Found problems: 85335
2007 Iran MO (3rd Round), 2
a) Let $ ABC$ be a triangle, and $ O$ be its circumcenter. $ BO$ and $ CO$ intersect with $ AC,AB$ at $ B',C'$. $ B'C'$ intersects the circumcircle at two points $ P,Q$. Prove that $ AP\equal{}AQ$ if and only if $ ABC$ is isosceles.
b) Prove the same statement if $ O$ is replaced by $ I$, the incenter.
2007 Stanford Mathematics Tournament, 14
Let there be 50 natural numbers $ a_i$ such that $ 0 < a_1 < a_2 < ... < a_{50} < 150$. What is the greatest possible sum of the differences $ d_j$ where each $ d_j \equal{} a_{j \plus{} 1} \minus{} a_j$?
2017 Ecuador Juniors, 1
An ancient Inca legend tells that a monster lives among the mountains that when wakes up, eats everyone who read this issue. After such a task, the monster returns to the mountains and sleeps for a number of years equal to the sum of its digits of the year in which you last woke up. The monster woke up for the first time in the year $234$.
a) Would the monster have woken up between the years $2005$ and $2015$?
b) Will we be safe in the next $10$ years?
2000 Kazakhstan National Olympiad, 7
Is there any function $ f : \mathbb{R}\to\mathbb{R}$ satisfying following conditions:
$1) f(0) = 1$
$2) f(x+f(y)) = f(x+y) + 1$, for all $x,y \to\mathbb{R} $
$3)$ there exist rational, but not integer $x_0$, such $f(x_0)$ is integer
2006 Stanford Mathematics Tournament, 1
After a cyclist has gone $ \frac{2}{3}$ of his route, he gets a flat tire. Finishing on foot, he spends twice as long walking as he did riding. How many times as fast does he ride as walk?
2009 Iran MO (3rd Round), 3
An arbitary triangle is partitioned to some triangles homothetic with itself. The ratio of homothety of the triangles can be positive or negative.
Prove that sum of all homothety ratios equals to $1$.
Time allowed for this problem was 45 minutes.
2007 Princeton University Math Competition, 6
A sphere of radius $\sqrt{85}$ is centered at the origin in three dimensions. A tetrahedron with vertices at integer lattice points is inscribed inside the sphere. What is the maximum possible volume of this tetrahedron?
2010 CHMMC Winter, 2
In the following diagram, points $E, F, G, H, I$, and $J$ lie on a circle. The triangle $ABC$ has side lengths $AB = 6$, $BC = 7$, and $CA = 9$. The three chords have lengths $EF = 12$, $GH = 15$, and $IJ = 16$. Compute $6 \cdot AE + 7 \cdot BG + 9 \cdot CI$.
[img]https://cdn.artofproblemsolving.com/attachments/2/7/661b3d6a0f0baac0cd3b8d57c4cd4c62eeab46.png[/img]
1961 Putnam, A1
The graph of the equation $x^y =y^x$ in the first quadrant consists of a straight line and a curve. Find the coordinates of the intersection of the line and the curve.
2009 AMC 12/AHSME, 4
A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. THe remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $ 15$ and $ 25$ meters. What fraction of the yard is occupied by the flower beds?
[asy]unitsize(2mm);
defaultpen(linewidth(.8pt));
fill((0,0)--(0,5)--(5,5)--cycle,gray);
fill((25,0)--(25,5)--(20,5)--cycle,gray);
draw((0,0)--(0,5)--(25,5)--(25,0)--cycle);
draw((0,0)--(5,5));
draw((20,5)--(25,0));[/asy]$ \textbf{(A)}\ \frac18\qquad
\textbf{(B)}\ \frac16\qquad
\textbf{(C)}\ \frac15\qquad
\textbf{(D)}\ \frac14\qquad
\textbf{(E)}\ \frac13$
2021 China Second Round Olympiad, Problem 15
Positive real numbers $x, y, z$ satisfy $\sqrt x + \sqrt y + \sqrt z = 1$. Prove that $$\frac{x^4+y^2z^2}{x^{\frac 52}(y+z)} + \frac{y^4+z^2x^2}{y^{\frac 52}(z+x)} + \frac{z^4+y^2x^2}{z^{\frac 52}(y+x)} \geq 1.$$
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 15)[/i]
2011 Belarus Team Selection Test, 2
Find the least positive integer $n$ for which there exists a set $\{s_1, s_2, \ldots , s_n\}$ consisting of $n$ distinct positive integers such that
\[ \left( 1 - \frac{1}{s_1} \right) \left( 1 - \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) = \frac{51}{2010}.\]
[i]Proposed by Daniel Brown, Canada[/i]
2023 Harvard-MIT Mathematics Tournament, 10
The number $$316990099009901=\frac{32016000000000001}{101}$$ is the product of two distinct prime numbers. Compute the smaller of these two primes.
2014 Greece National Olympiad, 4
We are given a circle $c(O,R)$ and two points $A,B$ so that $R<AB<2R$.The circle $c_1 (A,r)$ ($0<r<R$) crosses the circle $c$ at C,D ($C$ belongs to the short arc $AB$).From $B$ we consider the tangent lines $BE,BF$ to the circle $c_1$ ,in such way that $E$ lays out of the circle $c$.If $M\equiv EC\cap DF$ show that the quadrilateral $BCFM$ is cyclic.
2010 Stanford Mathematics Tournament, 2
Find the smallest prime $p$ such that the digits of $p$ (in base 10) add up to a prime number greater than $10$.
2004 Bulgaria Team Selection Test, 1
The points $P$ and $Q$ lie on the diagonals $AC$ and $BD$, respectively, of a quadrilateral $ABCD$ such that $\frac{AP}{AC} + \frac{BQ}{BD} =1$. The line $PQ$ meets the sides $AD$ and $BC$ at points $M$ and $N$. Prove that the circumcircles of the triangles $AMP$, $BNQ$, $DMQ$, and $CNP$ are concurrent.
1994 Poland - First Round, 3
A quadrilateral with sides $a,b,c,d$ is inscribed in a circle of radius $R$. Prove that if $a^2+b^2+c^2+d^2=8R^2$, then either one of the angles of the quadrilateral is right or the diagonals of the quadrilateral are perpendicular.
2006 AMC 10, 15
Odell and Kershaw run for 30 minutes on a circular track. Odell runs clockwise at 250 m/min and uses the inner lane with a radius of 50 meters. Kershaw runs counterclockwise at 300 m/min and uses the outer lane with a radius of 60 meters, starting on the same radial line as Odell. How many times after the start do they pass each other?
$ \textbf{(A) } 29 \qquad \textbf{(B) } 42 \qquad \textbf{(C) } 45 \qquad \textbf{(D) } 47 \qquad \textbf{(E) } 50$
2020 Balkan MO, 2
Denote $\mathbb{Z}_{>0}=\{1,2,3,...\}$ the set of all positive integers. Determine all functions $f:\mathbb{Z}_{>0}\rightarrow \mathbb{Z}_{>0}$ such that, for each positive integer $n$,
$\hspace{1cm}i) \sum_{k=1}^{n}f(k)$ is a perfect square, and
$\vspace{0.1cm}$
$\hspace{1cm}ii) f(n)$ divides $n^3$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2023 SAFEST Olympiad, 3
A binoku is a $9 \times 9$ grid that is divided into nine $3 \times 3$ subgrids with the following properties:
- each cell contains either a $0$ or a $1$,
- each row contains at least one $0$ and at least one $1$,
- each column contains at least one $0$ and at least one $1$, and
- each of the nine subgrids contains at least one $0$ and at least one $1$.
An incomplete binoku is obtained from a binoku by removing the numbers from some of the cells. What is the largest number of empty cells that an incomplete binoku can contain if it can be completed into a binoku in a unique way?
[i]Proposed by Stijn Cambie, South Korea[/i]
2002 Tuymaada Olympiad, 1
A positive integer $c$ is given. The sequence $\{p_{k}\}$ is constructed by the following rule: $p_{1}$ is arbitrary prime and for $k\geq 1$ the number $p_{k+1}$ is any prime divisor of $p_{k}+c$ not present among the numbers $p_{1}$, $p_{2}$, $\dots$, $p_{k}$. Prove that the sequence $\{p_{k}\}$ cannot be infinite.
[i]Proposed by A. Golovanov[/i]
PEN H Problems, 4
Find all pairs $(x, y)$ of positive rational numbers such that $x^{2}+3y^{2}=1$.
2022 Junior Balkan Team Selection Tests - Romania, P4
Let $n$ be a positive integer with $d^2$ positive divisors. We fill a $d\times d$ board with these divisors. At a move, we can choose a row, and shift the divisor from the $i^{\text{th}}$ column to the $(i+1)^{\text{th}}$ column, for all $i=1,2,\ldots, d$ (indices reduced modulo $d$).
A configuration of the $d\times d$ board is called [i]feasible[/i] if there exists a column with elements $a_1,a_2,\ldots,a_d,$ in this order, such that $a_1\mid a_2\mid\ldots\mid a_d$ or $a_d\mid a_{d-1}\mid\ldots\mid a_1.$ Determine all values of $n$ for which ragardless of how we initially fill the board, we can reach a feasible configuration after a finite number of moves.
2015 Caucasus Mathematical Olympiad, 3
Petya bought one cake, two cupcakes and three bagels, Apya bought three cakes and a bagel, and Kolya bought six cupcakes. They all paid the same amount of money for purchases. Lena bought two cakes and two bagels. And how many cupcakes could be bought for the same amount spent to her?
2014-2015 SDML (High School), 7
Let $S$ be a finite set of real numbers such that given any three distinct elements $x,y,z\in\mathbb{S}$, at least one of $x+y$, $x+z$, or $y+z$ is also contained in $S$. Find the largest possible number of elements that $S$ could have.