Found problems: 85335
2011 239 Open Mathematical Olympiad, 1
In the acute triangle $ABC$ on $AC$ point $P$ is chosen such that $2AP=BC$. Points $X$ and $Y$ are symmetric to $P$ wrt $A$ and $C$ respectively. It turned out that $BX=BY$. Find angle $C$.
1999 Brazil National Olympiad, 6
Given any triangle $ABC$, show how to construct $A'$ on the side $AB$, $B'$ on the side $BC$ and $C'$ on the side $CA$, such that $ABC$ and $A'B'C'$ are similar (with $\angle A = \angle A', \angle B = \angle B', \angle C = \angle C'$) and $A'B'C'$ has the least possible area.
2005 iTest, 28
Yoknapatawpha County has $500,000$ families. Each family is expected to continue to have children until it has a girl, at which point each family stops having children. If the probability of having a boy is $50\%$, and no families have either fertility problems or multiple children per birthing, how many families are expected to have at least $5$ children?
2020 Latvia Baltic Way TST, 6
For a natural number $n \ge 3$ we denote by $M(n)$ the minimum number of unit squares that must be coloured in a $6 \times n$ rectangle so that any possible $2 \times 3$ rectangle (it can be rotated, but it must be contained inside and cannot be cut) contains at least one coloured unit square. Is it true that for every natural $n \ge 3$ the number $M(n)$ can be expressed as $M(n)=p_n+k_n^3$, where $p_n$ is a prime and $k_n$ is a natural number?
2021 Science ON grade V, 2
There is a football championship with $6$ teams involved, such that for any $2$ teams $A$ and $B$, $A$ plays with $B$ and $B$ plays with $A$ ($2$ such games are distinct). After every match, the winning teams gains $3$ points, the loosing team gains $0$ points and if there is a draw, both teams gain $1$ point each.\\ \\
In the end, the team standing on the last place has $12$ points and there are no $2$ teams that scored the same amount of points.\\ \\
For all the remaining teams, find their final scores and provide an example with the outcomes of all matches for at least one of the possible final situations.
$\textit{(Andrei Bâra)}$
2020 CHMMC Winter (2020-21), 10
A research facility has $60$ rooms, numbered $1, 2, \dots 60$, arranged in a circle. The entrance is in room $1$ and the exit is in room $60$, and there are no other ways in and out of the facility. Each room, except for room $60$, has a teleporter equipped with an integer instruction $1 \leq i < 60$ such that it teleports a passenger exactly $i$ rooms clockwise.
On Monday, a researcher generates a random permutation of $1, 2, \dots, 60$ such that $1$ is the first integer in the permutation and $60$ is the last. Then, she configures the teleporters in the facility such that the rooms will be visited in the order of the permutation.
On Tuesday, however, a cyber criminal hacks into a randomly chosen teleporter, and he reconfigures its instruction by choosing a random integer $1 \leq j' < 60$ such that the hacked teleporter now teleports a passenger exactly $j'$ rooms clockwise (note that it is possible, albeit unlikely, that the hacked teleporter's instruction remains unchanged from Monday). This is a problem, since it is possible for the researcher, if she were to enter the facility, to be trapped in an endless \say{cycle} of rooms.
The probability that the researcher will be unable to exit the facility after entering in room $1$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2014 Belarus Team Selection Test, 2
Given positive real numbers $a,b,c$ with $ab+bc+ca\ge a+b+c$ , prove that $$(a + b + c)(ab + bc+ca) + 3abc \ge 4(ab + bc + ca).$$
(I. Gorodnin)
2023 Purple Comet Problems, 10
The figure below shows a smaller square within a larger square. Both squares have integer side lengths. The region inside the larger square but outside the smaller square has area $52$. Find the area of the larger square.
[img]https://cdn.artofproblemsolving.com/attachments/a/f/2cb8c70109196bf30f88aef0c53bbac07d6cc3.png[/img]
2021 2nd Memorial "Aleksandar Blazhevski-Cane", 4
Find all positive integers $n$ that have precisely $\sqrt{n+1}$ natural divisors.
2022 AIME Problems, 9
Let $\ell_A$ and $\ell_B$ be two distinct parallel lines. For positive integers $m$ and $n$, distinct points $A_1, A_2, \allowbreak A_3, \allowbreak \ldots, \allowbreak A_m$ lie on $\ell_A$, and distinct points $B_1, B_2, B_3, \ldots, B_n$ lie on $\ell_B$. Additionally, when segments $\overline{A_iB_j}$ are drawn for all $i=1,2,3,\ldots, m$ and $j=1,\allowbreak 2,\allowbreak 3, \ldots, \allowbreak n$, no point strictly between $\ell_A$ and $\ell_B$ lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when $m=7$ and $n=5$. The figure shows that there are 8 regions when $m=3$ and $n=2$.
[asy]
import geometry;
size(10cm);
draw((-2,0)--(13,0));
draw((0,4)--(10,4));
label("$\ell_A$",(-2,0),W);
label("$\ell_B$",(0,4),W);
point A1=(0,0),A2=(5,0),A3=(11,0),B1=(2,4),B2=(8,4),I1=extension(B1,A2,A1,B2),I2=extension(B1,A3,A1,B2),I3=extension(B1,A3,A2,B2);
draw(B1--A1--B2);
draw(B1--A2--B2);
draw(B1--A3--B2);
label("$A_1$",A1,S);
label("$A_2$",A2,S);
label("$A_3$",A3,S);
label("$B_1$",B1,N);
label("$B_2$",B2,N);
label("1",centroid(A1,B1,I1));
label("2",centroid(B1,I1,I3));
label("3",centroid(B1,B2,I3));
label("4",centroid(A1,A2,I1));
label("5",(A2+I1+I2+I3)/4);
label("6",centroid(B2,I2,I3));
label("7",centroid(A2,A3,I2));
label("8",centroid(A3,B2,I2));
dot(A1);
dot(A2);
dot(A3);
dot(B1);
dot(B2);
[/asy]
2012 China Western Mathematical Olympiad, 2
Define a sequence $\{a_n\}$ by\[a_0=\frac{1}{2},\ a_{n+1}=a_{n}+\frac{a_{n}^2}{2012}, (n=0,\ 1,\ 2,\ \cdots),\] find integer $k$ such that $a_{k}<1<a_{k+1}.$
(September 29, 2012, Hohhot)
2021 Alibaba Global Math Competition, 1
In a dance party initially there are $20$ girls and $22$ boys in the pool and infinitely many more girls and boys waiting outside. In each round, a participant is picked uniformly at random; if a girl is picked, then she invites a boy from the pool to dance and then both of them elave the party after the dance; while if a boy is picked, then he invites a girl and a boy from the waiting line and dance together. The three of them all stay after the dance. The party is over when there are only (two) boys left in the pool.
(a) What is the probability that the party never ends?
(b) Now the organizer of this party decides to reverse the rule, namely that if a girl is picked, then she invites a boy and a girl from the waiting line to dance and the three stay after the dance; while if a boy is picked, he invites a girl from the pool to dance and both leave after the dance. Still the party is over when there are only (two) boys left in the pool. What is the expected number of rounds until the party ends?
2014 Lusophon Mathematical Olympiad, 5
Find all quadruples of positive integers $(k,a,b,c)$ such that $2^k=a!+b!+c!$ and $a\geq b\geq c$.
2007 Nordic, 1
Find a solution to the equation $x^2-2x-2007y^2=0$ in positive integers.
2006 JBMO ShortLists, 8
Prove that there do not exist natural numbers $ n\ge 10$ having all digits different from zero, and such that all numbers which are obtained by permutations of its digits are perfect squares.
2017 ASDAN Math Tournament, 2
Let $f(x)=x^2$ and let $g(x)=x+1$. Let $h(x)=f(g(x))$. Compute $h'(1)$.
2011 Morocco National Olympiad, 3
Let $a$ and $b$ be two real numbers and let$M(a,b)=\max\left \{ 3a^{2}+2b; 3b^{2}+2a\right \}$. Find the values of $a$ and $b$ for which $M(a,b)$ is minimal.
2013 Harvard-MIT Mathematics Tournament, 30
How many positive integers $k$ are there such that \[\dfrac k{2013}(a+b)=lcm(a,b)\] has a solution in positive integers $(a,b)$?
2022 ITAMO, 2
Let $ABC$ be an acute triangle with $AB<AC$. Let then
• $D$ be the foot of the bisector of the angle in $A$,
• $E$ be the point on segment $BC$ (different from $B$) such that $AB=AE$,
• $F$ be the point on segment $BC$ (different from $B$) such that $BD=DF$,
• $G$ be the point on segment $AC$ such that $AB=AG$.
Prove that the circumcircle of triangle $EFG$ is tangent to line $AC$.
1999 Tournament Of Towns, 4
Points $K, L$ on sides $AC, CB$ respectively of a triangle $ABC$ are the points of contact of the excircles with the corresponding sides . Prove that the straight line through the midpoints of $KL$ and $AB$
(a) divides the perimeter of triangle $ABC$ in half,
(b) is parallel to the bisector of angle $ACB$.
( L Emelianov)
2012 AIME Problems, 1
Find the number of positive integers with three not necessarily distinct digits, $abc$, with $a \neq 0$, $c \neq 0$ such that both $abc$ and $cba$ are divisible by 4.
1967 IMO Longlists, 15
Suppose $\tan \alpha = \dfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove that the number $\tan \beta$ for which $\tan {2 \beta} = \tan {3 \alpha}$ is rational only when $p^2 + q^2$ is the square of an integer.
2021 Flanders Math Olympiad, 1
Johnny once saw plums hanging, like eggs so big and numbered according to the first natural numbers. He is the first to pick the plum with number $2$. After that, Jantje picks the plum each time with the smallest number $n$ that satisfies the following two conditions:
$\bullet$ $n$ is greater than all numbers on the already picked plums,
$\bullet$ $n$ is not the product of two equal or different numbers on already picked plums.
We call the numbers on the picked plums plum numbers. Is $100 000$ a plum number?
Justify your answer.
2016 Korea USCM, 8
For a $n\times n$ complex valued matrix $A$, show that the following two conditions are equivalent.
(i) There exists a $n\times n$ complex valued matrix $B$ such that $AB-BA=A$.
(ii) There exists a positive integer $k$ such that $A^k = O$. ($O$ is the zero matrix.)
2008 HMNT, 2
What is the units digit of $7^{2009}$?