Found problems: 85335
2015 IFYM, Sozopol, 1
Let ABCD be a convex quadrilateral such that $AB + CD = \sqrt{2}AC$ and $BC + DA = \sqrt{2}BD$. Prove that ABCD is a parallelogram.
2018 Online Math Open Problems, 20
For positive integers $k,n$ with $k\leq n$, we say that a $k$-tuple $\left(a_1,a_2,\ldots,a_k\right)$ of positive integers is [i]tasty[/i] if
[list]
[*] there exists a $k$-element subset $S$ of $[n]$ and a bijection $f:[k]\to S$ with $a_x\leq f\left(x\right)$ for each $x\in [k]$,
[*] $a_x=a_y$ for some distinct $x,y\in [k]$, and
[*] $a_i\leq a_j$ for any $i < j$.
[/list]
For some positive integer $n$, there are more than $2018$ tasty tuples as $k$ ranges through $2,3,\ldots,n$. Compute the least possible number of tasty tuples there can be.
Note: For a positive integer $m$, $[m]$ is taken to denote the set $\left\{1,2,\ldots,m\right\}$.
[i]Proposed by Vincent Huang and Tristan Shin[/i]
2015 Vietnam Team selection test, Problem 4
There are $100$ students who praticipate at exam.Also there are $25$ members of jury.Each student is checked by one jury.Known that every student likes $10$ jury
$a)$ Prove that we can select $7$ jury such that any student likes at least one jury.
$b)$ Prove that we can make this every student will be checked by the jury that he likes and every jury will check at most $10$ students.
2012 Tournament of Towns, 5
A car rides along a circular track in the clockwise direction. At noon Peter and Paul took their positions at two different points of the track. Some moment later they simultaneously ended their duties and compared their notes. The car passed each of them at least $30$ times. Peter noticed that each circle was passed by the car $1$ second faster than the preceding one while Paul’s observation was opposite: each circle was passed $1$ second slower than the preceding one.
Prove that their duty was at least an hour and a half long.
2011 Serbia National Math Olympiad, 3
Let $H$ be orthocenter and $O$ circumcenter of an acuted angled triangle $ABC$. $D$ and $E$ are feets of perpendiculars from $A$ and $B$ on $BC$ and $AC$ respectively. Let $OD$ and $OE$ intersect $BE$ and $AD$ in $K$ and $L$, respectively. Let $X$ be intersection of circumcircles of $HKD$ and $HLE$ different than $H$, and $M$ is midpoint of $AB$. Prove that $K, L, M$ are collinear iff $X$ is circumcenter of $EOD$.
1993 Greece National Olympiad, 11
Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the probability that he wins the sixth game is $m/n$, where $m$ and $n$ are relatively prime positive integers. What are the last three digits of $m + n$?
2025 Romania EGMO TST, P2
Prove that any finite set $H$ of lattice points on the plane has a subset $K$ with the following properties:
[list]
[*]any vertical or horizontal line in the plane cuts $K$ in at most $2$ points,
[*]any point of $H\setminus K$ is contained by a segment with endpoints from $K$.[/list]
2017 HMNT, 8
Marisa has a collection of $2^8-1=255$ distinct nonempty subsets of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. For each step she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to contain repeated sets.) She repeats this process $2^8-2=254$ times until there is only one set left in the collection. What is the expected size of this set?
2002 AIME Problems, 9
Harold, Tanya, and Ulysses paint a very long picket fence.
Harold starts with the first picket and paints every $h$th picket;
Tanya starts with the second picket and paints everth $t$th picket; and
Ulysses starts with the third picket and paints every $u$th picket.
Call the positive integer $100h+10t+u$ $\textit{paintable}$ when the triple $(h,t,u)$ of positive integers results in every picket being painted exaclty once. Find the sum of all the paintable integers.
2022 Bolivia Cono Sur TST, P2
On $\triangle ABC$ if there existed a point $D$ in $AC$ such that $\angle CBD=\angle ABD+60$ and $\angle BDC=30$ and $AB \cdot BC=BD^2$, then find the angles inside the triangle $\triangle ABC$
2004 AIME Problems, 8
Define a regular $n$-pointed star to be the union of $n$ line segments $P_1P_2, P_2P_3,\ldots, P_nP_1$ such that
$\bullet$ the points $P_1, P_2,\ldots, P_n$ are coplanar and no three of them are collinear,
$\bullet$ each of the $n$ line segments intersects at least one of the other line segments at a point other than an endpoint,
$\bullet$ all of the angles at $P_1, P_2,\ldots, P_n$ are congruent,
$\bullet$ all of the $n$ line segments $P_2P_3,\ldots, P_nP_1$ are congruent, and
$\bullet$ the path $P_1P_2, P_2P_3,\ldots, P_nP_1$ turns counterclockwise at an angle of less than 180 degrees at each vertex.
There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are similar, but there are two non-similar regular 7-pointed stars. How many non-similar regular 1000-pointed stars are there?
2015 Online Math Open Problems, 24
Let $ABC$ be an acute triangle with incenter $I$; ray $AI$ meets the circumcircle $\Omega$ of $ABC$ at $M \neq A$. Suppose $T$ lies on line $BC$ such that $\angle MIT=90^{\circ}$.
Let $K$ be the foot of the altitude from $I$ to $\overline{TM}$. Given that $\sin B = \frac{55}{73}$ and $\sin C = \frac{77}{85}$, and $\frac{BK}{CK} = \frac mn$ in lowest terms, compute $m+n$.
[i]Proposed by Evan Chen[/i]
2007 Stanford Mathematics Tournament, 5
Two disks of radius 1 are drawn so that each disk's circumference passes through the center of the other disk. What is the circumference of the region in which they overlap?
1992 All Soviet Union Mathematical Olympiad, 571
$ABCD$ is a parallelogram. The excircle of $ABC$ opposite $A$ has center $E$ and touches the line $AB$ at $X$. The excircle of $ADC$ opposite $A$ has center $F$ and touches the line $AD$ at $Y$. The line $FC$ meets the line$ AB$ at $W$, and the line $EC$ meets the line $AD$ at $Z$. Show that $WX = YZ$.
2012 Indonesia TST, 2
Suppose $S$ is a subset of $\{1,2,3,\ldots,2012\}$. If $S$ has at least $1000$ elements, prove that $S$ contains two different elements $a,b$, where $b$ divides $2a$.
2015 Indonesia MO, 7
Let $a,b,c$ be positive real numbers. Prove that
$\sqrt{\frac{a}{b+c}+\frac{b}{c+a}}+\sqrt{\frac{b}{c+a}+\frac{c}{a+b}}+\sqrt{\frac{c}{a+b}+\frac{a}{b+c}}\ge 3$
EMCC Guts Rounds, 2010
[u]Round 1[/u]
[b]p1.[/b] Define the operation $\clubsuit$ so that $a \,\clubsuit \, b = a^b + b^a$. Then, if $2 \,\clubsuit \,b = 32$, what is $b$?
[b]p2. [/b] A square is changed into a rectangle by increasing two of its sides by $p\%$ and decreasing the two other sides by $p\%$. The area is then reduced by $1\%$. What is the value of $p$?
[b]p3.[/b] What is the sum, in degrees, of the internal angles of a heptagon?
[b]p4.[/b] How many integers in between $\sqrt{47}$ and $\sqrt{8283}$ are divisible by $7$?
[u]Round 2[/u]
[b]p5.[/b] Some mutant green turkeys and pink elephants are grazing in a field. Mutant green turkeys have six legs and three heads. Pink elephants have $4$ legs and $1$ head. There are $100$ legs and $37$ heads in the field. How many animals are grazing?
[b]p6.[/b] Let $A = (0, 0)$, $B = (6, 8)$, $C = (20, 8)$, $D = (14, 0)$, $E = (21, -10)$, and $F = (7, -10)$. Find the area of the hexagon $ABCDEF$.
[b]p7.[/b] In Moscow, three men, Oleg, Igor, and Dima, are questioned on suspicion of stealing Vladimir Putin’s blankie. It is known that each man either always tells the truth or always lies. They make the following statements:
(a) Oleg: I am innocent!
(b) Igor: Dima stole the blankie!
(c) Dima: I am innocent!
(d) Igor: I am guilty!
(e) Oleg: Yes, Igor is indeed guilty!
If exactly one of Oleg, Igor, and Dima is guilty of the theft, who is the thief??
[b]p8.[/b] How many $11$-letter sequences of $E$’s and $M$’s have at least as many $E$’s as $M$’s?
[u]Round 3[/u]
[b]p9.[/b] John is entering the following summation $31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39$ in his calculator. However, he accidently leaves out a plus sign and the answer becomes $3582$. What is the number that comes before the missing plus sign?
[b]p10.[/b] Two circles of radius $6$ intersect such that they share a common chord of length $6$. The total area covered may be expressed as $a\pi + \sqrt{b}$, where $a$ and $b$ are integers. What is $a + b$?
[b]p11.[/b] Alice has a rectangular room with $6$ outlets lined up on one wall and $6$ lamps lined up on the opposite wall. She has $6$ distinct power cords (red, blue, green, purple, black, yellow). If the red and green power cords cannot cross, how many ways can she plug in all six lamps?
[b]p12.[/b] Tracy wants to jump through a line of $12$ tiles on the floor by either jumping onto the next block, or jumping onto the block two steps ahead. An example of a path through the $12$ tiles may be: $1$ step, $2$ steps, $2$ steps, $2$ steps, $1$ step, $2$ steps, $2$ steps. In how many ways can Tracy jump through these $12$ tiles?
PS. You should use hide for answers. Last rounds have been posted [url=https://artofproblemsolving.com/community/c4h2784268p24464984]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2013 NIMO Problems, 8
Find the number of positive integers $n$ for which there exists a sequence $x_1, x_2, \cdots, x_n$ of integers with the following property: if indices $1 \le i \le j \le n$ satisfy $i+j \le n$ and $x_i - x_j$ is divisible by $3$, then $x_{i+j} + x_i + x_j + 1$ is divisible by $3$.
[i]Based on a proposal by Ivan Koswara[/i]
2021 Caucasus Mathematical Olympiad, 7
4 tokens are placed in the plane. If the tokens are now at the vertices of a convex quadrilateral $P$, then the following move could be performed: choose one of the tokens and shift it in the direction perpendicular to the diagonal of $P$ not containing this token; while shifting tokens it is prohibited to get three collinear tokens.
Suppose that initially tokens were at the vertices of a rectangle $\Pi$, and after a number of moves tokens were at the vertices of one another rectangle $\Pi'$ such that $\Pi'$ is similar to $\Pi$ but not equal to $\Pi $. Prove that $\Pi$ is a square.
2013 Bogdan Stan, 1
Under composition, let be a group of linear polynomials that admit a fixed point . Show that all polynomials of this group have the same fixed point.
[i]Vasile Pop[/i]
2014 National Olympiad First Round, 8
In how many ways can $17$ identical red and $10$ identical white balls be distributed into $4$ distinct boxes such that the number of red balls is greater than the number of white balls in each box?
$
\textbf{(A)}\ 5462
\qquad\textbf{(B)}\ 5586
\qquad\textbf{(C)}\ 5664
\qquad\textbf{(D)}\ 5720
\qquad\textbf{(E)}\ 5848
$
2005 Poland - Second Round, 3
In space are given $n\ge 2$ points, no four of which are coplanar. Some of these points are connected by segments. Let $K$ be the number of segments $(K>1)$ and $T$ be the number of formed triangles. Prove that $9T^2<2K^3$.
2004 AMC 10, 25
A circle of radius $ 1$ is internally tangent to two circles of radius $ 2$ at points $ A$ and $ B$, where $ AB$ is a diameter of the smaller circle. What is the area of the region, shaded in the gure, that is outside the smaller circle and inside each of the two larger circles?
[asy]size(200);defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=4;
pair B = (0,1);
pair A = (0,-1);
label("$B$",B,NW);label("$A$",A,2S);
draw(Circle(A,2));draw(Circle(B,2));
fill((-sqrt(3),0)..B..(sqrt(3),0)--cycle,gray);
fill((-sqrt(3),0)..A..(sqrt(3),0)--cycle,gray);
draw((-sqrt(3),0)..B..(sqrt(3),0));
draw((-sqrt(3),0)..A..(sqrt(3),0));
path circ = Circle(origin,1);
fill(circ,white);
draw(circ);
dot(A);dot(B);
pair A1 = B + dir(45)*2;
pair A2 = dir(45);
pair A3 = dir(-135)*2 + A;
draw(B--A1,EndArrow(HookHead,2));
draw(origin--A2,EndArrow(HookHead,2));
draw(A--A3,EndArrow(HookHead,2));
label("$2$",midpoint(B--A1),NW);
label("$1$",midpoint(origin--A2),NW);
label("$2$",midpoint(A--A3),NW);[/asy]$ \textbf{(A)}\ \frac {5}{3}\pi \minus{} 3\sqrt {2}\qquad \textbf{(B)}\ \frac {5}{3}\pi \minus{} 2\sqrt {3}\qquad \textbf{(C)}\ \frac {8}{3}\pi \minus{} 3\sqrt {3}\qquad\textbf{(D)}\ \frac {8}{3}\pi \minus{} 3\sqrt {2}$
$ \textbf{(E)}\ \frac {8}{3}\pi \minus{} 2\sqrt {3}$
2017 Irish Math Olympiad, 4
An equilateral triangle of integer side length $n \geq 1$ is subdivided into small triangles of unit side length, as illustrated in the figure below for $n = 5$. In this diagram a subtriangle is a triangle of any size which is formed by connecting vertices of the small triangles along the grid lines.
[img]https://cdn.artofproblemsolving.com/attachments/e/9/17e83ad4872fcf9e97f0479104b9569bf75ad0.jpg[/img]
It is desired to color each vertex of the small triangles either red or blue in such a way that there is no subtriangle with all of its vertices having the same color. Let $f(n)$ denote the number of distinct colorings satisfying this condition.
Determine, with proof, $f(n)$ for every $n \geq 1$
2004 AMC 12/AHSME, 18
Square $ ABCD$ has side length $ 2$. A semicircle with diameter $ \overline{AB}$ is constructed inside the square, and the tangent to the semicricle from $ C$ intersects side $ \overline{AD}$ at $ E$. What is the length of $ \overline{CE}$?
[asy]
defaultpen(linewidth(0.8));
pair A=origin, B=(1,0), C=(1,1), D=(0,1), X=tangent(C, (0.5,0), 0.5, 1), F=C+2*dir(C--X), E=intersectionpoint(C--F, A--D);
draw(C--D--A--B--C--E);
draw(Arc((0.5,0), 0.5, 0, 180));
pair point=(0.5,0.5);
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));
label("$E$", E, dir(point--E));[/asy]
$ \textbf{(A)}\ \frac {2 \plus{} \sqrt5}{2} \qquad \textbf{(B)}\ \sqrt 5 \qquad \textbf{(C)}\ \sqrt 6 \qquad \textbf{(D)}\ \frac52 \qquad \textbf{(E)}\ 5 \minus{} \sqrt5$