Found problems: 85335
Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.
Let $ABC$ be a triangle with $AB = 13$, $AC = 14$, and $BC = 15$, and let $\Gamma$ be its incircle with incenter $ I$. Let $D$ and $E$ be the points of tangency between $\Gamma$ and $BC$ and $AC$ respectively, and let $\omega$ be the circle inscribed in $CDIE$. If $Q$ is the intersection point between $\Gamma$ and $\omega$ and $P$ is the intersection point between $CQ$ and $\omega$, compute the length of $P Q$.
Ana choses two real numbers $ y>0,x $ and Bogdan repeatedly tries to guess these in the following manner: at step $ j $ he choses a real number $ b_j, $ asks her if $ b_j=x+jy, $ and she tells him the truth.
[b]a)[/b] If $ x=0, $ can Bogdan find Ana's numbers in a finite number of steps?
[b]b)[/b] If $ x\neq 0, $ can Bogdan find Ana's numbers in a finite number of steps?
A round table has room for n diners ( $n\ge 2$). There are napkins in three different colours. In how many ways can the napkins be placed, one for each seat, so that no two neighbours get napkins of the same colour?
Given a regular hexagon with a side of $100$. Each side is divided into one hundred equal parts. Through the division points and vertices of the hexagon, all sorts of straight lines parallel to its sides are drawn. These lines divided the hexagon into single regular triangles. Consider covering a hexagon with equal rhombuses. Each rhombus is made up of two triangles. (These rhombuses cover the entire hexagon and do not overlap.) Among the lines that form our grid, we select those that intersect exactly to the rhombuses (intersect diagonally). How many such lines will there be if:
a) $k = 101$;
b) $k = 100$;
c) $k = 87$?
[i]20 problems for 25 minutes.[/i]
[b]p1.[/b] Evaluate $20 \times 21 + 2021$.
[b]p2.[/b] Let points $A$, $B$, $C$, and $D$ lie on a line in that order. Given that $AB = 5CD$ and $BD = 2BC$, compute $\frac{AC}{BD}$.
[b]p3.[/b] There are $18$ students in Vincent the Bug's math class. Given that $11$ of the students take U.S. History, $15$ of the students take English, and $2$ of the students take neither, how many students take both U.S. History and English?
[b]p4.[/b] Among all pairs of positive integers $(x, y)$ such that $xy = 12$, what is the least possible value of $x + y$?
[b]p5.[/b] What is the smallest positive integer $n$ such that $n! + 1$ is composite?
[b]p6.[/b] How many ordered triples of positive integers $(a, b,c)$ are there such that $a + b + c = 6$?
[b]p7.[/b] Thomas orders some pizzas and splits each into $8$ slices. Hungry Yunseo eats one slice and then finds that she is able to distribute all the remaining slices equally among the $29$ other math club students. What is the fewest number of pizzas that Thomas could have ordered?
[b]p8.[/b] Stephanie has two distinct prime numbers $a$ and $b$ such that $a^2-9b^2$ is also a prime. Compute $a + b$.
[b]p9.[/b] Let $ABCD$ be a unit square and $E$ be a point on diagonal $AC$ such that $AE = 1$. Compute $\angle BED$, in degrees.
[b]p10.[/b] Sheldon wants to trace each edge of a cube exactly once with a pen. What is the fewest number of continuous strokes that he needs to make? A continuous stroke is one that goes along the edges and does not leave the surface of the cube.
[b]p11.[/b] In base $b$, $130_b$ is equal to $3n$ in base ten, and $1300_b$ is equal to $n^2$ in base ten. What is the value of $n$, expressed in base ten?
[b]p12.[/b] Lin is writing a book with $n$ pages, numbered $1,2,..., n$. Given that $n > 20$, what is the least value of $n$ such that the average number of digits of the page numbers is an integer?
[b]p13.[/b] Max is playing bingo on a $5\times 5$ board. He needs to fill in four of the twelve rows, columns, and main diagonals of his bingo board to win. What is the minimum number of boxes he needs to fill in to win?
[b]p14.[/b] Given that $x$ and $y$ are distinct real numbers such that $x^2 + y = y^2 + x = 211$, compute the value of $|x - y|$.
[b]p15.[/b] How many ways are there to place 8 indistinguishable pieces on a $4\times 4$ checkerboard such that there are two pieces in each row and two pieces in each column?
[b]p16.[/b] The Manhattan distance between two points $(a, b)$ and $(c, d)$ in the plane is defined to be $|a - c| + |b - d|$. Suppose Neil, Neel, and Nail are at the points $(5, 3)$, $(-2,-2)$ and $(6, 0)$, respectively, and wish to meet at a point $(x, y)$ such that their Manhattan distances to$ (x, y)$ are equal. Find $10x + y$.
[b]p17.[/b] How many positive integers that have a composite number of divisors are there between $1$ and $100$, inclusive?
[b]p18.[/b] Find the number of distinct roots of the polynomial $$(x - 1)(x - 2) ... (x - 90)(x^2 - 1)(x^2 - 2) ... (x^2 - 90)(x^4 - 1)(x^4 - 2)...(x^4 - 90)$$.
[b]p19.[/b] In triangle $ABC$, let $D$ be the foot of the altitude from $ A$ to $BC$. Let $P,Q$ be points on $AB$, $AC$, respectively, such that $PQ$ is parallel to $BC$ and $\angle PDQ = 90^o$. Given that $AD = 25$, $BD = 9$, and $CD = 16$, compute $111 \times PQ$.
[b]p20.[/b] The simplified fraction with numerator less than $1000$ that is closest but not equal to $\frac{47}{18}$ is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Compute $p$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Do there exist positive integers $k$ and $n$ such that for any finite graph $G$ with diameter $k+1$ there exists a set $S$ of at most $n$ vertices such that for any $v\in V(G)\setminus S$, there exists a vertex $u\in S$ of distance at most $k$ from $v$?
[i]David Yang.[/i]
Find all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ such that $ \forall x,y,z \in \mathbb{R}$ we have: If
\[ x^3 \plus{} f(y) \cdot x \plus{} f(z) \equal{} 0,\]
then
\[ f(x)^3 \plus{} y \cdot f(x) \plus{} z \equal{} 0.\]
In the $100 \times 100$ table in every cell there is natural number. All numbers in same row or column are different.
Can be that for every square sum of numbers, that are in angle cells, is square number ?
Prove that the inequality $ r^2+r_a^2+r_b^2+ r_c^2 \ge 2S$ holds for an arbitrary triangle, where $r$ is the radius of the circle inscribed in the triangle, $r_a$, $r_b$, $r_c$ are the radii of its three excribed circles, $S$ is the area of the triangle.
If $ S \equal{} i^n \plus{} i^{\minus{}n}$, where $ i \equal{} \sqrt{\minus{}1}$ and $ n$ is an integer, then the total number of possible distinct values for $ S$ is:
$ \textbf{(A)}\ 1\qquad
\textbf{(B)}\ 2\qquad
\textbf{(C)}\ 3\qquad
\textbf{(D)}\ 4\qquad
\textbf{(E)}\ \text{more than 4}$
Given a triangle $ABC$ inscribed in circle $c(O,R)$ (with center $O$ and radius $R$) with $AB<AC<BC$ and let $BD$ be a diameter of the circle $c$. The perpendicular bisector of $BD$ intersects line $AC$ at point $M$ and line $AB$ at point $N$. Line $ND$ intersects the circle $c$ at point $T$. Let $S$ be the second intersection point of cicumcircles $c_1$ of triangle $OCM$, and $c_2$ of triangle $OAD$. Prove that lines $AD, CT$ and $OS$ pass through the same point.
Let $ABC$ be an acute triangle and let $A_{1}, B_{1}, C_{1}$ be the feet of its altitudes. The incircle of the triangle $A_{1}B_{1}C_{1}$ touches its sides at the points $A_{2}, B_{2}, C_{2}$. Prove that the Euler lines of triangles $ABC$ and $A_{2}B_{2}C_{2}$ coincide.
Let $ABC$ be a triangle with circumcircle $\omega$ and circumcenter $O$. The tangent line to from $A$ to $\omega$ intersects $BC$ at $K$. The tangent line to from $B$ to $\omega$ intersects $AC$ at $L$. Let $M,N$ be the midpoints of $AK,BL$ respectively. The line $MN$ is named by $\alpha$. The feet of perpendicular from $A,B,C$ to the edges of $\triangle ABC$ are named by $D,E,F$ respectively. The perpendicular bisectors of $EF,DF,DE$ intersect $\alpha$ at $X,Y,Z$ respectively. Let $AD,BE,CF$ intersect $\omega$ again at $D',E',F'$ respectively. If $H$ is the orthocenter of $ABC$, prove that the lines $XD',YE',ZF',OH$ are concurrent.
A given convex quadrilateral $ABCD$ is such that $\angle ABD + \angle ACD > \angle BAC + \angle BDC.$ Prove that
\[S_{ABD}+S_{ACD} > S_{BAC}+S_{BDC}.\]
The maximum number of solid regular tetrahedrons can be placed against each other so that one of their edges coincides with a given line segment in space?
[hide=original wording]Hoeveel massieve regelmatige viervlakken kan men maximaal tegen mekaar plaatsen
zodat ´e´en van hun ribben samenvalt met een gegeven lijnstuk in de ruimte?[/hide]
Amy has a six-sided die which always rolls values greater than or equal to the previous roll. She rolls the die repeatedly until she rolls a $6$. Find the expected value of the sum of all distinct values she has rolled when she finishes.
Let $K$ be a knot in the $3$-dimensional space (that is a differentiable injection of a circle into $\mathbb{R}^3$, and $D$ be the diagram of the knot (that is the projection of it to a plane so that in exception of the transversal double points it is also an injection of a circle). Let us color the complement of $D$ in black and color the diagram $D$ in a chessboard pattern, black and white so that zones which share an arc in common are coloured differently. Define $\Gamma_B(D)$ the black graph of the diagram, a graph which has vertices in the black areas and every two vertices which correspond to black areas which have a common point have an edge connecting them.
[list=a]
[*]Determine all knots having a diagram $D$ such that $\Gamma_B(D)$ has at most $3$ spanning trees. (Two knots are not considered distinct if they can be moved into each other with a one parameter set of the injection of the circle)[/*]
[*]Prove that for any knot and any diagram $D$, $\Gamma_B(D)$ has an odd number of spanning trees.[/*]
[/list]
An equilateral triangle of side length 2 is divided into four pieces by two perpendicular lines that intersect in the centroid of the triangle. What is the maximum possible area of a piece?
Quadrilateral $ABCD$ is given with $AD \nparallel BC$. The midpoints of $AD$ and $BC$ are denoted by $M$ and $N$, respectively. The line $MN$ intersects the diagonals $AC$ and $BD$ in points $K$ and $L$, respectively. Prove that the circumcircles of the triangles $AKM$ and $BNL$ have common point on the line $AB$.( Proposed by Emil Stoyanov )
[img]http://estoyanov.net/wp-content/uploads/2015/09/est.png[/img]
A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.
There are $n$ players, $n\ge 2$, which are playing a card game with $np$ cards in $p$ rounds. The cards are coloured in $n$ colours and each colour is labelled with the numbers $1,2,\ldots ,p$. The game submits to the following rules:
[list]each player receives $p$ cards.
the player who begins the first round throws a card and each player has to discard a card of the same colour, if he has one; otherwise they can give an arbitrary card.
the winner of the round is the player who has put the greatest card of the same colour as the first one.
the winner of the round starts the next round with a card that he selects and the play continues with the same rules.
the played cards are out of the game.[/list]
Show that if all cards labelled with number $1$ are winners, then $p\ge 2n$.
[i]Barbu Berceanu[/i]
On a circle are given $n\ge 3$ points. At most, how many parts can the segments with the endpoints at these $n$ points divide the interior of the circle into?
Let $i = \sqrt{-1}$. The product of the real parts of the roots of $z^2 - z = 5 - 5i$ is
$ \textbf{(A)}\ -25\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ -5\qquad\textbf{(D)}\ \frac{1}{4}\qquad\textbf{(E)}\ 25 $
Through point $P$, which lies on one of the sides of the triangle $ABC$, draw a line dividing its area in half.