Found problems: 85335
There are twenty-four 4-digit numbers that use each of the four digits 2, 5, 7, and 4exactly once. Listed in numerical order from smallest to largest, the number in the $17th$ position in the list is
$ \text{(A)}\ 4527\qquad\text{(B)}\ 5724\qquad\text{(C)}\ 5742\qquad\text{(D)}\ 7245\qquad\text{(E)}\ 7524 $
Find the smallest natural number n such that for all integers $m > n$ there are positive integers $x$ and $y$ for which the equality 1$7x + 23y = m$ holds
A [b]palindrome[/b] is a whole number that reads the same forwards and backwards. If one neglects the colon, certain times displayed on a digital watch are palindromes. Three examples are: $ \boxed{1:01} $, $ \boxed{12:21} $.
How many times during a 12-hour period will be palindromes?
$ \text{(A)}\ 57\qquad\text{(B)}\ 60\qquad\text{(C)}\ 63\qquad\text{(D)}\ 90\qquad\text{(E)}\ 93 $
Let $P$ be a polynomial of degree greater than or equal to $4$ with integer coefficients. An integer $x$ is called $P$-[i]representable[/i] if there exists integer numbers $a$ and $b$ such that $x = P(a) - P(b)$. Prove that, if for all $N \geq 0$, more than half of the integers of the set $\{0,1,\dots,N\}$ are $P$-[i]representable[/i], then all the even integers are $P$-[i]representable[/i] or all the odd integers are $P$-[i]representable[/i].
Proctors Andy and Kristin have a PUMaC team of eight students labelled $s_1, s_2, ... , s_8$ (the PUMaC staff being awful with names). The following occurs:
$1$. Andy tells the students to arrange themselves in a line in arbitrary order.
$2$. Kristin tells each student $s_i$ to move to the current spot of student $s_j$ , where $j \equiv 3i + 1$ (mod $8$).
$3$. Andy tells each student $s_i$ to move to the current spot of the student who was in the $i$th position of the line after step $1$.
How many possible orders can the students be in now?
If $n$ is a positive integer such that $1+2+3+\cdots+n=190$, then what is $n$?
Annisa has $n$ distinct textbooks, where $n > 6$. She has a different ways to pick a group of $4$ books, b different ways to pick $5$ books and c different ways to pick $6$ books. If Annisa buys two more (distinct) textbooks, how many ways will she be able to pick a group of $6$ books?
Let $O$ be the intersection of the diagonals of convex quadrilateral $ABCD$. The circumcircles of $\triangle{OAD}$ and $\triangle{OBC}$ meet at $O$ and $M$. Line $OM$ meets the circumcircles of $\triangle{OAB}$ and $\triangle{OCD}$ at $T$ and $S$ respectively.
Prove that $M$ is the midpoint of $ST$.
Four congruent semicircular half-disks are arranged inside a circle with radius $4$ so that each semicircle is internally tangent to the circle, and the diameters of the semicircles form a $2\times 2$ square centered at the center of the circle as shown. The radius of each semicircular half-disk is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[img]https://cdn.artofproblemsolving.com/attachments/f/e/8c0b9fdd69f6b54d39708da94ef2b2d039cb1e.png[/img]
Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcenter of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the center of the circle which passes through $B$ and is tangent to $\overline{AC}$ at $D_1$. Similarly, let $O_D$ be the center of the circle which passes through $D$ and is tangent to $\overline{AC}$ at $B_1$.
Assume that $\overline{BD_1} \parallel \overline{DB_1}$. Prove that $O$ lies on the line $\overline{O_BO_D}$.
The letters $R, M$, and $O$ represent whole numbers. If $R \times M \times O = 240$, $R \times O + M =46$ and $R + M \times O = 64$, what is the value of $R + M + O$?
Find all pairs $(b,c)$ of positive integers, such that the sequence defined by $a_1=b$, $a_2=c$ and $a_{n+2}= \left| 3a_{n+1}-2a_n \right|$ for $n \geq 1$ has only finite number of composite terms.
[i]Proposed by Oleg Mushkarov and Nikolai Nikolov[/i]
Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number?
Find out all the integer pairs $(m,n)$ such that there exist two monic polynomials $P(x)$ and $Q(x)$ ,with $\deg{P}=m$ and $\deg{Q}=n$,satisfy that $$P(Q(t))\not=Q(P(t))$$ holds for any real number $t$.
Circle with center $I$, inscribed in a triangle $ABC$ , touches the sides $BC$ and $AC$ at points $A_1$ and $B_1$ respectively. On rays $A_1I$ and $B_1I$, respectively, let be the points $A_2$ and $B_2$ such that $IA_2=IB_2=R$, where $R$is the radius of the circumscribed circle of the triangle $ABC$. Prove that:
a) $AA_2 = BB_2 = OI$ where $O$ is the center of the circumscribed circle of the triangle $ABC$,
b) lines $AA_2$ and $BB_2$ intersect on the circumcircle of the triangle $ABC$.
Find all positive integers $n$ such that
\[\cos\frac{\pi}{n}\cos\frac{2\pi}{n}\cos\frac{3\pi}{n}=\frac{1}{n+1}\]
Given a triangle, $AB=78, BC=50, AC=112,$ construct squares $ABXY, BCPQ, ACMN$ outside the triangle. Let $L_1, L_2, L_3$ be the midpoints of $\overline{MP}, \overline{QX}, \overline{NY},$ respectively. Find the area of $L_1L_2L_3.$
Find all functions $f:\mathbb{N}\to \mathbb{N}$ so that for every natural numbers $m,n$ :$f(n)+2mn+f(m)$ is a perfect square.
Three circles $\omega_1$, $\omega_2$, $\omega_3$ are given. Let $A_0$ and $A_1$ be the common points of $\omega_1$ and $\omega_2$, $B_0$ and $B_1$ be the common points of $\omega_2$ and $\omega_3$, $C_0$ and $C_1$ be the common points of $\omega_3$ and $\omega_1$. Let $O_{i,j,k}$ be the circumcenter of triangle $A_iB_jC_k$. Prove that the four lines of the form $O_{ijk}O_{1 - i,1 - j,1 - k}$ are concurrent or parallel.
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfy $yf(x)+f(y) \geq f(xy)$
How many ten-digit whole numbers satisfy the following property: they have only $2$ and $5$ as digits, and there are no consecutive $2$'s in the number (i.e. any two $2$'s are separated by at least one $5$)?
Let $O$ be the circumcenter of triangle $ABC$. Suppose the perpendicular bisectors of $\overline{OB}$ and $\overline{OC}$ intersect lines $AB$ and $AC$ at $D\neq A$ and $E\neq A$, respectively. Determine the maximum possible number of distinct intersection points between line $BC$ and the circumcircle of $\triangle ADE$.
[i]Andrew Wen[/i]
$\triangle ABC$ is a triangle with sides $a,b,c$. Each side of $\triangle ABC$ is divided in $n$ equal segments. Let $S$ be the sum of the squares of the distances from each vertex to each of the points of division on its opposite side. Show that $\frac{S}{a^2+b^2+c^2}$ is a rational number.
Show that there is an absolute constant $c < 1$ with the following property: whenever $\mathcal P$ is a polygon with area $1$ in the plane, one can translate it by a distance of $\frac{1}{100}$ in some direction to obtain a polygon $\mathcal Q$, for which the intersection of the interiors of $\mathcal P$ and $\mathcal Q$ has total area at most $c$.
[i]Linus Hamilton[/i]
In a scalene triangle $ABC$ with $\angle A = 90^\circ,$ the tangent line at $A$ to its circumcircle meets line $BC$ at $M$ and the incircle touches $AC$ at $S$ and $AB$ at $R.$
The lines $RS$ and $BC$ intersect at $N,$ while the lines $AM$ and $SR$ intersect at $U.$
Prove that the triangle $UMN$ is isosceles.