Found problems: 85335
2017 Junior Balkan Team Selection Tests - Romania, 2
Let $A$ be a point outside the circle $\omega$ . The tangents from $A$ touch the circle at $B$ and $C$. Let $P$ be an arbitrary point on extension of $AC$ towards $C$, $Q$ the projection of $C$ onto $PB$ and $E$ the second intersection point of the circumcircle of $ABP$ with the circle $\omega$ . Prove that $\angle PEQ = 2\angle APB$
2003 Indonesia Juniors, day 2
p1. It is known that $a_1=2$ , $a_2=3$ . For $k > 2$, define $a_k=\frac{1}{2}a_{k-2}+\frac{1}{3}a_{k-1}$.
Find the infinite sum of of $a_1+a_2+a_3+...$
p2. The [i]multiplied [/i] number is a natural number in two-digit form followed by the result time. For example, $7\times 8 = 56$, then $7856$ and $8756$ are multiplied numbers . $2\times 3 = 6$, then $236$ and $326$ are multiplied. $2\times 0 = 0$, then $200$ is the multiplied. For the record, the first digit of the number times can't be $0$.
a. What is the difference between the largest and the smallest multiplied number?
b. Find all the multiplied numbers that consist of three digits and each digit is square number.
c. Given the following "boxes" that must be filled with multiple numbers.
[img]https://cdn.artofproblemsolving.com/attachments/b/6/ac086a3d1a0549fae909c072224605430daf1d.png[/img]
Determine the contents of the shaded box. Is this content the only one?
d. Complete all the empty boxes above with multiplied numbers.
p3. Look at the picture of the arrangement of three squares below.
[img]https://cdn.artofproblemsolving.com/attachments/1/3/c0200abae77cc73260b083117bf4bafc707eea.png[/img]Prove that $\angle BAX + \angle CAX = 45^o$
p4. Prove that $(n-1)n (n^3 + 1)$ is always divisible by $6$ for all natural number $n$.
2022 Puerto Rico Team Selection Test, 1
Let's call a natural number [i] interesting[/i] if any of its two digits consecutive forms a number that is a multiple of $19$ or $21$. For example, The number $7638$ is interesting, because $76$ is a multiple of $19$, $63$ is multiple of $21$, and $38$ is a multiple of $19$. How many interesting numbers of $2022$ digits exist?
2021 Saudi Arabia Training Tests, 3
Let $ABC$ be an acute, non-isosceles triangle inscribed in (O) and $BB'$, $CC'$ are altitudes. Denote $E, F$ as the intersections of $BB'$, $CC'$ with $(O)$ and $D, P, Q$ are projections of $A$ on $BC$, $CE$, $BF$. Prove that the perpendicular bisectors of $PQ$ bisects two segments $AO$, $BC$.
2022 Austrian Junior Regional Competition, 2
You are given a rectangular playing field of size $13 \times 2$ and any number of dominoes of sizes $2\times 1$ and $3\times 1$. The playing field should be seamless with such dominoes and without overlapping, with no domino protruding beyond the playing field may. Furthermore, all dominoes must be aligned in the same way, i. e. their long sides must be parallel to each other. How many such coverings are possible?
(Walther Janous)
2008 AMC 8, 3
If February is a month that contains Friday the $13^{\text{th}}$, what day of the week is February 1?
$\textbf{(A)}\ \text{Sunday} \qquad
\textbf{(B)}\ \text{Monday} \qquad
\textbf{(C)}\ \text{Wednesday} \qquad
\textbf{(D)}\ \text{Thursday}\qquad
\textbf{(E)}\ \text{Saturday} $
2020 Iran Team Selection Test, 3
Given a triangle $ABC$ with circumcircle $\Gamma$. Points $E$ and $F$ are the foot of angle bisectors of $B$ and $C$, $I$ is incenter and $K$ is the intersection of $AI$ and $EF$. Suppose that $T$ be the midpoint of arc $BAC$. Circle $\Gamma$ intersects the $A$-median and circumcircle of $AEF$ for the second time at $X$ and $S$. Let $S'$ be the reflection of $S$ across $AI$ and $J$ be the second intersection of circumcircle of $AS'K$ and $AX$. Prove that quadrilateral $TJIX$ is cyclic.
[i]Proposed by Alireza Dadgarnia and Amir Parsa Hosseini[/i]
2017 Princeton University Math Competition, B1
If $x$ is a positive number such that $x^{x^{x^{x}}} = ((x^{x})^{x})^{x}$, find $(x^{x})^{(x^{x})}$.
2002 Finnish National High School Mathematics Competition, 4
Convex figure $\mathcal{K}$ has the following property:
if one looks at $\mathcal{K}$ from any point of the certain circle $\mathcal{Y}$, then $\mathcal{K}$ is seen in the right angle.
Show that the figure is symmetric with respect to the centre of $\mathcal{Y.}$
2022 MIG, 13
Sarah is leading a class of $35$ students. Initially, all students are standing. Each time Sarah waves her hands, a prime number of standing students sit down. If no one is left standing after Sarah waves her hands $3$ times, what is the greatest possible number of students that could have been standing before her third wave?
$\textbf{(A) }23\qquad\textbf{(B) }27\qquad\textbf{(C) }29\qquad\textbf{(D) }31\qquad\textbf{(E) }33$
2011 AMC 12/AHSME, 22
Let $T_1$ be a triangle with sides $2011, 2012,$ and $2013$. For $n \ge 1$, if $T_n=\triangle ABC$ and $D,E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB,BC$ and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD,BE,$ and $CF$, if it exists. What is the perimeter of the last triangle in the sequence $(T_n)$?
$ \textbf{(A)}\ \frac{1509}{8} \qquad
\textbf{(B)}\ \frac{1509}{32} \qquad
\textbf{(C)}\ \frac{1509}{64} \qquad
\textbf{(D)}\ \frac{1509}{128} \qquad
\textbf{(E)}\ \frac{1509}{256} $
2012 Princeton University Math Competition, A5 / B7
$5$ people stand in a line facing one direction. In every round, the person at the front moves randomly to any position in the line, including the front or the end. Suppose that $\frac{m}{n}$ is the expected number of rounds needed for the last person of the initial line to appear at the front of the line, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?
2019 District Olympiad, 3
Consider the rectangular parallelepiped $ABCDA'B'C'D' $ as such the measure of the dihedral angle formed by the planes $(A'BD)$ and $(C'BD)$ is $90^o$ and the measure of the dihedral angle formed by the planes $(AB'C)$ and $(D'B'C)$ is $60^o$. Determine and measure the dihedral angle formed by the planes $(BC'D)$ and $(A'C'D)$.
2009 Putnam, A2
Functions $ f,g,h$ are differentiable on some open interval around $ 0$ and satisfy the equations and initial conditions
\begin{align*}f'&=2f^2gh+\frac1{gh},\ f(0)=1,\\
g'&=fg^2h+\frac4{fh},\ g(0)=1,\\
h'&=3fgh^2+\frac1{fg},\ h(0)=1.\end{align*}
Find an explicit formula for $ f(x),$ valid in some open interval around $ 0.$
2015 AoPS Mathematical Olympiad, 3
A small apartment building has four doors, with door numbers $1, 2, 3, 4.$ John has $2^4-1=15$ keys, label with of possible nonempty subsets of $\{1,2,3,4\}$, but he forgot which key is which. If an element on the key matches the door number, the key can open the door (e.g. key $\{1,2,4\}$ can open Door 4). He picks a key at random and tries to open Door 1, which fails, so he discards it. John then randomly picks one of his remaining 14 keys and tries to open Door 2, but it doesn't open, so he throws away that key as well. He then randomly selects one of the remaining 13 keys, and tests it on Door 3. What is the probability that it will open?
[i]Proposed by dantx5[/i]
MathLinks Contest 7th, 7.3
Let $ n$ be a positive integer, and let $ M \equal{} \{1,2,\ldots, 2n\}$. Find the minimal positive integer $ m$, such that no matter how we choose the subsets $ A_i \subset M$, $ 1\leq i\leq m$, with the properties:
(1) $ |A_i\minus{}A_j|\geq 1$, for all $ i\neq j$,
(2) $ \bigcup_{i\equal{}1}^m A_i \equal{} M$,
we can always find two subsets $ A_k$ and $ A_l$ such that $ A_k \cup A_l \equal{} M$ (here $ |X|$ represents the number of elements in the set $ X$.)
2011 Belarus Team Selection Test, 2
Let $A_1A_2 \ldots A_n$ be a convex polygon. Point $P$ inside this polygon is chosen so that its projections $P_1, \ldots , P_n$ onto lines $A_1A_2, \ldots , A_nA_1$ respectively lie on the sides of the polygon. Prove that for points $X_1, \ldots , X_n$ on sides $A_1A_2, \ldots , A_nA_1$ respectively,
\[\max \left\{ \frac{X_1X_2}{P_1P_2}, \ldots, \frac{X_nX_1}{P_nP_1} \right\} \geq 1.\] if
a) $X_1, \ldots , X_n$ are the midpoints of the corressponding sides,
b) $X_1, \ldots , X_n$ are the feet of the corressponding altitudes,
c) $X_1, \ldots , X_n$ are arbitrary points on the corressponding lines.
Modified version of [url=https://artofproblemsolving.com/community/c6h418634p2361975]IMO 2010 SL G3[/url] (it was question c)
2025 Kyiv City MO Round 2, Problem 2
Mykhailo chose three distinct positive real numbers \( a, b, c \) and wrote the following numbers on the board:
\[
a + b, \quad b + c, \quad c + a, \quad ab, \quad bc, \quad ca.
\]
What is the minimum possible number of distinct numbers that can be written on the board?
[i]Proposed by Anton Trygub[/i]
2004 Postal Coaching, 18
Let $0 = a_1 < a_2 < a_3 < \cdots < a_n < 1$ and $0 = b_1 < b_2 < b_3 \cdots < b_m < 1$ be real numbers such that for no $a_j$ and $b_k$ the relation $a_j + b_k = 1$ is satisfied. Prove that if the $mn$ numbers ${\ a_j + b_k : 1 \leq j \leq n , 1 \leq k \leq m \}}$ are reduced modulo $1$, then at least $m+n -1$ residues will be distinct.
1997 AMC 8, 22
A two-inch cube $(2\times 2\times 2)$ of silver weighs 3 pounds and is worth \$200. How much is a three-inch cube of silver worth?
$\textbf{(A)}\ 300\text{ dollars} \qquad \textbf{(B)}\ 375\text{ dollars} \qquad \textbf{(C)}\ 450\text{ dollars} \qquad \textbf{(D)}\ 560\text{ dollars} \qquad \textbf{(E)}\ 675\text{ dollars}$
2006 AMC 12/AHSME, 18
An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?
$ \textbf{(A) } 120 \qquad \textbf{(B) } 121 \qquad \textbf{(C) } 221 \qquad \textbf{(D) } 230 \qquad \textbf{(E) } 231$
2010 District Olympiad, 2
Consider the matrix $ A,B\in \mathcal l{M}_3(\mathbb{C})$ with $ A=-^tA$ and $ B=^tB$. Prove that if the polinomial function defined by
\[ f(x)=\det(A+xB)\]
has a multiple root, then $ \det(A+B)=\det B$.
1997 All-Russian Olympiad Regional Round, 8.3
On sides $AB$ and $BC$ of an equilateral triangle $ABC$ are taken points $D$ and $K$, and on the side $AC$ , points $E$ and $M$ so that $DA + AE = KC +CM = AB$. Prove that the angle between lines $DM$ and $KE$ is equal to $60^o$.
2015 Federal Competition For Advanced Students, P2, 5
Let I be the incenter of triangle $ABC$ and let $k$ be a circle through the points $A$ and $B$. The circle intersects
* the line $AI$ in points $A$ and $P$
* the line $BI$ in points $B$ and $Q$
* the line $AC$ in points $A$ and $R$
* the line $BC$ in points $B$ and $S$
with none of the points $A,B,P,Q,R$ and $S$ coinciding and such that $R$ and $S$ are interior points of the line segments $AC$ and $BC$, respectively.
Prove that the lines $PS$, $QR$, and $CI$ meet in a single point.
(Stephan Wagner)
2002 Austrian-Polish Competition, 2
Let $P_{1}P_{2}\dots P_{2n}$ be a convex polygon with an even number of corners. Prove that there exists a diagonal $P_{i}P_{j}$ which is not parallel to any side of the polygon.