Found problems: 85335
2016 Junior Regional Olympiad - FBH, 5
Pages of some book are numerated with numbers $1$ to $100$. From the book several double pages were ripped out and sum of enumerations of that pages is equal to $4949$. How many double pages were ripped out?
2007 Mathematics for Its Sake, 2
For a given natural number $ n\ge 2, $ find all $ \text{n-tuples} $ of nonnegative real numbers which have the property that each one of the numbers forming the $ \text{n-tuple} $ is the square of the sum of the other $ n-1 $ ones.
[i]Mugur Acu[/i]
2024 BMT, 4
Two circles, $\omega_1$ and $\omega_2$, are internally tangent at $A.$ Let $B$ be the point on $\omega_2$ opposite of $A.$ The radius of $\omega_1$ is $4$ times the radius of $\omega_2.$ Point $P$ is chosen on the circumference of $\omega_1$ such that the ratio $\tfrac{AP}{BP}=\tfrac{2\sqrt{3}}{\sqrt{7}}.$ Let $O$ denote the center of $\omega_2.$ Determine $\tfrac{OP}{AO}.$
2024 Princeton University Math Competition, 11
Austen has a regular icosahedron ($20$-sided polyhedron with all triangular faces). He randomly chooses $3$ distinct points among the vertices and constructs the circle through these three points. The expected value of the total number of the icosahedron’s vertices that lie on this circle can be written as $m/n$ for relatively prime positive integers $m$ and $n.$ Find $m + n.$
Today's calculation of integrals, 858
On the plane $S$ in a space, given are unit circle $C$ with radius 1 and the line $L$. Find the volume of the solid bounded by the curved surface formed by the point $P$ satifying the following condition $(a),\ (b)$.
$(a)$ The point of intersection $Q$ of the line passing through $P$ and perpendicular to $S$ are on the perimeter or the inside of $C$.
$(b)$ If $A,\ B$ are the points of intersection of the line passing through $Q$ and pararell to $L$, then $\overline{PQ}=\overline{AQ}\cdot \overline{BQ}$.
2002 Denmark MO - Mohr Contest, 1
An interior point in a rectangle is connected by line segments to the midpoints of its four sides. Thus four domains (polygons) with the areas $a, b, c$ and $d$ appear (see the figure). Prove that $a + c = b + d$.
[img]https://1.bp.blogspot.com/-BipDNHELjJI/XzcCa68P3HI/AAAAAAAAMXY/H2Iqya9VItMLXrRqsdyxHLTXCAZ02nEtgCLcBGAsYHQ/s0/2002%2BMohr%2Bp1.png[/img]
2023-IMOC, C3
Graph $G$ has $n\geq 2$ vertices. Find the largest $m$ such that one of the following is true for always:
1. There exists a cycle with $k\geq m$ vertices.
2. There exists an independent set with $m$ vertices.
2017 F = ma, 12
A small hard solid sphere of mass m and negligible radius is connected to a thin rod of length L and mass 2m. A second small hard solid sphere, of mass M and negligible radius, is fired perpendicularly at the rod at a distance h above the sphere attached to the rod, and sticks to it.
In order for the rod not to rotate after the collision, the second sphere should have a mass M given by which of the following?
$\textbf{(A)} M = m\qquad
\textbf{(B)} M = 1.5m\qquad
\textbf{(C)} M = 2m\qquad
\textbf{(D)} M = 3m\qquad
\textbf{(E)}\text{Any mass M will work}$
2015 Cuba MO, 4
Let $A$ and $B$ be two subsets of $\{1, 2, 3, 4, ..., 100\}$, such that $|A| = |B|$ and $A\cap B =\emptyset$. If $n \in A$ implies that $2n + 2 \in B$, determine the largest possible value of $ |A \cup B|$.
2021 Oral Moscow Geometry Olympiad, 4
On the diagonal $AC$ of cyclic quadrilateral $ABCD$ a point $E$ is chosen such that $\angle ABE = \angle CBD$. Points $O,O_1,O_2$ are the circumcircles of triangles $ABC, ABE$ and $CBE$ respectively. Prove that lines $DO,AO_{1}$ and $CO_{2}$ are concurrent.
2012 AMC 10, 17
Let $a$ and $b$ be relatively prime integers with $a>b>0$ and $\tfrac{a^3-b^3}{(a-b)^3}=\tfrac{73}{3}$. What is $a-b$?
$ \textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 5
$
1987 Flanders Math Olympiad, 1
A rectangle $ABCD$ is given. On the side $AB$, $n$ different points are chosen strictly between $A$ and $B$. Similarly, $m$ different points are chosen on the side $AD$. Lines are drawn from the points parallel to the sides. How many rectangles are formed in this way? (One possibility is shown in the figure.)
[img]https://cdn.artofproblemsolving.com/attachments/0/1/dcf48e4ce318fdcb8c7088a34fac226e26e246.png[/img]
1952 AMC 12/AHSME, 48
Two cyclists, $ k$ miles apart, and starting at the same time, would be together in $ r$ hours if they traveled in the same direction, but would pass each other in $ t$ hours if they traveled in opposite directions. The ratio of the speed of the faster cyclist to that of the slower is:
$ \textbf{(A)}\ \frac {r \plus{} t}{r \minus{} t} \qquad\textbf{(B)}\ \frac {r}{r \minus{} t} \qquad\textbf{(C)}\ \frac {r \plus{} t}{r} \qquad\textbf{(D)}\ \frac {r}{t} \qquad\textbf{(E)}\ \frac {r \plus{} k}{t \minus{} k}$
2022 Malaysian IMO Team Selection Test, 1
Given an acute triangle $ABC$, mark $3$ points $X, Y, Z$ in the interior of the triangle. Let $X_1, X_2, X_3$ be the projections of $X$ to $BC, CA, AB$ respectively, and define the points $Y_i, Z_i$ similarly for $i=1, 2, 3$.
a) Suppose that $X_iY_i<X_iZ_i$ for all $i=1,2,3$, prove that $XY<XZ$.
b) Prove that this is not neccesarily true, if triangle $ABC$ is allowed to be obtuse.
[i]Proposed by Ivan Chan Kai Chin[/i]
2017 Harvard-MIT Mathematics Tournament, 1
[b]T[/b]wo ordered pairs $(a,b)$ and $(c,d)$, where $a,b,c,d$ are real numbers, form a basis of the coordinate plane if $ad \neq bc$. Determine the number of ordered quadruples $(a,b,c)$ of integers between $1$ and $3$ inclusive for which $(a,b)$ and $(c,d)$ form a basis for the coordinate plane.
1993 Tournament Of Towns, (380) 2
Vertices $A$, $B$ and $C$ of a triangle are connected with points $A'$ , $B'$ and $C'$ lying in the opposite sides of the triangle (not at vertices). Can the midpoints of the segments $AA'$, $BB'$ and $CC'$ lie in a straight line?
(Folklore)
Russian TST 2019, P1
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$
2018 Harvard-MIT Mathematics Tournament, 9
How many ordered sequences of $36$ digits have the property that summing the digits to get a number and taking the last digit of the sum results in a digit which is not in our original sequence? (Digits range from $0$ to $9$.)
1990 Poland - Second Round, 6
For any convex polygon $ W $ with area 1, let us denote by $ f(W) $ the area of the convex polygon whose vertices are the centers of all sides of the polygon $ W $. For each natural number $ n \geq 3 $, determine the lower limit and the upper limit of the set of numbers $ f(W) $ when $ W $ runs through the set of all $ n $ convex angles with area 1.
1990 India Regional Mathematical Olympiad, 2
For all positive real numbers $ a,b,c$, prove that
\[ \frac {a}{b \plus{} c} \plus{} \frac {b}{c \plus{} a} \plus{} \frac {c}{a \plus{} b} \geq \frac {3}{2}.\]
1987 AMC 8, 15
The sale ad read: "Buy three tires at the regular price and get the fourth tire for $\$3$." Sam paid $\$240$ for a set of four tires at the sale. What was the regular price of one tire?
$\text{(A)}\ 59.25\text{ dollars} \qquad \text{(B)}\ 60\text{ dollars} \qquad \text{(C)}\ 70\text{ dollars} \qquad \text{(D)}\ 79\text{ dollars} \qquad \text{(E)}\ 80\text{ dollars}$
2009 Portugal MO, 3
Duarte wants to draw a square whose side's length is $2009$ cm and which is divided in $2009\times2009$ squares whose side's length is $1$ cm and whose sides are parallel to the original square's one, without taking the pencil out of the paper. Starting on one of the vertex of the giant square, what is the length of the shortest line that allows him to make this drawing?
2021 Saudi Arabia Training Tests, 16
Let $ABC$ be an acute, non-isosceles triangle with circumcenter $O$, incenter $I$ and $(I)$ tangent to $BC$, $CA$, $AB$ at $D, E, F$ respectively. Suppose that $EF$ cuts $(O)$ at $P, Q$. Prove that $(PQD)$ bisects segment $BC$.
2018 Kazakhstan National Olympiad, 2
The natural number $m\geq 2$ is given.Sequence of natural numbers $(b_0,b_1,\ldots,b_m)$ is called concave if $b_k+b_{k-2}\le2b_{k-1}$ for all $2\le k\le m.$ Prove that there exist not greater than $2^m$ concave sequences starting with $b_0 =1$ or $b_0 =2$
2015 Dutch Mathematical Olympiad, 1
We make groups of numbers. Each group consists of [i]fi ve[/i] distinct numbers. A number may occur in multiple groups. For any two groups, there are exactly four numbers that occur in both groups.
(a) Determine whether it is possible to make $2015$ groups.
(b) If all groups together must contain exactly [i]six [/i] distinct numbers, what is the greatest number of groups that you can make?
(c) If all groups together must contain exactly [i]seven [/i] distinct numbers, what is the greatest number of groups that you can make?