Olympiad problems

1986 National High School Mathematics League, 1

Tags: function , inequalities

Let $-1<a<0$, $\theta=\arcsin a$. Then the solution set to the inequality $\sin x<a$ is $\text{(A)}\{x|2n\pi+\theta<x<(2n+1)\pi-\theta,n\in\mathbb{Z}\}$ $\text{(B)}\{x|2n\pi-\theta<x<(2n+1)\pi+\theta,n\in\mathbb{Z}\}$ $\text{(C)}\{x|(2n-1)\pi+\theta<x<2n\pi-\theta,n\in\mathbb{Z}\}$ $\text{(D)}\{x|(2n-1)\pi-\theta<x<2n\pi+\theta,n\in\mathbb{Z}\}$

1970 AMC 12/AHSME, 17

Tags: inequalities

If $r\ge 0$, then for all $p$ and $q$ such that $pq\neq 0$ and $pr>qr$, we have $\textbf{(A) }-p>-q\qquad\textbf{(B) }-p>q\qquad\textbf{(C) }1>-q/p\qquad$ $\textbf{(D) }1<q/p\qquad \textbf{(E) }\text{None of These}$

2000 Canada National Olympiad, 5

Tags: inequalities

Suppose that the real numbers $a_1, a_2, \ldots, a_{100}$ satisfy \begin{eqnarray*} 0 \leq a_{100} \leq a_{99} \leq \cdots \leq a_2 &\leq& a_1 , \\ a_1+a_2 & \leq & 100 \\ a_3+a_4+\cdots+a_{100} &\leq & 100. \end{eqnarray*} Determine the maximum possible value of $a_1^2 + a_2^2 + \cdots + a_{100}^2$, and find all possible sequences $a_1, a_2, \ldots , a_{100}$ which achieve this maximum.

2003 China Team Selection Test, 3

Tags: polynomial , inequalities , algebra

The $ n$ roots of a complex coefficient polynomial $ f(z) \equal{} z^n \plus{} a_1z^{n \minus{} 1} \plus{} \cdots \plus{} a_{n \minus{} 1}z \plus{} a_n$ are $ z_1, z_2, \cdots, z_n$. If $ \sum_{k \equal{} 1}^n |a_k|^2 \leq 1$, then prove that $ \sum_{k \equal{} 1}^n |z_k|^2 \leq n$.

2013 USAJMO, 6

Tags: parameterization , inequalities

Find all real numbers $x,y,z\geq 1$ satisfying \[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]

1997 Iran MO (3rd Round), 2

Tags: inequalities , trigonometry , circumcircle , geometry

Show that for any arbitrary triangle $ABC$, we have \[\sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \leq \frac{abc}{(a+b)(b+c)(c+a)}.\]

2019 Romania National Olympiad, 1

Tags: inequalities

If $a,b,c>0$ then $$\frac{1}{abc}+1\ge3\left(\frac{1}{a^2+b^2+c^2}+\frac{1}{a+b+c}\right)$$

2005 SNSB Admission, 1

Tags: function , vectorial spaces , linear algebra , inequalities

[b]a)[/b] Let be three vectorial spaces $ E,F,G, $ where $ F $ has finite dimension, and $ E $ is a subspace of $ F. $ Prove that if the function $ T:F\longrightarrow G $ is linear, then $$ \dim TF -\dim TE\le \dim F-\dim E. $$ [b]b)[/b] Let $ A,B,C $ be matrices of real numbers. Prove that $$ \text{rang} (AB) +\text{rang} (BC) \le \text{rang} (ABC) +\text{rang} (B) . $$

2008 Switzerland - Final Round, 10

Tags: algebra , inequalities

Find all pairs$ (a, b)$ of positive real numbers with the following properties: (i) For all positive real numbers $x, y, z,w$ holds $x + y^2 + z^3 + w^6 \ge a (xyzw)^{b}$ . (ii) There is a quadruple $(x, y, z,w)$ of positive real numbers such that in equality (i) applies.

2022 Canadian Mathematical Olympiad Qualification, 3

Tags: algebra , inequalities

Consider n real numbers $x_0, x_1, . . . , x_{n-1}$ for an integer $n \ge 2$. Moreover, suppose that for any integer $i$, $x_{i+n} = x_i$ . Prove that $$\sum^{n-1}_{i=0} x_i(3x_i - 4x_{i+1} + x_{i+2}) \ge 0.$$

1973 Bulgaria National Olympiad, Problem 6

Tags: geometry , 3d geometry , tetrahedron , inequalities , geometric inequalities

In the tetrahedron $ABCD$, $E$ and $F$ are the midpoints of $BC$ and $AD$, $G$ is the midpoint of the segment $EF$. Construct a plane through $G$ intersecting the segments $AB$, $AC$, $AD$ in the points $M,N,P$ respectively in such a way that the sum of the volumes of the tetrahedrons $BMNP$, $CMNP$ and $DMNP$ to be minimal. [i]H. Lesov[/i]

2011 Today's Calculation Of Integral, 744

Tags: calculus , integration , inequalities , calculus computations

Let $a,\ b$ be real numbers. If $\int_0^3 (ax-b)^2dx\leq 3$ holds, then find the values of $a,\ b$ such that $\int_0^3 (x-3)(ax-b)dx$ is minimized.

2010 Contests, 3

Tags: inequalities

Let $p$ and $q$ be integers such that $q$ is nonzero. Prove that \[ \Bigl\lvert \frac{p}{q} - \sqrt{7} \Bigr\rvert \ge \frac{24 - 9\sqrt{7}}{q^2} \, . \]

1993 Greece National Olympiad, 4

Tags: symmetry , algebra , polynomial , linear algebra , matrix , inequalities

How many ordered four-tuples of integers $(a,b,c,d)$ with $0 < a < b < c < d < 500$ satisfy $a + d = b + c$ and $bc - ad = 93$?

1980 Vietnam National Olympiad, 2

Tags: function , inequalities

Let $m_1, m_2, \cdots ,m_k$ be positive numbers with the sum $S$. Prove that \[\displaystyle\sum_{i=1}^k\left(m_i +\frac{1}{m_i}\right)^2 \ge k\left(\frac{k}{S}+\frac{S}{k}\right)^2\]

2005 National High School Mathematics League, 2

Tags: inequalities

Positive numbers $a, b, c, x, y, z$ satisfy that $cy + bz = a$, $az + cx = b$, and $bx + ay = c$. Find the minimum value of the function $f(x,y,z) =\frac{x^2}{x+1}+\frac{y^2}{y+1}+\frac{z^2}{z+1}$.

1975 Polish MO Finals, 3

Tags: limit , inequalities

consider $0<u<1$. find $\alpha > 0$ minimum such that there exists $\beta > 0$ satisfying $(1+x)^u +(1-x)^u \leq 2 - \frac{x^\alpha}{\beta} \forall 0<x<1$

2017 Stars of Mathematics, 2

Tags: inequalities

Let $ x,y,z $ be three positive real numbers such that $ x^2+y^2+z^2+3=2(xy+yz+zx) . $ Show that $$ \sqrt{xy}+\sqrt{yz}+\sqrt{zx}\ge 3, $$ and determine in which circumstances equality happens. [i]Vlad Robu[/i]

2022 VJIMC, 1

Tags: calculus , integration , inequalities , function

Determine whether there exists a differentiable function $f:[0,1]\to\mathbb R$ such that $$f(0)=f(1)=1,\qquad|f'(x)|\le2\text{ for all }x\in[0,1]\qquad\text{and}\qquad\left|\int^1_0f(x)dx\right|\le\frac12.$$

1999 Romania Team Selection Test, 5

Tags: inequalities

Let $x_1,x_2,\ldots,x_n$ be distinct positive integers. Prove that \[ x_1^2+x_2^2 + \cdots + x_n^2 \geq \frac {2n+1}3 ( x_1+x_2+\cdots + x_n). \] [i]Laurentiu Panaitopol[/i]

2012 JBMO ShortLists, 5

Tags: inequalities

Find the largest positive integer $n$ for which the inequality \[ \frac{a+b+c}{abc+1}+\sqrt[n]{abc} \leq \frac{5}{2}\] holds true for all $a, b, c \in [0,1]$. Here we make the convention $\sqrt[1]{abc}=abc$.

2009 Jozsef Wildt International Math Competition, W. 13

Tags: inequalities

If $a_k >0$ [ $k=$1, 2, $\cdots$, $n$], then prove the following inequality $$\left (\sum \limits_{k=1}^n a_k^5 \right )^4 \geq \frac{1}{n} \left (\frac{2}{n-1} \right )^5 \left (\sum \limits_{1\leq i<j\leq n} a_i^2a_j^2 \right )^5$$

2003 China National Olympiad, 3

Tags: trigonometry , inequalities

Given a positive integer $n$, find the least $\lambda>0$ such that for any $x_1,\ldots x_n\in \left(0,\frac{\pi}{2}\right)$, the condition $\prod_{i=1}^{n}\tan x_i=2^{\frac{n}{2}}$ implies $\sum_{i=1}^{n}\cos x_i\le\lambda$. [i]Huang Yumin[/i]

1979 Vietnam National Olympiad, 1

Tags: algebra , inequalities

Show that for all $x > 1$ there is a triangle with sides, $x^4 + x^3 + 2x^2 + x + 1, 2x^3 + x^2 + 2x + 1, x^4 - 1.$

2018 Balkan MO Shortlist, A3

Tags: inequalities

Show that for every positive integer $n$ we have: $$\sum_{k=0}^{n}\left(\frac{2n+1-k}{k+1}\right)^k=\left(\frac{2n+1}{1}\right)^0+\left(\frac{2n}{2}\right)^1+...+\left(\frac{n+1}{n+1}\right)^n\leq 2^n$$ [i]Proposed by Dorlir Ahmeti, Albania[/i]