} a+\sqrt{2} b+1}+\sqrt{a^2+b^2-\sqrt{2} a-\sqrt{2} b+1} \geq C\)
S30. $C = 2$
P31. Let $a, b > 0$ such that $a + b = 1$. Determine the largest constant $C$ such that the following inequality holds for all $a, b$:
\[\left(a+\frac{1}{a}\right)^2+\left(b+\frac{1}{b}\right)^2 \geq C.\]S31. $C = \frac{25}{2}$
P32. Given four real numbers $a, b, c, d \geq 0$ satisfying $a \geq b + 7c + d$, find the largest constant $C$ such that the following inequality holds for all $a, b, c, d$:
\[7(a+b)(b+c)(c+d)(d+a) \geq C a b c d\]S32. $C = 240$
P33. Let $x, y \in \mathbb{R}$ such that $(x+1)(y+2) = 8$. Find the largest constant $C$ such that the following inequality holds for all $x, y$ satisfying the given constraint: \(|x y - 10| \geq C\)
S33. $C = 8$
P34. Let $a, b, c$ be positive real numbers such that $abc = 1$. Determine the largest constant $C$ such that the following inequality holds for all $a, b, c \in \mathbb{R}^{+}$: \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq C.\)
S34. $C = 3$
P35. Let $a, b, c > 0$. Find the largest constant $C$ such that the following inequality holds for all positive $a, b, c$: \(\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq \frac{C \sqrt{a^2+b^2+c^2}}{\sqrt[3]{a b c}}\)
S35. $C = \sqrt{3}$
P36. Let $x, y, z$ be positive real numbers such that $xy + yz + zx \geq 3$. Find the largest constant $C$ such that the following inequality holds for all $x, y, z$ satisfying the given constraint: \(\frac{x}{\sqrt{4x+5y}} + \frac{y}{\sqrt{4y+5z}} + \frac{z}{\sqrt{4z+5x}} \geq C\)
S36. $C = 1$
P37. Given the real number $b \leq 2$, find the largest constant $C$ such that for all positive real numbers $x, y$ where $xy = 1$, the following inequality holds: \(\frac{\sqrt{x^2+y^2}}{C}+\frac{b}{x+y} \geq 1+\frac{b}{2}\)
S37. $C = \sqrt{2}$
P38. Let $a, b, c > 0$. Find the smallest constant $C$ such that the following inequality holds for all positive $a, b, c$: \(\sqrt{\frac{2 a b\left(a^2-a b+b^2\right)}{a^4+b^4}}+\sqrt{\frac{2 b c\left(b^2-b c+c^2\right)}{b^4+c^4}}+\sqrt{\frac{2 c a\left(c^2-c a+a^2\right)}{c^4+a^4}} \leq C \frac{(\sqrt{a}+\sqrt{b}+\sqrt{c})^2}{a+b+c}\)
S38. $C = 1$
P39. Let $a, b, c \in \mathbb{R}^{+}$ with $abc = 1$. Determine the largest constant $C$ such that the following inequality holds for all $a, b, c$:
\[a^3 + b^3 + c^3 + \frac{16}{(b+c)(c+a)(a+b)} \geq C.\]S39. $C = 5$
P40. Let $a > 1$ be a real number. Determine the largest constant $C$ such that for all $n \in \mathbb{N}$, the following inequality holds: \(\frac{a^n-1}{n} \geq C\sqrt{a}^{n+1}-\sqrt{a}^{n-1}.\)
S40. $C = 1$
P41. Let $a, b, c \geq 0$ and $a+b+c=3$. Determine the largest constant $C$ such that the following inequality holds for all $a, b, c$:
\[\frac{a+1}{a+b+1}+\frac{b+1}{b+c+1}+\frac{c+1}{c+a+1} \geq C.\]S41. $C = 2$
P42. Let $a, b, c$ be positive real numbers. Find the smallest constant $C$ such that the following inequality holds for all $a, b, c \in \mathbb{R}^{+}$: \(\frac{a}{\sqrt{3 a+2 b+c}}+\frac{b}{\sqrt{3 b+2 c+a}}+\frac{c}{\sqrt{3 c+2 a+b}} \leq C \sqrt{a+b+c}\)
S42. $C = \frac{1}{\sqrt{2}}$
P43. Given $k > 1$ and $a, b, c > 0$ with $abc = k^3$, find the largest constant $C$ such that the following inequality holds for all $a, b, c$:
\[\sum_{\text{cyc}}\left(\frac{1}{1+a}+\frac{1}{1+bc}\right) \geq C\left(\frac{1}{1+k}+\frac{1}{1+k^2}\right).\]S43. $C = 3
P44. Let $a, b, c$ be real numbers such that $a^2 + bc = 1$. Find the smallest constant $C$ such that the following inequality holds for all $a, b, c$ satisfying the given constraint: \(\frac{1}{a^2+1} + \frac{1}{b^2+1} + \frac{1}{c^2+1} \leq C\)
S44. $C = \frac{5}{2}$
P45. Let $a, b, c \geq 0$ and $a^2 + b + c^2 = 1$. Determine the largest constant $C$ such that the following inequality holds for all $a, b, c$:
\[\frac{1}{1+a^2} + \frac{1}{3(1+b)} + \frac{1}{1+c^2} \geq C.\]S45. $C = \frac{5}{3}$
P46. Let $x > 4$. Determine the largest constant $C$ such that the following inequality holds for all $x$:
\[\frac{x^3}{x-4} \geq C.\]S46. $C = 108$
P47. Let $a, b, c \geq 1$. Determine the minimal constant $C$ such that the following inequality holds for all $a, b, c$:
\[\sqrt{\frac{a b+b c+c a}{3}}-\sqrt[3]{a b c} \geq \frac{\sqrt{\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c a}}}{C}-\frac{1}{\sqrt[3]{a b c}}.\]S47. $C = \sqrt{3}$
P48. Let $a, b, c \geq 0$ with $ab+bc+ca > 0$. Determine the largest constant $C$ such that the following inequality holds for all $a, b, c$:
\[\frac{a}{\sqrt{b+c+Ca}} + \frac{b}{\sqrt{a+c+Cb}} + \frac{c}{\sqrt{b+a+Cc}} \geq \sqrt{\frac{ab+bc+ca}{a+b+c}}\]S48. $C = 7$
P49. Let $x, y, z \in [-1, 3]$ such that $x + y + z = 3$. Find the smallest constant $C$ such that the following inequality holds for all $x, y, z$: \(x^2 + y^2 + z^2 \leq C.\)
S49. $C = 11$
P50. Let $p > q > 0$. Determine the largest constants $C$ such that the following inequality holds for all $x \in \mathbb{R}$: \(\frac{p+q}{p-q} \geq C\frac{2(x^2-2 q x+p^2)}{x^2+2 q x+p^2}\)
S50. $C = \frac{1}{2}$
Input: 2025.12.08 15:51