Found problems: 85335
1985 Greece National Olympiad, 1
Inside triangle $ABC$ consider random point $O$. Prove that: $$E_A \overrightarrow{OA}+E_B \overrightarrow{OB}+E_C\overrightarrow{OC}=\overrightarrow{O}$$
where $E_A,E_B,E_C$ the areas of triangle $BOC, COB, AOB$ respectively
2020 Bangladesh Mathematical Olympiad National, Problem 7
$f$ is a function on the set of complex numbers such that $f(z)=1/(z*)$, where $z*$ is the complex conjugate of $z$. $S$ is the set of complex numbers $z$ such that the real part of $f(z)$ lies between $1/2020$ and $1/2018$. If $S$ is treated as a subset of the complex plane, the area of $S$ can be expressed as $m× \pi$ where $m$ is an integer. What is the value of $m$?
2024 Switzerland Team Selection Test, 10
Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.
2004 Poland - Second Round, 1
Find all positive integers $n$ which have exactly $\sqrt{n}$ positive divisors.
2001 Putnam, 4
Let $S$ denote the set of rational numbers different from $ \{ -1, 0, 1 \} $. Define $f: S \rightarrow S $ by $f(x)=x-1/x$. Prove or disprove that \[ \cap_{n=1}^{\infty} f^{(n)} (S) = \emptyset \] where $f^{(n)}$ denotes $f$ composed with itself $n$ times.
2013 Lusophon Mathematical Olympiad, 6
Consider a triangle $ABC$. Let $S$ be a circumference in the interior of the triangle that is tangent to the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$ respectively. In the exterior of the triangle we draw three circumferences $S_A$, $S_B$, $S_C$. The circumference $S_A$ is tangent to $BC$ at $L$ and to the prolongation of the lines $AB$, $AC$ at the points $M$, $N$ respectively. The circumference $S_B$ is tangent to $AC$ at $E$ and to the prolongation of the line $BC$ at $P$. The circumference $S_C$ is tangent to $AB$ at $F$ and to the prolongation of the line $BC$ at $Q$. Show that the lines $EP$, $FQ$ and $AL$ meet at a point of the circumference $S$.
2001 Croatia Team Selection Test, 1
Consider $A = \{1, 2, ..., 16\}$. A partition of $A$ into nonempty sets $A_1, A_2,..., A_n$ is said to be good if none of the Ai contains elements $a, b, c$ (not necessarily distinct) such that $a = b + c$.
(a) Find a good partition $\{A_1, A_2, A_3, A_4\}$ of $A$.
(b) Prove that no partition $\{A_1, A_2, A_3\}$ of $A$ is good
2005 Thailand Mathematical Olympiad, 5
A die is thrown six times. How many ways are there for the six rolls to sum to $21$?
1967 IMO Longlists, 6
Solve the system of equations:
$
\begin{matrix}
|x+y| + |1-x| = 6 \\ |x+y+1| + |1-y| = 4.
\end{matrix}
$
LMT Guts Rounds, 2021 S
[u]Round 5[/u]
[b]p13.[/b] Pieck the Frog hops on Pascal’s Triangle, where she starts at the number $1$ at the top. In a hop, Pieck can hop to one of the two numbers directly below the number she is currently on with equal probability. Given that the expected value of the number she is on after $7$ hops is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m+n$.
[b]p14.[/b] Maisy chooses a random set $(x, y)$ that satisfies $$x^2 + y^2 -26x -10y \le 482.$$ The probability that $y>0$ can be expressed as $\frac{A\pi -B\sqrt{C}}{D \pi}$. Find $A+B +C +D$.
[color=#f00]Due to the problem having a typo, all teams who inputted answers received points[/color]
[b]p15.[/b] $6$ points are located on a circle. How many ways are there to draw any number of line segments between the points such that none of the line segments overlap and none of the points are on more than one line segment? (It is possible to draw no line segments).
[u]Round 6[/u]
[b]p16.[/b] Find the number of $3$ by $3$ grids such that each square in the grid is colored white or black and no two black squares share an edge.
[b]p17.[/b] Let $ABC$ be a triangle with side lengths $AB = 20$, $BC = 25$, and $AC = 15$. Let $D$ be the point on BC such that $CD = 4$. Let $E$ be the foot of the altitude from $A$ to $BC$. Let $F$ be the intersection of $AE$ with the circle of radius $7$ centered at $A$ such that $F$ is outside of triangle $ABC$. $DF$ can be expressed as $\sqrt{m}$, where $m$ is a positive integer. Find $m$.
[b]p18.[/b] Bill and Frank were arrested under suspicion for committing a crime and face the classic Prisoner’s Dilemma. They are both given the choice whether to rat out the other and walk away, leaving their partner to face a $9$ year prison sentence. Given that neither of them talk, they both face a $3$ year sentence. If both of them talk, they both will serve a $6$ year sentence. Both Bill and Frank talk or do not talk with the same probabilities. Given the probability that at least one of them talks is $\frac{11}{36}$ , find the expected duration of Bill’s sentence in months.
[u]Round 7[/u]
[b]p19.[/b] Rectangle $ABCD$ has point $E$ on side $\overline{CD}$. Point $F$ is the intersection of $\overline{AC}$ and $\overline{BE}$. Given that the area of $\vartriangle AFB$ is $175$ and the area of $\vartriangle CFE$ is $28$, find the area of $ADEF$.
[b]p20.[/b] Real numbers $x, y$, and $z$ satisfy the system of equations
$$5x+ 13y -z = 100,$$
$$25x^2 +169y^2 -z2 +130x y= 16000,$$
$$80x +208y-2z = 2020.$$
Find the value of $x yz$.
[color=#f00]Due to the problem having infinitely many solutions, all teams who inputted answers received points.
[/color]
[b]p21.[/b] Bob is standing at the number $1$ on the number line. If Bob is standing at the number $n$, he can move to $n +1$, $n +2$, or $n +4$. In howmany different ways can he move to the number $10$?
[u]Round 8[/u]
[b]p22.[/b] A sequence $a_1,a_2,a_3, ...$ of positive integers is defined such that $a_1 = 4$, and for each integer $k \ge 2$, $$2(a_{k-1} +a_k +a_{k+1}) = a_ka_{k-1} +8.$$ Given that $a_6 = 488$, find $a_2 +a_3 +a_4 +a_5$.
[b]p23.[/b] $\overline{PQ}$ is a diameter of circle $\omega$ with radius $1$ and center $O$. Let $A$ be a point such that $AP$ is tangent to $\omega$. Let $\gamma$ be a circle with diameter $AP$. Let $A'$ be where $AQ$ hits the circle with diameter $AP$ and $A''$ be where $AO$ hits the circle with diameter $OP$. Let $A'A''$ hit $PQ$ at $R$. Given that the value of the length $RA'$ is is always less than $k$ and $k$ is minimized, find the greatest integer less than or equal to $1000k$.
[b]p24.[/b] You have cards numbered $1,2,3, ... ,100$ all in a line, in that order. You may swap any two adjacent cards at any time. Given that you make ${100 \choose 2}$ total swaps, where you swap each distinct pair of cards exactly once, and do not do any swaps simultaneously, find the total number of distinct possible final orderings of the cards.
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166472p28814057]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166480p28814155]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2023 Switzerland Team Selection Test, 7
Find all monic polynomials $P(x)=x^{2023}+a_{2022}x^{2022}+\ldots+a_1x+a_0$ with real coefficients such that $a_{2022}=0$, $P(1)=1$ and all roots of $P$ are real and less than $1$.
1998 IMC, 1
$V$ is a real vector space and $ f, f_{i}: V \rightarrow \mathbb{R} $ are linear for $i = 1, 2, ... , k.$ Also $f $ is zero at all points for which all of $ f_{i }$ are zero. Show that $ f $ is a linear combination of the $f_{i}$.
2017 China Western Mathematical Olympiad, 8
Let $a_1,a_2,\cdots,a_n>0$ $(n\geq 2)$. Prove that$$\sum_{i=1}^n max\{a_1,a_2,\cdots,a_i \} \cdot min
\{a_i,a_{i+1},\cdots,a_n\}\leq \frac{n}{2\sqrt{n-1}}\sum_{i=1}^n a^2_i$$
Kvant 2020, M2617
The points in the plane are painted in 100 colors. Prove that there are three points of the same color that are the vertices of a triangle of area 1.
[i]Proposed by V. Bragin[/i]
2021-IMOC, G2
Let the midline of $\triangle ABC$ parallel to $BC$ intersect the circumcircle $\Gamma$ of $\triangle ABC$ at $P$, $Q$, and the tangent of $\Gamma$ at $A$ intersects $BC$ at $T$. Show that $\measuredangle BTQ = \measuredangle PTA$.
2009 Stanford Mathematics Tournament, 8
Three points are randomly placed on a circle. What is the probability that they lie on the same semicircle
2023 Grosman Mathematical Olympiad, 2
A "Hishgad" lottery ticket contains the numbers $1$ to $mn$, arranged in some order in a table with $n$ rows and $m$ columns. It is known that the numbers in each row increase from left to right and the numbers in each column increase from top to bottom. An example for $n=3$ and $m=4$:
[asy]
size(3cm);
Label[][] numbers = {{"$1$", "$2$", "$3$", "$9$"}, {"$4$", "$6$", "$7$", "$10$"}, {"$5$", "$8$", "$11$", "$12$"}};
for (int i=0; i<5;++i) {
draw((i,0)--(i,3));
}
for (int i=0; i<4;++i) {
draw((0,i)--(4,i));
}
for (int i=0; i<4;++i){
for (int j=0; j<3;++j){
label(numbers[2-j][i], (i+0.5, j+0.5));
}}
[/asy]
When the ticket is bought the numbers are hidden, and one must "scratch" the ticket to reveal them. How many cells does it always suffice to reveal in order to determine the whole table with certainty?
2016 USAMO, 2
Prove that for any positive integer $k$, \[(k^2)!\cdot\displaystyle\prod_{j=0}^{k-1}\frac{j!}{(j+k)!}\]is an integer.
2006 Switzerland Team Selection Test, 1
In the triangle $A,B,C$, let $D$ be the middle of $BC$ and $E$ the projection of $C$ on $AD$. Suppose $\angle ACE = \angle ABC$. Show that the triangle $ABC$ is isosceles or rectangle.
2011 Argentina National Olympiad Level 2, 5
Let $a$ and $b$ be integers such that the remainder of dividing $a$ by $17$ is equal to the remainder of dividing $b$ by $19$, and the remainder of dividing $a$ by $19$ is equal to the remainder of dividing $b$ by $17$. Determine the possible values of the remainder of $a + b$ when divided by $323$.
2003 Balkan MO, 4
A rectangle $ABCD$ has side lengths $AB = m$, $AD = n$, with $m$ and $n$ relatively prime and both odd. It is divided into unit squares and the diagonal AC intersects the sides of the unit squares at the points $A_1 = A, A_2, A_3, \ldots , A_k = C$. Show that \[ A_1A_2 - A_2A_3 + A_3A_4 - \cdots + A_{k-1}A_k = {\sqrt{m^2+n^2}\over mn}. \]
1991 All Soviet Union Mathematical Olympiad, 553
The chords $AB$ and $CD$ of a sphere intersect at $X. A, C$ and $X$ are equidistant from a point $Y$ on the sphere. Show that $BD$ and $XY$ are perpendicular.
1992 IMO Longlists, 47
Evaluate
\[\left \lfloor \ \prod_{n=1}^{1992} \frac{3n+2}{3n+1} \ \right \rfloor\]
2017 Bosnia And Herzegovina - Regional Olympiad, 3
Find prime numbers $p$, $q$, $r$ and $s$, pairwise distinct, such that their sum is prime number and numbers $p^2+qr$ and $p^2+qs$ are perfect squares
2020 AMC 10, 6
Driving along a highway, Megan noticed that her odometer showed $15951$ (miles). This number is a palindrome—it reads the same forward and backward. Then $2$ hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this $2$-hour period?
$\textbf{(A) }50 \qquad \textbf{(B) }55 \qquad \textbf{(C) }60\qquad \textbf{(D) }65\qquad\textbf{(E) }70$