Found problems: 85335
Inside the acute-angled triangle $ABC$ we take $P$ and $Q$ two isogonal conjugate points. The perpendicular lines on the interior angle-bisector of $\angle BAC$ passing through $P$ and $Q$ intersect the segments $AC$ and $AB$ at the points $B_p\in AC$, $B_q\in AC$, $C_p\in AB$ and $C_q\in AB$, respectively. Let $W$ be the midpoint of the arc $BAC$ of the circle $(ABC)$. The line $WP$ intersects the circle $(ABC)$ again at $P_1$ and the line $WQ$ intersects the circle $(ABC)$ again at $Q_1$. Prove that the points $P_1$, $Q_1$, $B_p$, $B_q$, $C_p$ and $C_q$ lie on a circle.
[i]Proposed by P. Bibikov[/i]
Is that true that there exist a polynomial $f(x)$ with rational coefficients, not all integers, with degree $n>0$, a polynomial $g(x)$, with integer coefficients, and a set $S$ with $n+1$ integers such that $f(t)=g(t)$ for all $t \in S$?
[u]Round 1[/u]
[b]p1.[/b] Alex is writing a sequence of $A$’s and $B$’s on a chalkboard. Any $20$ consecutive letters must have an equal number of $A$’s and $B$’s, but any 22 consecutive letters must have a different number of $A$’s and $B$’s. What is the length of the longest sequence Alex can write?.
[b]p2.[/b] A positive number is placed on each of the $10$ circles in this picture. It turns out that for each of the nine little equilateral triangles, the number on one of its corners is the sum of the numbers on the other two corners. Is it possible that all $10$ numbers are different?
[img]https://cdn.artofproblemsolving.com/attachments/b/f/c501362211d1c2a577e718d2b1ed1f1eb77af1.png[/img]
[b]p3.[/b] Pablo and Nina take turns entering integers into the cells of a $3 \times 3$ table. Pablo goes first. The person who fills the last empty cell in a row must make the numbers in that row add to $0$. Can Nina ensure at least two of the columns have a negative sum, no matter what Pablo does?
[b]p4. [/b]All possible simplified fractions greater than $0$ and less than $1$ with denominators less than or equal to $100$ are written in a row with a space before each number (including the first).
Zeke and Qing play a game, taking turns choosing a blank space and writing a “$+$” or “$-$” sign in it. Zeke goes first. After all the spaces have been filled, Zeke wins if the value of the resulting expression is an integer.
Can Zeke win no matter what Qing does?
[img]https://cdn.artofproblemsolving.com/attachments/3/6/15484835686fbc2aa092e8afc6f11cd1d1fb88.png[/img]
[b]p5.[/b] A police officer patrols a town whose map is shown. The officer must walk down every street segment at least once and return to the starting point, only changing direction at intersections and corners. It takes the officer one minute to walk each segment. What is the fastest the officer can complete a patrol?
[img]https://cdn.artofproblemsolving.com/attachments/0/c/d827cf26c8eaabfd5b0deb92612a6e6ebffb47.png[/img]
[u]Round 2[/u]
[b]p6.[/b] Prove that among any $3^{2022}$ integers, it is possible to find exactly $3^{2021}$ of them whose sum is divisible by $3^{2021}$.
[b]p7.[/b] Given a list of three numbers, a zap consists of picking two of the numbers and decreasing each of them by their average. For example, if the list is $(5, 7, 10)$ and you zap $5$ and $10$, whose average is $7.5$, the new list is $(-2.5, 7, 2.5)$.
Is it possible to start with the list $(3, 1, 4)$ and, through some sequence of zaps, end with a list in which the sum of the three numbers is $0$?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Let $A=\{1,2,3,4\}$, and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$. The probability that the range of $f$ and the range of $g$ are disjoint is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.
Let $N{}$ be the number of positive integers with $10$ digits $\overline{d_9d_8\cdots d_0}$ in base $10$ (where $0\le d_i\le9$ for all $i$ and $d_9>0$) such that the polynomial
\[d_9x^9+d_8x^8+\cdots+d_1x+d_0\]
is irreducible in $\Bbb Q$. Prove that $N$ is even.
(A polynomial is irreducible in $\Bbb Q$ if it cannot be factored into two non-constant polynomials with rational coefficients.)
Minivan chooses a prime number. Then every second, he adds either the digit $1$ or the digit $3$ to the right end of his number (after the unit digit), such that the new number is also a prime. Can he continue indefinitely?
[i](Proposed by Wong Jer Ren)[/i]
An ordered quadruple of numbers is called [i]ten-esque[/i] if it is composed of 4 nonnegative integers whose sum is equal to $10$. Ana chooses a ten-esque quadruple $(a_1, a_2, a_3, a_4)$ and Banana tries to guess it. At each stage Banana offers a ten-esque quadtruple $(x_1,x_2,x_3,x_4)$ and Ana tells her the value of
\[|a_1-x_1|+|a_2-x_2|+|a_3-x_3|+|a_4-x_4|\]
How many guesses are needed for Banana to figure out the quadruple Ana chose?
In a circle with center $ O$, chord $ \overline{AB}$ equals chord $ \overline{AC}$. Chord $ \overline{AD}$ cuts $ \overline{BC}$ in $ E$. If $ AC \equal{} 12$ and $ AE \equal{} 8$, then $ AD$ equals:
$ \textbf{(A)}\ 27\qquad
\textbf{(B)}\ 24\qquad
\textbf{(C)}\ 21\qquad
\textbf{(D)}\ 20\qquad
\textbf{(E)}\ 18$
For some integers $u$,$ v$, and $w$, the equation
$$26ab - 51bc + 74ca = 12(a^2 + b^2 + c^2)$$
holds for all real numbers a, b, and c that satisfy
$$au + bv + cw = 0$$
Find the minimum possible value of $u^2 + v^2 + w^2$.
Let $n$ be a prime number. Show that there is a permutation $a_1,a_2,...,a_n$ of $1,2,...,n$ so that $a_1,a_1a_2,...,a_1a_2...a_n$ leave distinct remainders when divided by $n$
Let $X=\{A_{1},...,A_{n}\}$ be a set of distinct 3-element subsets of the set $\{1,2,...,36\}$ such that
(a) $A_{i},A_{j}$ have nonempty intersections for all $i,j$
(b) The intersection of all elements of $X$ is the empty set.
Show that $n\leq 100$.
Determine the number of such sets $X$ when $n=100$
There are $10$ girls in a class, all with different heights. They want to form a queue so that no girl stands directly between two girls shorter than her. How many ways are there to form the queue?
Using an escalator to go down to the Metro station and walking with a regular pace, I find that I need $50$ steps to go down. if i come back later to run up it, at a speed $5$ times my previous normal pace, I check that I need $125$ steps to reach the top. How many visible steps does the mechanical staircase have when it is stopped?
Mary chose an even $4$-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,...,\tfrac{n}{2},n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$?
$\textbf{(A) } 324 \qquad \textbf{(B) } 330 \qquad \textbf{(C) } 340 \qquad \textbf{(D) } 361 \qquad \textbf{(E) } 646$
Let $ a,b,n $ be three natural numbers. Prove that there exists a natural number $ c $ satisfying:
$$ \left( \sqrt{a} +\sqrt{b} \right)^n =\sqrt{ c+(a-b)^n} +\sqrt{c} $$
[i]Dan Popescu[/i]
2. Prove that for any $2\le n \in \mathbb{N}$ there exists positive integers $a_1,a_2,\cdots,a_n$ such that $\forall i\neq j: \text{gcd}(a_i,a_j) = 1$ and $\forall i: a_i \ge 1402$ and the given relation holds.
$$[\frac{a_1}{a_2}]+[\frac{a_2}{a_3}]+\cdots+[\frac{a_n}{a_1}] = [\frac{a_2}{a_1}]+[\frac{a_3}{a_2}]+\cdots+[\frac{a_1}{a_n}]$$
On the Cartesian coordinate plane, the graph of the parabola $y = x^2$ is drawn. Three distinct points $A$, $B$, and $C$ are marked on the graph with $A$ lying between $B$ and $C$. Point $N$ is marked on $BC$ so that $AN$ is parallel to the y-axis. Let $K_1$ and $K_2$ are the areas of triangles $ABN$ and $ACN$, respectively. Express $AN$ in terms of $K_1$ and $K_2$.
For positive integers $a$, $b$, and $c$, define \[ f(a,b,c)=\frac{abc}{\text{gcd}(a,b,c)\cdot\text{lcm}(a,b,c)}. \] We say that a positive integer $n$ is $f@$ if there exist pairwise distinct positive integers $x,y,z\leq60$ that satisfy $f(x,y,z)=n$. How many $f@$ integers are there?
[i]Proposed by Michael Ren[/i]
Determine all monotone functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$, so that for all $x, y \in \mathbb{Z}$
\[f(x^{2005} + y^{2005}) = (f(x))^{2005} + (f(y))^{2005}\]
Find all integers $n$ for which $9n + 16$ and $16n + 9$ are both perfect squares.
Let $O$ be the circumcircle of acute triangle $ABC$, $AD$-altitude of $ABC$ ($ D \in BC$), $ AD \cap CO =E$, $M$-midpoint of $AE$, $F$-feet of perpendicular from $C$ to $AO$.
Proved that point of intersection $OM$ and $BC$ lies on circumcircle of triangle $BOF$
Let $ABC$ be acute-angled triangle . A circle $\omega_1(O_1,R_1)$ passes through points $B$ and $C$ and meets the sides $AB$ and $AC$ at points $D$ and $E$ ,respectively .
Let $\omega_2(O_2,R_2)$ be the circumcircle of triangle $ADE$ . Prove that $O_1O_2$ is equal to the circumradius of triangle $ABC$ .
If $N$, written in base $2$, is $11000$, the integer immediately preceeding $N$, written in base $2$, is:
$\textbf{(A) }10001\qquad
\textbf{(B) }10010\qquad
\textbf{(C) }10011\qquad
\textbf{(D) }10110\qquad
\textbf{(E) }10111$
Instead of walking along two adjacent sides of a rectangular field, a boy took a shortcut along the diagonal of the field and saved a distance equal to $ \frac{1}{2}$ the longer side. The ratio of the shorter side of the rectangle to the longer side was:
$ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{2}{3} \qquad\textbf{(C)}\ \frac{1}{4} \qquad\textbf{(D)}\ \frac{3}{4} \qquad\textbf{(E)}\ \frac{2}{5}$
Let $OABC$ be a tetrahedon with $\angle BOC=\alpha,\angle COA =\beta$ and $\angle AOB =\gamma$. The angle between the faces $OAB$ and $OAC$ is $\sigma$ and the angle between the faces $OAB$ and $OBC$ is $\rho$.
Show that $\gamma > \beta \cos\sigma + \alpha \cos\rho$.