Found problems: 85335
2017 Polish MO Finals, 6
Three sequences $(a_0, a_1, \ldots, a_n)$, $(b_0, b_1, \ldots, b_{n})$, $(c_0, c_1, \ldots, c_{2n})$ of non-negative real numbers are given such that for all $0\leq i,j\leq n$ we have $a_ib_j\leq (c_{i+j})^2$. Prove that
$$\sum_{i=0}^n a_i\cdot\sum_{j=0}^n b_j\leq \left( \sum_{k=0}^{2n} c_k\right)^2.$$
2009 Brazil Team Selection Test, 2
Be $x_1, x_2, x_3, x_4, x_5$ be positive reais with $x_1x_2x_3x_4x_5=1$. Prove that
$$\frac{x_1+x_1x_2x_3}{1+x_1x_2+x_1x_2x_3x_4}+\frac{x_2+x_2x_3x_4}{1+x_2x_3+x_2x_3x_4x_5}+\frac{x_3+x_3x_4x_5}{1+x_3x_4+x_3x_4x_5x_1}+\frac{x_4+x_4x_5x_1}{1+x_4x_5+x_4x_5x_1x_2}+\frac{x_5+x_5x_1x_2}{1+x_5x_1+x_5x_1x_2x_3} \ge \frac{10}{3}$$
2009 Korea - Final Round, 1
$a,b,c$ are the length of three sides of a triangle. Let $A= \frac{a^2 +bc}{b+c}+\frac{b^2 +ca}{c+a}+\frac{c^2 +ab}{a+b}$, $B=\frac{1}{\sqrt{(a+b-c)(b+c-a)}}+\frac{1}{\sqrt{(b+c-a)(c+a-b)}}$$+\frac{1}{\sqrt{(c+a-b)(a+b-c)}}$. Prove that $AB \ge 9$.
2020 BMT Fall, 2
There are $38$ people in the California Baseball League (CBL). The CBL cannot start playing games until people are split into teams of exactly $9$ people (with each person in exactly one team). Moreover, there must be an even number of teams. What is the fewest number of people who must join the CBL such that the CBL can start playing games? The CBL may not revoke membership of the $38$ people already in the CBL.
2013 Tournament of Towns, 2
On the sides of triangle $ABC$, three similar triangles are constructed with triangle $YBA$ and triangle $ZAC$ in the exterior and triangle $XBC$ in the interior. (Above, the vertices of the triangles are ordered so that the similarities take vertices to corresponding vertices, for example, the similarity between triangle $YBA$ and triangle $ZAC$ takes $Y$ to $Z, B$ to $A$ and $A$ to $C$). Prove that $AYXZ$ is a parallelogram
2008 Singapore Team Selection Test, 3
Fifty teams participate in a round robin competition over 50 days. Moreover, all the teams (at least two) that show up in any day must play against each other. Prove that on every pair of consecutive days, there is a team that has to play on those two days.
2013 India Regional Mathematical Olympiad, 6
Let $P(x)=x^3+ax^2+b$ and $Q(x)=x^3+bx+a$, where $a$ and $b$ are nonzero real numbers. Suppose that the roots of the equation $P(x)=0$ are the reciprocals of the roots of the equation $Q(x)=0$. Prove that $a$ and $b$ are integers. Find the greatest common divisor of $P(2013!+1)$ and $Q(2013!+1)$.
2007 Stanford Mathematics Tournament, 8
Tina writes four letters to her friends Silas, Jessica, Katie, and Lekan. She prepares an envelope for Silas, an envelope for Jessica, an envelope for Katie, and an envelope for Lekan. However, she puts each letter into a random envelope. What is the probability that no one receives the letter they are supposed to receive?
1994 USAMO, 5
Let $\, |U|, \, \sigma(U) \,$ and $\, \pi(U) \,$ denote the number of elements, the sum, and the product, respectively, of a finite set $\, U \,$ of positive integers. (If $\, U \,$ is the empty set, $\, |U| = 0, \, \sigma(U) = 0, \, \pi(U) = 1$.) Let $\, S \,$ be a finite set of positive integers. As usual, let $\, \binom{n}{k} \,$ denote $\, n! \over k! \, (n-k)!$. Prove that \[ \sum_{U \subseteq S} (-1)^{|U|} \binom{m - \sigma(U)}{|S|} = \pi(S) \] for all integers $\, m \geq \sigma(S)$.
2023 Canada National Olympiad, 1
William is thinking of an integer between 1 and 50, inclusive. Victor can choose a positive integer $m$ and ask William: "does $m$ divide your number?", to which William must answer truthfully. Victor continues asking these questions until he determines William's number. What is the minimum number of questions that Victor needs to guarantee this?
2008 District Olympiad, 1
Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a countinuous function such that
$$ \int_0^1 f(x)dx=\int_0^1 xf(x)dx. $$
Show that there is a $ c\in (0,1) $ such that $ f(c)=\int_0^c f(x)dx. $
1998 Chile National Olympiad, 2
Given a semicircle of diameter $ AB $, with $ AB = 2r $, be $ CD $ a variable string, but of fixed length $ c $. Let $ E $ be the intersection point of lines $ AC $ and $ BD $, and let $ F $ be the intersection point of lines $ AD $ and $ BC $.
a) Prove that the lines $ EF $ and $ AB $ are perpendicular.
b) Determine the locus of the point $ E $.
c) Prove that $ EF $ has a constant measure, and determine it based on $ c $ and $ r $.
2011 Today's Calculation Of Integral, 681
Evaluate $\int_0^{\frac{\pi}{2}} \sqrt{1-2\sin 2x+3\cos ^ 2 x}\ dx.$
[i]2011 University of Occupational and Environmental Health/Medicine　entrance exam[/i]
2006 Turkey Team Selection Test, 2
How many ways are there to divide a $2\times n$ rectangle into rectangles having integral sides, where $n$ is a positive integer?
1965 AMC 12/AHSME, 27
When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \minus{} 1$ the quotient is $ f(y)$ and the remainder is $ R_1$. When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \plus{} 1$ the quotient is $ g(y)$ and the remainder is $ R_2$. If $ R_1 \equal{} R_2$ then $ m$ is:
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \minus{} 1 \qquad \textbf{(E)}\ \text{an undetermined constant}$
1988 Polish MO Finals, 3
Find the largest possible volume for a tetrahedron which lies inside a hemisphere of radius $1$.
2004 USA Team Selection Test, 2
Assume $n$ is a positive integer. Considers sequences $a_0, a_1, \ldots, a_n$ for which $a_i \in \{1, 2, \ldots , n\}$ for all $i$ and $a_n = a_0$.
(a) Suppose $n$ is odd. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i \pmod{n}$ for all $i = 1, 2, \ldots, n$.
(b) Suppose $n$ is an odd prime. Find the number of such sequences if $a_i - a_{i-1} \not \equiv i, 2i \pmod{n}$ for all $i = 1, 2, \ldots, n$.
1982 IMO, 3
Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$.
[b]a)[/b] Prove that for every such sequence there is an $n\ge1$ such that: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999. \]
[b]b)[/b] Find such a sequence such that for all $n$: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4. \]
2021 SYMO, Q6
Let $P(x)$ and $Q(x)$ be non-constant integer-coefficient polynomials such that for any integer $x\in \mathbb Z$, there exists integer $y\in \mathbb Z$ such that $P(x)=Q(y)$. Prove that the degree of $Q$ divides the degree of $P$.
2021 Purple Comet Problems, 12
Let $L_1$ and $L_2$ be perpendicular lines, and let $F$ be a point at a distance $18$ from line $L_1$ and a distance $25$ from line $L_2$. There are two distinct points, $P$ and $Q$, that are each equidistant from $F$, from line $L_1$, and from line $L_2$. Find the area of $\triangle{FPQ}$.
2006 AMC 12/AHSME, 9
Oscar buys 13 pencils and 3 erasers for $ \$1.00$. A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser?
$ \textbf{(A) } 10\qquad \textbf{(B) } 12\qquad \textbf{(C) } 15\qquad \textbf{(D) } 18\qquad \textbf{(E) } 20$
Kyiv City MO 1984-93 - geometry, 1984.8.3
Construct a right triangle given the lengths of segments of the medians $m_a,m_b$ corresponding on its legs.
2021 The Chinese Mathematics Competition, Problem 7
Let $f(x)$ be a bounded continuous function on $[0,+\infty)$. Prove that every solutions of the equation $y''+14y'+13y=f(x)$ are bounded continuous functions on $[0,+\infty)$
1961 Poland - Second Round, 3
Prove that for any angles $x,y,z$ holds the equality $$1-\cos^2x-\cos^2y- y-\cos^2z +2 \cos x \cos y \cos z= 4 \sin \frac{x+y+z}{2} \sin \frac{x+y-z}{2} \sin \frac{x-y+z}{2} \sin\frac{-x-y+z}{2}. $$
2017 ASDAN Math Tournament, 19
How many ways can you tile a $2\times5$ rectangle with $2\times1$ dominoes of $4$ different colors if no two dominoes of the same color may be adjacent?