Found problems: 85335
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both
$$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$
are integers.
One hundred natural numbers whose greatest common divisor is $1$ are arranged around a circle. An allowed operation is to add to a number the greatest common divisor of its two neighhbors. Prove that we can make all the numbers pairwise copirme in a finite number of moves.
In a convex quadrilateral $ABCD$, $E$ is the midpoint of $CD$, $F$ is midpoint of $AD$, $K$ is the intersection point of $AC$ with $BE$. Prove that the area of triangle $BKF$ is half the area of triangle $ABC$.
Let $ABCD$ be a rectangle, $AB = a, BC = b$. Consider the family of parallel and equidistant straight lines (the distance between two consecutive lines being $d$) that are at an the angle $\phi, 0 \leq \phi \leq 90^{\circ},$ with respect to $AB$. Let $L$ be the sum of the lengths of all the segments intersecting the rectangle. Find:
[i](a)[/i] how $L $ varies,
[i](b)[/i] a necessary and sufficient condition for $L$ to be a constant, and
[i](c)[/i] the value of this constant.
$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 \equal{} 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.
At the beginning of the school year, $ 50\%$ of all students in Mr. Well's math class answered "Yes" to the question "Do you love math", and $ 50\%$ answered "No." At the end of the school year, $ 70\%$ answered "Yes" and $ 30\%$ answered "No." Altogether, $ x\%$ of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of $ x$?
$ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 80$
In Venetiania, the smallest currency is the ducat. The finance minister instructs his officials as follows: "I wish six kinds of banknotes, each worth a whole number of ducats. Those six values must be such that there exists a number N with the following property:
Any amount of money of $n$ ducats ($n$ positive and integer) where $n \le N$ may be paid in such a way that no more than two copies of each kind are required either to pay or to return. I also wish those six values to be as large as possible for $N$. Determine those six values and provide proof that all conditions have been met."
Solve the problem of those officials
Say that an integer $n \ge 2$ is [i]delicious[/i] if there exist $n$ positive integers adding up to 2014 that have distinct remainders when divided by $n$. What is the smallest delicious integer?
Nine weights are placed in a scale with the respective values $1kg,2kg,...,9kg$. In how many ways can we place six weights in the left side and three weights in the right side such that the right side is heavier than the left one?
In an acute triangle $ABC$ the angle bisector $AL$, $L \in BC$, intersects its circumcircle at $N$. Let $K$ and $M$ be the projections of $L$ onto sides $AB$ and $AC$. Prove that triangle $ABC$ and quadrilateral $A K N M$ have equal areas.
Let $a,b$ be real numbers such that $a<b$.
Evaluate
\[\lim_{b\rightarrow a} \frac{\displaystyle\int_a^b \ln |1+(x-a)(b-x)|dx}{(b-a)^3}\].
Given are a positive integer $n$ and an infinite sequence of proper fractions $x_0 = \frac{a_0}{n}$, $\ldots$, $x_i=\frac{a_i}{n+i}$, with $a_i < n+i$. Prove that there exist a positive integer $k$ and integers $c_1$, $\ldots$, $c_k$ such that \[ c_1 x_1 + \ldots + c_k x_k = 1. \]
[i]Proposed by M. Dubashinsky[/i]
Let $ABC$ be a triangle and let $M$ be the midpoint $BC$. On the exterior of the triangle, consider the parallelogram $BCDE$ such that $BE//AM$ and $BE=AM/2$ . Prove that line $EM$ passes through the midpoint of segment $AD$.
Place the 21 two-digit prime numbers in the white squares of the grid on the right so that each two-digit prime is used exactly once. Two white squares sharing a side must contain two numbers with either the same tens digit or ones digit. A given digit in a white square must equal at least one of the two digits of that square’s prime number.
[asy]
size(10cm);
real s= 10.0;
int[][] x = {
{0,0,0,0,0},
{0,0,0,0,0},
{0,0,0,0,0},
{0,0,0,0,0},
{0,0,0,0,0}};
void square(int a, int b)
{
fill(s*(a,b)--s*(a+1,b)--s*(a+1,b+1)--s*(a,b+1)--cycle);
}
square(1,2);
square(1,3);
square(3,1);
square(3,2);
for(int i = 0; i < 6; ++i) {
draw(s*(i,0)--s*(i,5));
}
for(int i = 0; i < 6; ++i) {
draw(s*(0,i)--s*(5,i));
}
for(int k = 0; k<5; ++k){
for(int l = 0; l<5; ++l){
if(x[k][l]!=0){
label(scale(5.0)*string(x[k][l]),s*(l+0.5,-k+4.5));
}
}
}
void sudokuLabel(int p, int q, int r) {
label(string(r), s*(p, q) + (1, -1));
}
sudokuLabel(1, 1, 4);
sudokuLabel(2, 1, 1);
sudokuLabel(3, 1, 1);
sudokuLabel(4, 1, 3);
sudokuLabel(0, 3, 9);
sudokuLabel(2, 3, 9);
sudokuLabel(4, 3, 5);
sudokuLabel(0, 5, 3);
sudokuLabel(1, 5, 1);
sudokuLabel(2, 5, 3);
sudokuLabel(3, 5, 2);[/asy]
There is a unique solution, but you do not need to prove that your answer is the only one possible. You merely need to find an answer that satisfies the constraints above.
(Note: in any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)
Given two forests $A$ and $B$ with \(V(A) = V(B)\), that is the graphs are over same vertex set. Suppose $A$ has [b]strictly more[/b] edges than $B$. Prove that there exists an edge of $A$ which if included in the edge set of $B$, then $B$ will still remain a forest. Graphs are undirected
Compute the sum of all positive integers $n$ such that the median of the $n$ smallest prime numbers is $n$.
[i]Proposed by Luke Robitaille[/i]
Let $n>3$ be an integer. Integers $a_1, \dots, a_n$ are given so that $a_k\in \{k, -k\}$ for all $1\leq k\leq n$. Prove that there is a sequence of indices $1\leq k_1, k_2, \dots, k_n\leq n$, not necessarily distinct, for which the sums
\[a_{k_1}\]
\[a_{k_1}+a_{k_2}\]
\[a_{k_1}+a_{k_2}+a_{k_3}\]
\[\vdots\]
\[a_{k_1}+a_{k_2}+\cdots+a_{k_n}\]
have distinct residues modulo $2n+1$, and so that the last one is divisible by $2n+1$.
$\textbf{(a)}$ On the sides of triangle $ABC$ we consider the points $M\in \overline{BC}$, $N\in \overline{AC}$ and $P\in \overline{AB}$ such that the quadrilateral $MNAP$ with right angles $\angle MNA$ and $\angle MPA$ has an inscribed circle. Prove that $MNAP$ has to be a kite.
$\textbf{(b)}$ Is it possible for an isosceles trapezoid to be orthodiagonal and circumscribed too?
[i] (Călin Udrea) [/i]
Show how to construct a chord $BPC$ of a given angle $A$ through a given point $P$ such that $\tfrac{1}{BP}\plus{} \tfrac{1}{PC}$ is a maximum.
[asy]
size(200);
defaultpen(linewidth(0.7));
pair A = origin, B = (5,0), C = (4.2,3), P = waypoint(B--C,0.65);
pair Bp = 1.3 * B, Cp = 1.2 * C;
draw(A--B--C--A^^Bp--A--Cp);
dot(P);
label("$A$",A,W);
label("$B$",B,S);
label("$C$",C,dir(B--C));
label("$P$",P,dir(A--P));
[/asy]
Given trapezoid $ABCD$ with parallel sides $AB$ and $CD$, let $E$ be a point on line $BC$ outside segment $BC$, such that segment $AE$ intersects segment $CD$. Assume that there exists a point $F$ inside segment $AD$ such that $\angle EAD=\angle CBF$. Denote by $I$ the point of intersection of $CD$ and $EF$, and by $J$ the point of intersection of $AB$ and $EF$. Let $K$ be the midpoint of segment $EF$, and assume that $K$ is different from $I$ and $J$.
Prove that $K$ belongs to the circumcircle of $\triangle ABI$ if and only if $K$ belongs to the circumcircle of $\triangle CDJ$.
At each of the given $n$ points on a circle, either $+1$ or $-1$ is written. The following operation is performed: between any two consecutive numbers on the circle their product is written, and the initial $n$ numbers are deleted. Suppose that, for any initial arrangement of $+1$ and $-1$ on the circle, after finitely many operations all the numbers on the circle will be equal to $+1$. Prove that $n$ is a power of two.
Let $n$ be a positive integer. Suppose that $A$ and $B$ are invertible $n \times n$ matrices with complex entries such that $A+B=I$ (where $I$ is the identity matrix) and
\[(A^2+B^2)(A^4+B^4)=A^5+B^5.\]
Find all possible values of $\det(AB)$ for the given $n$.
Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$.
Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$.
(a) Show that there are only finitely many pairs of consecutive highly divisible
integers of the form $(a, b)$ with $a\mid b$.
(b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.
Let $ a$, $ b$, $ c$ and $ n$ be positive integers such that $ a^n$ is divisible by $ b$, such that $ b^n$ is divisible by $ c$, and such that $ c^n$ is divisible by $ a$.
Prove that $ \left(a \plus{} b \plus{} c\right)^{n^2 \plus{} n \plus{} 1}$ is divisible by $ abc$.
An even broader [i]generalization[/i], though not part of the QEDMO problem and not quite number theory either:
If $ u$ and $ n$ are positive integers, and $ a_1$, $ a_2$, ..., $ a_u$ are integers such that $ a_i^n$ is divisible by $ a_{i \plus{} 1}$ for every $ i$ such that $ 1\leq i\leq u$ (we set $ a_{u \plus{} 1} \equal{} a_1$ here), then show that $ \left(a_1 \plus{} a_2 \plus{} ... \plus{} a_u\right)^{n^{u \minus{} 1} \plus{} n^{u \minus{} 2} \plus{} ... \plus{} n \plus{} 1}$ is divisible by $ a_1a_2...a_u$.
A container in the shape of a right circular cone is 12 inches tall and its base has a 5-inch radius. The liquid that is sealed inside is 9 inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the liquid is $m-n\sqrt[3]{p},$ where $m,$ $n,$ and $p$ are positive integers and $p$ is not divisible by the cube of any prime number. Find $m+n+p.$