Found problems: 85335
2021 Azerbaijan IMO TST, 1
Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$.
Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.
2020 ABMC, 2020 Nov
[b]p1.[/b] A large square is cut into four smaller, congruent squares. If each of the smaller squares has perimeter $4$, what was the perimeter of the original square?
[b]p2.[/b] Pie loves to bake apples so much that he spends $24$ hours a day baking them. If Pie bakes a dozen apples in one day, how many minutes does it take Pie to bake one apple, on average?
[b]p3.[/b] Bames Jond is sent to spy on James Pond. One day, Bames sees James type in his $4$-digit phone password. Bames remembers that James used the digits $0$, $5$, and $9$, and no other digits, but he does not remember the order. How many possible phone passwords satisfy this condition?
[b]p4.[/b] What do you get if you square the answer to this question, add $256$ to it, and then divide by $32$?
[b]p5.[/b] Chloe the Horse and Flower the Chicken are best friends. When Chloe gets sad for any reason, she calls Flower, so Chloe must remember Flower's $3$ digit phone number, which can consist of any digits $0-5$. Given that the phone number's digits are unique and add to $5$, the number does not start with $0$, and the $3$ digit number is prime, what is the sum of all possible phone numbers?
[b]p6.[/b] Anuj has a circular pizza with diameter $A$ inches, which is cut into $B$ congruent slices, where $A$,$B$ are positive integers. If one of Anuj's pizza slices has a perimeter of $3\pi + 30$ inches, find $A + B$.
[b]p7.[/b] Bob really likes to study math. Unfortunately, he gets easily distracted by messages sent by friends. At the beginning of every minute, there is an $\frac{6}{10}$ chance that he will get a message from a friend. If Bob does get a message from a friend, there is a $\frac{9}{10}$ chance that he will look at the message, causing him to waste $30$ seconds before resuming his studying. If Bob doesn't get a message from a friend, there is a $\frac{3}{10}$ chance Bob will still check his messages hoping for a message from his friends, wasting $10$ seconds before he resumes his studying. What is the expected number of minutes in $100$ minutes for which Bob will be studying math?
[b]p8.[/b] Suppose there is a positive integer $n$ with $225$ distinct positive integer divisors. What is the minimum possible number of divisors of n that are perfect squares?
[b]p9.[/b] Let $a, b, c$ be positive integers. $a$ has $12$ divisors, $b$ has $8$ divisors, $c$ has $6$ divisors, and $lcm(a, b, c) = abc$. Let $d$ be the number of divisors of $a^2bc$. Find the sum of all possible values of $d$.
[b]p10.[/b] Let $\vartriangle ABC$ be a triangle with side lengths $AB = 17$, $BC = 28$, $AC = 25$. Let the altitude from $A$ to $BC$ and the angle bisector of angle $B$ meet at $P$. Given the length of $BP$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, $c$ where $gcd(a, c) = 1$ and $b$ is not divisible by the square of any prime, find $a + b + c$.
[b]p11.[/b] Let $a$, $b$, and $c$ be the roots of the cubic equation $x^3-5x+3 = 0$. Let $S = a^4b+ab^4+a^4c+ac^4+b^4c+bc^4$. Find $|S|$.
[b]p12.[/b] Call a number palindromeish if changing a single digit of the number into a different digit results in a new six-digit palindrome. For example, the number $110012$ is a palindromeish number since you can change the last digit into a $1$, which results in the palindrome $110011$. Find the number of $6$ digit palindromeish numbers.
[b]p13.[/b] Let $P(x)$ be a polynomial of degree $3$ with real coecients and leading coecient $1$. Let the roots of $P(x)$ be $a$, $b$, $c$. Given that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}= 4$ and $a^2 + b^2 + c^2 = 36$, the coefficient of $x^2$ is negative, and $P(1) = 2$, let the $S$ be the sum of possible values of $P(0)$. Then $|S|$ can be expressed as $\frac{a + b\sqrt{c}}{d}$ for positive integers $a$, $b$, $c$, $d$ such that $gcd(a, b, d) = 1$ and $c$ is not divisible by the square of any prime. Find $a + b + c + d$.
[b]p14.[/b] Let $ABC$ be a triangle with side lengths $AB = 7$, $BC = 8$, $AC = 9$. Draw a circle tangent to $AB$ at $B$ and passing through $C$. Let the center of the circle be $O$. The length of $AO$ can be expressed as $\frac{a\sqrt{b}}{c\sqrt{d}}$ for positive integers $a$, $b$, $c$, $d$ where $gcd(a, c) = gcd(b, d) = 1$ and $b$,$ d$ are not divisible by the square of any prime. Find $a + b + c + d$.
[b]p15.[/b] Many students in Mr. Noeth's BC Calculus class missed their first test, and to avoid taking a makeup, have decided to never leave their houses again. As a result, Mr. Noeth decides that he will have to visit their houses to deliver the makeup tests. Conveniently, the $17$ absent students in his class live in consecutive houses on the same street. Mr. Noeth chooses at least three of every four people in consecutive houses to take a makeup. How many ways can Mr. Noeth select students to take makeups?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2022 Yasinsky Geometry Olympiad, 6
Let $AD$, $BE$ and $CF$ be the diameters of the circle circumscribed around the acute angle triangle $ABC$. Point $N$ is the midpoint of the arc $CAD$, and point $M$ is the midpoint of arc $BAD$. Prove that the lines $EN$ and $MF$ intersect at the angle bisector of $\angle BAC$.
(Matvii Kurskyi)
2006 AMC 10, 19
A circle of radius 2 is centered at $ O$. Square $ OABC$ has side length 1. Sides $ \overline{AB}$ and $ \overline{CB}$ are extended past $ b$ to meet the circle at $ D$ and $ E$, respectively. What is the area of the shaded region in the figure, which is bounded by $ \overline{BD}$, $ \overline{BE}$, and the minor arc connecting $ D$ and $ E$?
[asy]
defaultpen(linewidth(0.8));
pair O=origin, A=(1,0), C=(0,1), B=(1,1), D=(1, sqrt(3)), E=(sqrt(3), 1), point=B;
fill(Arc(O, 2, 0, 90)--O--cycle, mediumgray);
clip(B--Arc(O, 2, 30, 60)--cycle);
draw(Circle(origin, 2));
draw((-2,0)--(2,0)^^(0,-2)--(0,2));
draw(A--D^^C--E);
label("$A$", A, dir(point--A));
label("$C$", C, dir(point--C));
label("$O$", O, dir(point--O));
label("$D$", D, dir(point--D));
label("$E$", E, dir(point--E));
label("$B$", B, SW);[/asy]
$ \textbf{(A) } \frac {\pi}3 \plus{} 1 \minus{} \sqrt {3} \qquad \textbf{(B) } \frac {\pi}2\left( 2 \minus{} \sqrt {3}\right) \qquad \textbf{(C) } \pi\left(2 \minus{} \sqrt {3}\right) \qquad \textbf{(D) } \frac {\pi}{6} \plus{} \frac {\sqrt {3} \minus{} 1}{2} \\
\qquad \indent \textbf{(E) } \frac {\pi}{3} \minus{} 1 \plus{} \sqrt {3}$
Cono Sur Shortlist - geometry, 2021.G2
Let $ABC$ be an acute triangle. Define $A_1$ the midpoint of the largest arc $BC$ of the circumcircle of $ABC$ . Let $A_2$ and $A_3$ be the feet of the perpendiculars from $A_1$ on the lines $AB$ and $AC$ , respectively. Define $B_1$, $B_2$, $B_3$, $C_1$, $C_2$, and $C_3$ analogously. Show that the lines $A_2A_3$, $B_2B_3$, $C_2C_3$ are concurrent.
2018 Thailand TST, 2
Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.
2010 Middle European Mathematical Olympiad, 5
Three strictly increasing sequences
\[a_1, a_2, a_3, \ldots,\qquad b_1, b_2, b_3, \ldots,\qquad c_1, c_2, c_3, \ldots\]
of positive integers are given. Every positive integer belongs to exactly one of the three sequences. For every positive integer $n$, the following conditions hold:
(a) $c_{a_n}=b_n+1$;
(b) $a_{n+1}>b_n$;
(c) the number $c_{n+1}c_{n}-(n+1)c_{n+1}-nc_n$ is even.
Find $a_{2010}$, $b_{2010}$ and $c_{2010}$.
[i](4th Middle European Mathematical Olympiad, Team Competition, Problem 1)[/i]
2005 Portugal MO, 3
On a board with $a$ rows and $b$ columns, each square has a switch and an unlit light bulb. By pressing the switch of a house, the lamp in that house changes state, along with the lamps in the same row and those in the same column (those that are on go out and the that are off light up). For what values of $a$ and $b$ is it possible to have just one lamp on, by pressing a series of switches?
2008 AMC 10, 3
For the positive integer $n$, let $\left< n \right>$ denote the sum of all the positive divisors of $n$ with the exception of $n$ itself. For example, $\left<4\right> = 1+2=3$ and $\left<12\right>=1+2+3+4+6=16$ What is $\left< \left< \left< 6 \right>\right>\right>$?
$ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 32 \qquad \textbf{(E)}\ 36$
2009 AMC 10, 7
A carton contains milk that is $ 2\%$ fat, and amount that is $ 40\%$ less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk?
$ \textbf{(A)}\ \frac{12}{5} \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ \frac{10}{3} \qquad
\textbf{(D)}\ 38 \qquad
\textbf{(E)}\ 42$
2000 National High School Mathematics League, 14
Function $f(x)=-\frac{1}{2}x^2+\frac{13}{2}$. If the minumum and maximum value of $f(x)$ are $2a$ and $2b$ respectively on $[a,b]$. Find $a,b$.
1995 IMO Shortlist, 4
An acute triangle $ ABC$ is given. Points $ A_1$ and $ A_2$ are taken on the side $ BC$ (with $ A_2$ between $ A_1$ and $ C$), $ B_1$ and $ B_2$ on the side $ AC$ (with $ B_2$ between $ B_1$ and $ A$), and $ C_1$ and $ C_2$ on the side $ AB$ (with $ C_2$ between $ C_1$ and $ B$) so that
\[ \angle AA_1A_2 \equal{} \angle AA_2A_1 \equal{} \angle BB_1B_2 \equal{} \angle BB_2B_1 \equal{} \angle CC_1C_2 \equal{} \angle CC_2C_1.\]
The lines $ AA_1,BB_1,$ and $ CC_1$ bound a triangle, and the lines $ AA_2,BB_2,$ and $ CC_2$ bound a second triangle. Prove that all six vertices of these two triangles lie on a single circle.
2011 Poland - Second Round, 1
For $x,y\in\mathbb{R}$, solve the system of equations
\[ \begin{cases} (x-y)(x^3+y^3)=7 \\ (x+y)(x^3-y^3)=3 \end{cases} \]
1979 IMO Longlists, 78
Denote the number of different prime divisors of the number $n$ by $\omega (n)$, where $n$ is an integer greater than $1$. Prove that there exist infinitely many numbers $n$ for which $\omega (n)< \omega (n+1)<\omega (n+2)$ holds.
2015 BMT Spring, 13
On a $2\times 40$ chessboard colored black and white in the standard alternating pattern, $20$ rooks are placed randomly on the black squares. The expected number of white squares with only rooks as neighbors can be expressed as $a/b$, where $a$ and $b$ are coprime positive integers. What is $a + b$? (Two squares are said to be neighbors if they share an edge.)
2018 IFYM, Sozopol, 8
The row $x_1, x_2,…$ is defined by the following recursion
$x_1=1$ and $x_{n+1}=x_n+\sqrt{x_n}$
Prove that
$\sum_{n=1}^{2018}{\frac{1}{x_n}}<3$.
2011 Middle European Mathematical Olympiad, 4
Let $k$ and $m$, with $k > m$, be positive integers such that the number $km(k^2 - m^2)$ is divisible by $k^3 - m^3$. Prove that $(k - m)^3 > 3km$.
2007 Portugal MO, 3
Determines the largest integer $n$ that is a multiple of all positive integers less than $\sqrt{n}$.
2016 District Olympiad, 2
For any natural number $ n, $ denote $ x_n $ as being the number of natural numbers of $ n $ digits that are divisible by $ 4 $ and formed only with the digits $ 0,1,2 $ or $ 6. $
[b]a)[/b] Calculate $ x_1,x_2,x_3,x_4. $
[b]b)[/b] Find the natural number $ m $ such that
$$ 1+\left\lfloor \frac{x_2}{x_1}\right\rfloor +\left\lfloor \frac{x_3}{x_2}\right\rfloor +\left\lfloor \frac{x_4}{x_3}\right\rfloor +\cdots +\left\lfloor \frac{x_{m+1}}{x_m}\right\rfloor =2016 , $$
where $ \lfloor\rfloor $ is the usual integer part.
2019 Harvard-MIT Mathematics Tournament, 4
Let $\mathbb{N}$ be the set of positive integers, and let $f: \mathbb{N} \to \mathbb{N}$ be a function satisfying
[list]
[*] $f(1) = 1$,
[*] for $n \in \mathbb{N}$, $f(2n) = 2f(n)$ and $f(2n+1) = 2f(n) - 1$.
[/list]
Determine the sum of all positive integer solutions to $f(x) = 19$ that do not exceed 2019.
1950 Miklós Schweitzer, 3
Let $ E$ be a system of $ n^2 \plus{} 1$ closed intervals of the real line. Show that $ E$ has either a subsystem consisting of $ n \plus{} 1$ elements which are monotonically ordered with respect to inclusion or a subsystem consisting of $ n \plus{} 1$ elements none of which contains another element of the subsystem.
1968 Leningrad Math Olympiad, grade 8
[b]8.1[/b] In the parallelogram $ABCD$ , the diagonal $AC$ is greater than the diagonal $BD$. The point $M$ on the diagonal $AC$ is such that around the quadrilateral $BCDM$ one can circumscribe a circle. Prove that $BD$ is the common tangent of the circles circumscribed around the triangles $ABM$ and $ADM$.
[img]https://cdn.artofproblemsolving.com/attachments/b/3/9f77ff1f2198c201e5c270ec5b091a9da4d0bc.png[/img]
[b]8.2 [/b] $A$ is an odd integer, $x$ and $y$ are roots of equation $t^2+At-1=0$. Prove that $x^4 + y^4$ and $x^5+ y^5$ are coprime integer numbers.
[b]8.3[/b] A regular triangle is reflected symmetrically relative to one of its sides. The new triangle is again reflected symmetrically about one of its sides. This is repeated several times. It turned out that the resulting triangle coincides with the original one. Prove that an even number of reflections were made.
[b]8.4 /7.6[/b] Several circles are arbitrarily placed in a circle of radius $3$, the sum of their radii is $25$. Prove that there is a straight line that intersects at least $9$ of these circles.
[b]8.5 [/b] All two-digit numbers that do not end in zero are written one after another so that each subsequent number begins with that the same digit with which the previous number ends. Prove that you can do this and find the sum of the largest and smallest of all multi-digit numbers that can be obtained in this way.
[url=https://artofproblemsolving.com/community/c6h3390996p32049528]8,6*[/url] (asterisk problems in separate posts)
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here[/url].
Kvant 2022, M2712
Let $ABC$ be a triangle, with $\angle A=\alpha,\angle B=\beta$ and $\angle C=\gamma$. Prove that \[\sum_{\text{cyc}}\tan \frac{\alpha}{2}\tan\frac{\beta}{2}\cot\frac{\gamma}{2}\geqslant\sqrt{3}.\][i]Proposed by R. Regimov (Azerbaijan)[/i]
2015 All-Russian Olympiad, 6
Let a,b,c,d be real numbers satisfying $|a|,|b|,|c|,|d|>1$ and $abc+abd+acd+bcd+a+b+c+d=0$. Prove that $\frac {1} {a-1}+\frac {1} {b-1}+ \frac {1} {c-1}+ \frac {1} {d-1} >0$
1985 Traian Lălescu, 1.4
Let $ ABC $ a right triangle in $ A. $ Let $ D $ a point on the segment $ AC, $ and $ E,F $ the projections of $ A $ upon the lines $ BD, $ respectively, $ BC. $ Show that the quadrilateral $ CDEF $ is concyclic.