Found problems: 85335
Let $\alpha$ and $\beta$ be positive integers such that
$$
\frac{16}{37}<\frac{\alpha}{\beta}<\frac{7}{16} .
$$
Find the smallest possible value of $\beta$.
Kilua and Ndoti play the following game in a square $ABCD$: Kilua chooses one of the sides of the square and draws a point $X$ at this side. Ndoti chooses one of the other three sides and draws a point Y. Kilua chooses another side that hasn't been chosen and draws a point Z. Finally, Ndoti chooses the last side that hasn't been chosen yet and draws a point W. Each one of the players can draw his point at a vertex of $ABCD$, but they have to choose the side of the square that is going to be used to do that. For example, if Kilua chooses $AB$, he can draws $X$ at the point $B$ and it doesn't impede Ndoti of choosing $BC$. A vertex cannot de chosen twice. Kilua wins if the area of the convex quadrilateral formed by $X$, $Y$, $Z$, and $W$ is greater or equal than a half of the area of $ABCD$. Otherwise, Ndoti wins. Which player has a winning strategy? How can he play?
The number $125$ can be written as a sum of some pairwise coprime integers larger than $1$. Determine the largest number of terms that the sum may have.
Find the area bounded by $ y\equal{}x^2\minus{}|x^2\minus{}1|\plus{}|2|x|\minus{}2|\plus{}2|x|\minus{}7$ and the $ x$ axis.
A positive integer $m$ is called triangular if $m = 1 + 2 + ... + n$, for an integer $n$. Show that a positive integer $m$ is triangular if and only if $8m + 1$ is the square of an integer.
Let $ABC$ be a triangle with $\angle C = 90^o$ and $D$ a point on the ray $CB$ such that $|AC| \cdot |CD| = |BC|^2$. A parallel line to $AB$ through $D$ intersects the ray $CA$ at $E$. Find $\angle BEC$.
Show that the fraction $ \frac{n^2+n-1}{n^2+2n}$ is irreducible for every positive integer n.
Consider the following three lines in the Cartesian plane: $$\begin{cases}
\ell_1: & 2x - y = 7\\
\ell_2: & 5x + y = 42\\
\ell_3: & x + y = 14
\end{cases}$$
and let $f_i(P)$ correspond to the reflection of the point $P$ across $\ell_i$. Suppose $X$ and $Y$ are points on the $x$ and $y$ axes, respectively, such that $f_1(f_2(f_3(X)))= Y$. Let $t$ be the length of segment $XY$; what is the sum of all possible values of $t^2$?
Let $h_a,h_b,h_c$ be the altitudes, and let $m_a,m_b,m_c$ be the medians of the acute triangle drawn to the sides $a, b, c$ respectively. Let $r$ and $R$ be the radii of the inscribed and circumscribed circles. Prove that $$\frac{m_a}{h_a}+\frac{m_b}{h_b}+\frac{m_c}{h_c} <1+\frac{R}{r}.$$
In $\triangle ABC$ points $D$ and $E$ lie on $\overline{AB}$ so that $AD < AE < AB$, while points $F$ and $G$ lie on $\overline{AC}$ so that $AF < AG < AC$. Suppose $AD = 4$, $DE = 16$, $EB = 8$, $AF = 13$, $FG = 52$, and $GC = 26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G$ through $E$. The area of quadrilateral $DEGF$ is $288$. Find the area of heptagon $AFNBCEM$, as shown in the figure below.
[asy]
unitsize(14);
pair A = (0, 9), B = (-6, 0), C = (12, 0), D = (5A + 2B)/7, E = (2A + 5B)/7, F = (5A + 2C)/7, G = (2A + 5C)/7, M = 2F - D, N = 2E - G;
filldraw(A--F--N--B--C--E--M--cycle, lightgray);
draw(A--B--C--cycle);
draw(D--M);
draw(N--G);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
dot(F);
dot(G);
dot(M);
dot(N);
label("$A$", A, dir(90));
label("$B$", B, dir(225));
label("$C$", C, dir(315));
label("$D$", D, dir(135));
label("$E$", E, dir(135));
label("$F$", F, dir(45));
label("$G$", G, dir(45));
label("$M$", M, dir(45));
label("$N$", N, dir(135));
[/asy]
Find the smallest positive integer $n$ such that a cube with sides of length $n$ can be divided up into exactly $2007$ smaller cubes, each of whose sides is of integer length.
Let $n$ be the number of ways that $10$ dollars can be changed into dimes and quarters, with at least one of each coin being used. Then $n$ equals:
$\textbf{(A)}\ 40 \qquad
\textbf{(B)}\ 38 \qquad
\textbf{(C)}\ 21 \qquad
\textbf{(D)}\ 20 \qquad
\textbf{(E)}\ 19 $
The degree measures of the six interior angles of a convex hexagon form an arithmetic sequence (not necessarily in cyclic order). The common difference of this arithmetic sequence can be any real number in the open interval $(-D, D)$. Compute the greatest possible value of $D$.
There are $m \geq 3$ positive integers, not necessarily distinct, that are arranged in a circle so that any positive integer divides the sum of its neighbours.
Show that if there is exactly one $1$, then for any positive integer $n$, there are at most $\phi(n)$ copies
of $n$.
[i]Proposed By- (usjl, adapted from 2014 Taiwan TST)[/i]
Let $A$ be a set $A=\{1,2,3,...,2017\}$. Subset $S$ of set $A$ is [i]good [/i] if for all $x\in A$ sum of remaining elements of set $S$ has same last digit as $x$. Prove that [i]good[/i] subset with $405$ elements is not possible.
In the Cartesian plane let $A = (1,0)$ and $B = \left( 2, 2\sqrt{3} \right)$. Equilateral triangle $ABC$ is constructed so that $C$ lies in the first quadrant. Let $P=(x,y)$ be the center of $\triangle ABC$. Then $x \cdot y$ can be written as $\tfrac{p\sqrt{q}}{r}$, where $p$ and $r$ are relatively prime positive integers and $q$ is an integer that is not divisible by the square of any prime. Find $p+q+r$.
Given a positive integer $n \geq 2$. Let $a_{ij}$ $(1 \leq i,j \leq n)$ be $n^2$ non-negative reals and their sum is $1$. For $1\leq i \leq n$, define $R_i=max_{1\leq k \leq n}(a_{ik})$. For $1\leq j \leq n$, define $C_j=min_{1\leq k \leq n}(a_{kj})$
Find the maximum value of $C_1C_2 \cdots C_n(R_1+R_2+ \cdots +R_n)$
Let $M$ be the midpoint of side $AB$ of the triangle $ABC$. Let$P$ be a point on $AB$ between $A$ and $M$, and let $MD$ be drawn parallel to $PC$ and intersecting $BC$ at $D$. If the ratio of the area of the triangle $BPD$ to that of triangle $ABC$ is denoted by $r$, then
$\text{(A)}\ \tfrac{1}{2}<r<1\text{ depending upon the position of }P \qquad\\
\text{(B)}\ r=\tfrac{1}{2}\text{ independent of the position of }P\qquad\\
\text{(C)}\ \tfrac{1}{2}\le r<1\text{ depending upon the position of }P \qquad\\
\text{(D)}\ \tfrac{1}{3}<r<\tfrac{2}{3}\text{ depending upon the position of }P \qquad\\
\text{(E)}\ r=\tfrac{1}{3} \text{ independent of the position of }P$
For each positive integer $n$ let the set $A_n$ consist of all numbers $\pm 1 \pm 2 \pm ...\pm n$. For example, $$A_1 = \{-1,1\}, A_2 = \{ -3 ,-1 ,1 ,3 \} , A_3 = \{ -6 ,-4 ,-2 ,0 ,2 ,4 ,6 \}.$$
Find the number of elements in $A_n$ .
Let $0\leq p_i \leq 1$ for $i=1,2, \dots, n.$ Show that $$\sum_{i=1}^{n} \frac{1}{|x-p_i|} \leq 8n(1+1/3+1/5+\dots +\frac{1}{2n-1})$$ for some $x$ satisfying $0\leq x \leq 1.$
There are two different versions of this problem, each with different solutions. You must find bounds for each of these problems.
[list]
[*] Alice and Bob are playing a collaborative game in which they agree upon an encoding/strategy. Alice shuffles a deck of 52 cards (numbered 1-52) (*) and takes the first $n$ cards off the top of the deck. Alice then chooses one card to be put to the side, and chooses an ordering of the other $n-1$ cards. Bob then walks into the room seeing the $n-1$ cards in the order Alice put them in. For example, if Alice was given 9-4-10-51-7-8 and chose to put 8 to the side, she could put 5 cards in order of 9-4-10-51-7 which Bob would see.
Find the lowest $n$ for which Bob could guarantee that he could use Alice's encoding to find the card placed to the side.
(*) (If you would prefer, you can write your solution in terms of the deck being a standard deck with 4 suits and 13 ranks, there's a way to move between them that we can handle.)
[*] Alice and Bob are playing a collaborative game in which they agree upon an encoding/strategy. Alice shuffles a deck of 52 cards (numbered 1-52) and takes the first $n$ cards off the top of the deck. Alice then chooses one card to be put to the side, and chooses an ordering of the other $n-1$ cards. Carol then comes in to the room and randomly discards one of the $n-1$ cards, and places the cards back in a way that preserves the order Alice had. Bob then walks into the room seeing the $n-2$ cards in the order Alice put them in. For example, if Alice was given 1-2-3-4-5-6 and chose to put 6 to the side, she would put 5 cards in order of 1-2-3-4-5, and Carol removed 4, Bob would see 1-2-3-5.
Find the lowest $n$ for which Bob could guarantee that he could use Alice's encoding to find the card placed to the side.
[/list]
[i]Proposed by Eric Oh[/i]
As shown in the figure, it is known that $BC = AC$ in $ABC$, $M$ is the midpoint of $AB$, points $D$ and $E$ lie on $AB$ satisfying $\angle DCE = \angle MCB$, the circumscribed circle of $\vartriangle BDC$ and the circumscribed circle of $\vartriangle AEC$ intersect at point $F$ (different from point $C$), point $H$ lies on $AB$ such that the straight line $CM$ bisects the line segment $HF$. Let the circumcenters of $\vartriangle HFE$ and $\vartriangle BFM$ be $O_1$ and $O_2$ respectively. Prove that $O_1O_2\perp CF$.
[img]https://cdn.artofproblemsolving.com/attachments/e/4/e8fc62735b8cfbd382e490617f26d335c46823.png[/img]
An integer \(n \geq 2\) is said to be [i]tuanis[/i] if, when you add the smallest prime divisor of \(n\) and the largest prime divisor of \(n\) (these divisors can be the same), you obtain an odd result. Calculate the sum of all [i]tuanis[/i] numbers that are less or equal to \(2023\).
Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. What is the probability that no two ants arrive at the same vertex?
$ \textbf{(A)}\ \frac {5}{256} \qquad
\textbf{(B)}\ \frac {21}{1024} \qquad
\textbf{(C)}\ \frac {11}{512} \qquad
\textbf{(D)}\ \frac {23}{1024} \qquad
\textbf{(E)}\ \frac {3}{128}$
Let $ABC$ be an isosceles triangle ($AB = AC$) with its circumcenter $O$. Point $N$ is the midpoint of the segment $BC$ and point $M$ is the reflection of the point $N$ with respect to the side $AC$. Suppose that $T$ is a point so that $ANBT$ is a rectangle. Prove that $\angle OMT = \frac{1}{2} \angle BAC$.
[i]Proposed by Ali Zamani[/i]