Found problems: 85335
1999 Spain Mathematical Olympiad, 6
A plane is divided into $N$ regions by three families of parallel lines. No three lines pass through the same point. What is the smallest number of lines needed so that $N > 1999$?
1983 Bulgaria National Olympiad, Problem 2
Let $b_1\ge b_2\ge\ldots\ge b_n$ be nonnegative numbers, and $(a_1,a_2,\ldots,a_n)$ be an arbitrary permutation of these numbers. Prove that for every $t\ge0$,
$$(a_1a_2+t)(a_3a_4+t)\cdots(a_{2n-1}a_{2n}+t)\le(b_1b_2+t)(b_3b_4+t)\cdots(b_{2n-1}b_{2n}+t).$$
2004 Tournament Of Towns, 3
The perpendicular projection of a triangular pyramid on some plane has the largest possible area. Prove that this plane is parallel to either a face or two opposite edges of the pyramid.
2017 Indonesia MO, 5
A polynomial $P$ has integral coefficients, and it has at least 9 different integral roots. Let $n$ be an integer such that $|P(n)| < 2017$. Prove that $P(n) = 0$.
2000 Canada National Olympiad, 1
At 12:00 noon, Anne, Beth and Carmen begin running laps around a circular track of length $300$ meters, all starting from the same point on the track. Each jogger maintains a constant speed in one of the two possible directions for an indefinite period of time. Show that if Anne's speed is different from the other two speeds, then at some later time Anne will be at least $100$ meters from each of the other runners. (Here, distance is measured along the shorter of the two arcs separating two runners.)
2021 Brazil National Olympiad, 4
A set \(A\) of real numbers is framed when it is bounded and, for all \(a, b \in A\), not necessarily distinct, \((a-b)^{2} \in A\). What is the smallest real number that belongs to some framed set?
2008 AMC 8, 17
Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of $50$ units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles?
$\textbf{(A)}\ 76\qquad
\textbf{(B)}\ 120\qquad
\textbf{(C)}\ 128\qquad
\textbf{(D)}\ 132\qquad
\textbf{(E)}\ 136$
2015 Azerbaijan JBMO TST, 3
Acute-angled $\triangle{ABC}$ triangle with condition $AB<AC<BC$ has cimcumcircle $C^,$ with center $O$ and radius $R$.And $BD$ and $CE$ diametrs drawn.Circle with center $O$ and radius $R$ intersects $AC$ at $K$.And circle with center $A$ and radius $AD$ intersects $BA$ at $L$.Prove that $EK$ and $DL$ lines intersects at circle $C^,$.
2023 Indonesia TST, 3
Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.
2011 AMC 12/AHSME, 17
Circles with radii $1, 2$, and $3$ are mutually externally tangent. What is the area of the triangle determined by the points of tangency?
$ \textbf{(A)}\ \frac{3}{5} \qquad
\textbf{(B)}\ \frac{4}{5} \qquad
\textbf{(C)}\ 1 \qquad
\textbf{(D)}\ \frac{6}{5} \qquad
\textbf{(E)}\ \frac{4}{3}
$
2020 Iranian Our MO, 6
Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ and plynomials $P(x),Q(x),R(x)$ with positive real coefficients such that $Q(-1)=-1$ and for all positive reals $x,y$:$$f(\frac{x}{y}+R(y))=\frac{f(x)}{Q(y)}+P(y).$$
[i]Proposed by Alireza Danaie, Ali Mirazaie Anari[/i] [b]Rated 2[/b]
2015 Online Math Open Problems, 12
Let $a$, $b$, $c$ be the distinct roots of the polynomial $P(x) = x^3 - 10x^2 + x - 2015$.
The cubic polynomial $Q(x)$ is monic and has distinct roots $bc-a^2$, $ca-b^2$, $ab-c^2$.
What is the sum of the coefficients of $Q$?
[i]Proposed by Evan Chen[/i]
1963 AMC 12/AHSME, 35
The lengths of the sides of a triangle are integers, and its area is also an integer. One side is $21$ and the perimeter is $48$. The shortest side is:
$\textbf{(A)}\ 8 \qquad
\textbf{(B)}\ 10\qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 14 \qquad
\textbf{(E)}\ 16$
2007 Moldova Team Selection Test, 1
Show that the plane cannot be represented as the union of the inner regions of a finite number of parabolas.
1979 Bulgaria National Olympiad, Problem 6
The set $M=\{1,2,\ldots,2n\}~(n\ge2)$ is partitioned into $k$ nonintersecting subsets $M_1,M_2,\ldots,M_k$, where $k^3+1\le n$. Prove that there exist $k+1$ even numbers $2j_1,2j_2,\ldots,2j_{k+1}$ in $M$ that are in one and the same subset $M_j$ $(1\le j\le k)$ such that the numbers $2j_1-1,2j_2-1,\ldots,2j_{k+1}-1$ are also in one and the same subset $M_r$ $(1\le r\le k)$.
2024 All-Russian Olympiad, 4
Let $ABCD$ be a convex quadrilateral with $\angle A+\angle D=90^\circ$ and $E$ the point of intersection of its diagonals. The line $\ell$ cuts the segments $AB$, $CD$, $AE$ and $ED$ in points $X,Y,Z,T$, respectively. Suppose that $AZ=CE$ and $BE=DT$. Prove that the length of the segment $XY$ is not larger than the diameter of the the circumcircle of $ETZ$.
[i]Proposed by A. Kuznetsov, I. Frolov[/i]
1969 IMO Longlists, 56
Let $a$ and $b$ be two natural numbers that have an equal number $n$ of digits in their decimal expansions. The first $m$ digits (from left to right) of the numbers $a$ and $b$ are equal. Prove that if $m >\frac{n}{2},$ then $a^{\frac{1}{n}} -b^{\frac{1}{n}} <\frac{1}{n}$
2009 Tournament Of Towns, 6
On an in finite chessboard are placed $2009 \ n \times n$ cardboard pieces such that each of them covers exactly $n^2$ cells of the chessboard. Prove that the number of cells of the chessboard which are covered by odd numbers of cardboard pieces is at least $n^2.$
[i](9 points)[/i]
2011 USA TSTST, 7
Let $ABC$ be a triangle. Its excircles touch sides $BC, CA, AB$ at $D, E, F$, respectively. Prove that the perimeter of triangle $ABC$ is at most twice that of triangle $DEF$.
2016 PUMaC Geometry B, 5
Let $V$ be the volume of the octahedron $ABCDEF$ with $A$ and $F$ opposite, $B$ and $E$ opposite, and $C$ and $D$ opposite, such that $AB = AE = EF = BF = 13$, $BC = DE = BD = CE = 14$, and $CF = CA = AD = FD = 15$. If $V = a\sqrt{b}$ for positive integers $a$ and $b$, where $b$ is not divisible by the square of any prime, find $a + b$.
1996 May Olympiad, 3
Natalia and Marcela count $1$ by $1$ starting together at $1$, but Marcela's speed is triple that of Natalia (when Natalia says her second number, Marcela says the fourth number). When the difference of the numbers that they say in unison is any of the multiples of $ 29$, between $500$ and $600$, Natalia continues counting normally and Marcela begins to count downwards in such a way that, at one point, the two say in unison the same number. What is said number?
2009 Moldova Team Selection Test, 2
$ f(x)$ and $ g(x)$ are two polynomials with nonzero degrees and integer coefficients, such that $ g(x)$ is a divisor of $ f(x)$ and the polynomial $ f(x)\plus{}2009$ has $ 50$ integer roots. Prove that the degree of $ g(x)$ is at least $ 5$.
2021 AMC 10 Spring, 25
How many ways are there to place $3$ indistinguishable red chips, $3$ indistinguishable blue chips, and $3$ indistinguishable green chips in the squares of a $3 \times 3$ grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally?
$\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 30 \qquad\textbf{(E)}\ 36$
2010 Contests, 3
Circles $W_1,W_2$ meet at $D$and $P$. $A$ and $B$ are on $W_1,W_2$ respectively, such that $AB$ is tangent to $W_1$ and $W_2$. Suppose $D$ is closer than $P$ to the line $AB$. $AD$ meet circle $W_2$ for second time at $C$. Let $M$ be the midpoint of $BC$. Prove that $\angle{DPM}=\angle{BDC}$.
1966 Bulgaria National Olympiad, Problem 2
Prove that for every four positive numbers $a,b,c,d$ the following inequality is true:
$$\sqrt{\frac{a^2+b^2+c^2+d^2}4}\ge\sqrt[3]{\frac{abc+abd+acd+bcd}4}.$$