Found problems: 85335
Let $\Delta ABC$ be a triangle with $AB=AC$ and $\angle A = \alpha$, and let $O,H$ be its circumcenter and orthocenter, respectively. If $P,Q$ are points on $AB$ and $AC$, respectively, such that $APHQ$ forms a rhombus, determine $\angle POQ$ in terms of $\alpha$.
All diagonals of a convex pentagon are drawn, dividing it in one smaller pentagon and $10$ triangles. Find the maximum number of triangles with the same area that may exist in the division.
Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^2$
For an integer $n>3$ denote by $n?$ the product of all primes less than $n$. Solve the equation $n?=2n+16$.
[i]V. Senderov [/i]
Sam dumps tea for $6$ hours at a constant rate of $60$ tea crates per hour. Eddie takes $4$ hours to dump the same
amount of tea at a different constant rate. How many tea crates does Eddie dump per hour?
[i]Proposed by Samuel Tsui[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{90}$
Sam dumps a total of $6 \cdot 60 = 360$ tea crates and since it takes Eddie $4$ hours to dump that many he dumps at a rate of $\dfrac{360}{4}= \boxed{90}$ tea crates per hour.
[/hide]
The polynomial $f(x) = x^3 + rx^2 + sx + t$ has $r, s$, and $t$ as its roots (with multiplicity), where $f(1)$ is rational and $ t \ne 0$. Compute $|f(0)|$.
Fatima and Asma are playing the following game. First, Fatima chooses $2013$ pairwise different numbers, called $a_1, a_2, ..., a_{2013}$. Then, Asma tries to know the value of each number $a_1, a_2, ..., a_{2013}$.. At each time, Asma chooses $1 \le i < j \le 2013$ and asks Fatima ''[i]What is the set $\{a_i,a_j\}$?[/i]'' (For example, if Asma asks what is the set $\{a_i,a_j\}$, and $a_1 = 17$ and $a_2 = 13$, Fatima will answer $\{13. 17\}$). Find the least number of questions Asma needs to ask, to know the value of all the numbers $a_1, a_2, ..., a_{2013}$.
A $300\times 300$ board is arbitrarily filled with $2\times 1$ dominoes with no overflow, underflow, or overlap. (Tokens can be placed vertically or horizontally.)
Decide if it is possible to paint the tiles with three different colors, so that the following conditions are met:
$\bullet$ Each token is painted in one and only one of the colors.
$\bullet$ The same number of tiles are painted in each color.
$\bullet$ No piece is a neighbor of more than two pieces of the same color.
Note: Two dominoes are [i]neighbors [/i]if they share an edge.
Let $s_m$ be the number $66\cdots 6$ with $m$ digits $6$. Find
\[
s_1 + s_2 + \cdots + s_n
\]
Twelve candidates for mayor participate in a TV talk show. At some point a candidate said: "One lie has been told." Another said: "Now two lies have been told". "Now three lies," said a third. This continued until the twelfth said: "Now twelve lies have been told". At this point the moderator ended the discussion. It turned out that at least one of the candidates correctly stated the number of lies told before he made the claim. How many lies were actually told by the candidates?
On a plane several straight lines are drawn in such a way that each of them intersects exactly $15$ other lines. How many lines are drawn on the plane? Find all possibilities and justify your answer.
Poland
Let $n$ be a positive integer, and let $A$ and $B$ be $n\times n$ matrices with complex entries such that $A^2=B^2$. Show that there exists an $n\times n$ invertible matrix $S$ with complex entries that satisfies $S(AB-BA)=(BA-AB)S$.
Monic quadratic polynomials $ P(x)$ and $ Q(x)$ have the property that $ P(Q(x))$ has zeroes at $ x\equal{}\minus{}23,\minus{}21,\minus{}17, \text{and} \minus{}15$, and $ Q(P(x))$ has zeroes at $ x\equal{}\minus{}59, \minus{}57, \minus{}51, \text{and} \minus{}49$. What is the sum of the minimum values of $ P(x)$ and $ Q(x)$?
$ \textbf{(A)}\ \text{\minus{}100} \qquad \textbf{(B)}\ \text{\minus{}82} \qquad \textbf{(C)}\ \text{\minus{}73} \qquad \textbf{(D)}\ \text{\minus{}64} \qquad \textbf{(E)}\ 0$
A "letter $ T$" erected at point $ A$ of the $ x$-axis in the $ xy$-plane is the union of a segment $ AB$ in the upper half-plane perpendicular to the $ x$-axis and a segment $ CD$ containing $ B$ in its interior and parallel to the $ x$-axis. Show that it is impossible to erect a letter $ T$ at every point of the $ x$-axis so that the union of those erected at rational points is disjoint from the union of those erected at irrational points.
[i]A.Csaszar[/i]
The increasing sequence of positive integers $a_{1},a_{2},a_{3},\ldots$ has the property that $a_{n+2} = a_{n} + a_{n+1}$ for all $n \ge 1$. If $a_{7} = 120$, then $a_{8}$ is
$ \textbf{(A)}\ 128\qquad\textbf{(B)}\ 168\qquad\textbf{(C)}\ 193\qquad\textbf{(D)}\ 194\qquad\textbf{(E)}\ 210 $
On the blackboard there are written $100$ different positive integers . To each of these numbers is added the $\gcd$ of the $99$ other numbers . In the new $100$ numbers , is it possible for $3$ of them to be equal.
[i] (С. Берлов)[/i]
Let $a$, $b$, and $c$ be real numbers such that $0\le a,b,c\le 5$ and $2a + b + c = 10$. Over all possible values of $a$, $b$, and $c$, determine the maximum possible value of $a + 2b + 3c$.
[i]Proposed by Andrew Wen[/i]
A badminton club consists of $2n$ members who are n couples. The club plans to arrange a round of mixed doubles matches where spouses neither play together nor against each other. Requirements are:
$\cdot$ Each pair of members of the same gender meets exactly once as opponents in a mixed doubles match.
$\cdot$ Any two members of the opposite gender who are not spouses meet exactly once as partners and also as opponents in a mixed doubles match.
Given that $(n,6)=1$, can you arrange a round of mixed doubles matches that meets the above specifications and requirements?
Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$, $\angle ABC=\angle DEA=90^\circ$ and $BC+DE=1$. Compute the area of the pentagon.
[i]Greece[/i]
a) Factorize $A= x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2$
b) Prove that there are no integers $x,y,z$ such that $x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=2000 $
In an international meeting of $n \geq 3$ participants, 14 languages are spoken. We know that:
- Any 3 participants speak a common language.
- No language is spoken more that by the half of the participants.
What is the least value of $n$?
Let $ p(x) \equal{} x^6 \plus{} ax^5 \plus{} bx^4 \plus{} cx^3 \plus{} dx^2 \plus{} ex \plus{} f$ be a polynomial such that $ p(1) \equal{} 1, p(2) \equal{} 2, p(3) \equal{} 3, p(4) \equal{} 4, p(5) \equal{} 5,$ and $ p(6) \equal{} 6.$ What is $ p(7)$?
A. 0
B. 7
C. 14
D. 49
E. 727
If the sum of any $10$ of $21$ real numbers is less than the sum of remaining $11$ of them, at least how many of these $21$ numbers are positive?
$
\textbf{(A)}\ 18
\qquad\textbf{(B)}\ 19
\qquad\textbf{(C)}\ 20
\qquad\textbf{(D)}\ 21
\qquad\textbf{(E)}\ \text{None of the preceding}
$
As shown in the figure below a regular dodecahedron (the polyhedron consisting of 12 congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring?
[asy]
import graph;
unitsize(4.5cm);
pair A = (0.082, 0.378);
pair B = (0.091, 0.649);
pair C = (0.249, 0.899);
pair D = (0.479, 0.939);
pair E = (0.758, 0.893);
pair F = (0.862, 0.658);
pair G = (0.924, 0.403);
pair H = (0.747, 0.194);
pair I = (0.526, 0.075);
pair J = (0.251, 0.170);
pair K = (0.568, 0.234);
pair L = (0.262, 0.449);
pair M = (0.373, 0.813);
pair N = (0.731, 0.813);
pair O = (0.851, 0.461);
path[] f;
f[0] = A--B--C--M--L--cycle;
f[1] = C--D--E--N--M--cycle;
f[2] = E--F--G--O--N--cycle;
f[3] = G--H--I--K--O--cycle;
f[4] = I--J--A--L--K--cycle;
f[5] = K--L--M--N--O--cycle;
draw(f[0]);
axialshade(f[1], white, M, gray(0.5), (C+2*D)/3);
draw(f[1]);
filldraw(f[2], gray);
filldraw(f[3], gray);
axialshade(f[4], white, L, gray(0.7), J);
draw(f[4]);
draw(f[5]);
[/asy]
$\textbf{(A) } 125 \qquad \textbf{(B) } 250 \qquad \textbf{(C) } 405 \qquad \textbf{(D) } 640 \qquad \textbf{(E) } 810$
Prove that the positive integers $n$ that cannot be written as a sum of $r$ consecutive positive integers, with $r>1$, are of the form $n=2^l~$ for some $l\geqslant 0$.