Found problems: 85335
We consider triangle $ABC$ and variables points $M$ on the half-line $BC$, $N$ on the half-line $CA$, and $P$ on the half-line $AB$, each start simultaneously from $B,C$ and respectively $A$, moving with constant speeds $ v_1, v_2, v_3 > 0 $, where $v_1$, $v_2$, and $v_3$ are expressed in the same unit of measure.
a) Given that there exist three distinct moments in which triangle $MNP$ is equilateral, prove that triangle $ABC$ is equilateral and that $v_1 = v_2 = v_3$.
b) Prove that if $v_1 = v_2 = v_3$ and there exists a moment in which triangle $MNP$ is equilateral, then triangle $ABC$ is also equilateral.
When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to $0$ and is the same as that of getting heads exactly twice. Let $\frac ij$, in lowest terms, be the probability that the coin comes up heads in exactly $3$ out of $5$ flips. Find $i+j$.
Circle $\omega$ is tangent to the interior of the circle $\Omega$ at the point C. Chord $AB$ of circle $\Omega$ is tangent to $\omega$. Chords $CF$ and $BG$ of circle $\Omega$ intersect at point $E$ lying on $\omega$. Prove that the circumcircle of triangle $CGE$ is tangent to straight line $AF$.
(I. Kukharchuk)
Initially, one cow is located on every negative integer on the number line. Each day, Farmer John chooses an integer $k$ where the interval $[k, k+5000]$ has cows. First, he moves each cow in $[k, k+5000]$ to another integer in the interval, so that no two cows move to the same integer. Then, he chooses a cow in the interval and removes it.
Can Farmer John get a cow on $100,000,000$ after some time?
We are given a certain number of identical sets of weights; each set consists of four different weights expressed by natural numbers (of weight units). Using these weights we are able to weigh out every integer mass up to $1985$ (inclusive). How many ways are there to compose such a set of weight sets given that the joint mass of all weights is the least possible?
Let $k$ and $N$ be integers such that $k > 1$ and $N > 2k + 1$. A number of $N$ persons sit around the Round Table, equally spaced. Each person is either a knight (always telling the truth) or a liar (who always lies). Each person sees the nearest k persons clockwise, and the nearest $k$ persons anticlockwise. Each person says: ''I see equally many knights to my left and to my right." Establish, in terms of $k$ and $N$, whether the persons around the Table are necessarily all knights.
Sergey Berlov, Russia
Suppose that $x$ and $y$ are positive reals such that \[x-y^2=3, \qquad x^2+y^4=13.\] Find $x$.
Albert rolls a fair six-sided die thirteen times. For each time he rolls a number that is strictly greater than the previous number he rolled, he gains a point, where his first roll does not gain him a point. Find the expected number of points that Albert receives.
[i]Proposed by Nathan Ramesh
For how many triples of positive integers $(a,b,c)$ with $1\leq a,b,c\leq 5$ is the quantity \[(a+b)(a+c)(b+c)\] not divisible by $4$?
Given the Fibonacci sequence with $f_0=f_1=1$and for $n\geq 1, f_{n+1}=f_n+f_{n-1}$, find all real solutions to the equation: $$x^{2024}=f_{2023}x+f_{2022}.$$
Problem 4. A chessboard with $2n \times 2n$ tiles is coloured such that every tile is coloured with one out of $n$ colours. Prove that there exists 2 tiles in either the same column or row such that if the colours of both tiles are swapped, then there exists a rectangle where all its four corner tiles have the same colour.
Let $x\ne 1,y\ne 1$ and $x\ne y.$ Show that if \[\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y},\] then \[\frac{yz-x^2}{1-x}=\frac{zx-y^2}{1-y}=x+y+z.\]
Let $P$ be a right prism whose two bases are equilateral triangles with side length 2. The height of $P$ is $2\sqrt{3}$. Let $l$ be the line connecting the centroids of the bases. Remove the solid, keeping only the bases. Rotate one of the bases $180^\circ$ about $l$. Let $T$ be the convex hull of the two current triangles. What is the volume of $T$?
Divide a cube into three equal pyramids.
A convex flat quadrilateral is given. Prove that for the ratio $q$ of the largest to the smallest of all distances, for any two vertices: $q \ge \sqrt2$.
[hide=original wording]Gegeben sei ein konvexes ebenes Viereck. Es ist zu beweisen, dass fur den Quotienten q aus dem großten und dem kleinsten aller Abstande zweier beliebiger Eckpunkte voneinander stets gilt: q >= \sqrt2.[/hide]
Let $a_1, a_2, \ldots, a_n$ be $n$ positive integers, and let $b_1, b_2, \ldots, b_m$ be $m$ positive integers such that $a_1 a_2 \cdots a_n = b_1 b_2 \cdots b_m$. Prove that a rectangular table with $n$ rows and $m$ columns can be filled with positive integer entries in such a way that
* the product of the entries in the $i$-th row is $a_i$ (for each $i \in \left\{1,2,\ldots,n\right\}$);
* the product of the entries in the $j$-th row is $b_j$ (for each $i \in \left\{1,2,\ldots,m\right\}$).
Fix $a_1, . . . , a_n \in (0, 1)$ and define $$f(I) = \prod_{i \in I} a_i \cdot \prod_{j \notin I} (1 - a_j)$$
for each $I \subseteq \{1, . . . , n\}$. Assuming that $$\sum_{I\subseteq \{1,...,n\}, |I| odd} {f(I)} = \frac12,$$
show that at least one $a_i$ has to be equal to $\frac12$.
(Paolo Leonetti)
Let $n$ be a positive integer.
Determine the number of sequences $a_0, a_1, \ldots, a_n$ with terms in the set $\{0,1,2,3\}$ such that $$n=a_0+2a_1+2^2a_2+\ldots+2^na_n.$$
For positive integers $m$ and $n$, the decimal representation for the fraction $\tfrac{m}{n}$ begins $0.711$ followed by other digits. Find the least possible value for $n$.
The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
$\textbf{(A) } 20
\qquad\textbf{(B) } \dfrac{360}{17}
\qquad\textbf{(C) } \dfrac{107}{5}
\qquad\textbf{(D) } \dfrac{43}{2}
\qquad\textbf{(E) } \dfrac{281}{13}
$
In a handbook of plants each plant is characterized by $100$ attributes (each attribute may either be present in a plant or not). Two plants are called [i]dissimilar [/i] if they differ by no less than $51$ attributes.
(a) Prove that the handbook cannot describe more than $50$ pair-wise dissimilar plants.
(b) Can it describe $50$ pairwise dissimilar plants?
(Dima Tereshin)
A right cylinder is given with a height of $20$ and a circular base of radius $5$. A vertical planar cut is made into this base of radius $5$. A vertical planar cut, perpendicular to the base, is made into this cylinder, splitting the cylinder into two pieces. Suppose the area the cut leaves behind on one of the pieces is $100\sqrt2$. Then the volume of the larger piece can be written as $a + b\pi$, where $a, b$ are positive integers. Find $a + b$.
Compute
$$\prod^{12}_{k=1} \left(\prod^{10}_{j=1} \left(e^{2\pi ji/11} - e^{2\pi ki/13}\right) \right) .$$ (The notation $\prod^{b}_{k=a}f(k)$means the product $f(a)f(a + 1)... f(b)$.)
How many three-digit numbers have at least one $2$ and at least one $3$?
$\textbf{(A) }52\qquad\textbf{(B) }54\qquad\textbf{(C) }56\qquad\textbf{(D) }58\qquad\textbf{(E) }60$
It is possible to express the sum
$$\sum_{n=1}^{24}\frac{1}{\sqrt{n+\sqrt{n^2-1}}}$$
as $a\sqrt{2}+b\sqrt{3}$, for some integers $a$ and $b$. Compute the ordered pair $(a,b)$.