Found problems: 85335
2024 Brazil EGMO TST, 1
Decide whether there exists a positive real number \( a < 1 \) such that, for any positive real numbers \( x \) and \( y \), the inequality
\[
\frac{2xy^2}{x^2 + y^2} \leq (1 - a)x + ay
\]
holds true.
MBMT Team Rounds, 2019
[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide]
[b]D1.[/b] What is the solution to the equation $3 \cdot x \cdot 5 = 4 \cdot 5 \cdot 6$?
[b]D2.[/b] Mr. Rose is making Platonic solids! If there are five different types of Platonic solids, and each Platonic solid can be one of three colors, how many different colored Platonic solids can Mr. Rose make?
[b]D3.[/b] What fraction of the multiples of $5$ between $1$ and $100$ inclusive are also multiples of $20$?
[b]D4.[/b] What is the maximum number of times a circle can intersect a triangle?
[b]D5 / L1.[/b] At an interesting supermarket, the nth apple you purchase costs $n$ dollars, while pears are $3$ dollars each. Given that Layla has exactly enough money to purchase either $k$ apples or $2k$ pears for $k > 0$, how much money does Layla have?
[b]D6 / L3.[/b] For how many positive integers $1 \le n \le 10$ does there exist a prime $p$ such that the sum of the digits of $p$ is $n$?
[b]D7 / L2.[/b] Real numbers $a, b, c$ are selected uniformly and independently at random between $0$ and $1$. What is the probability that $a \ge b \le c$?
[b]D8.[/b] How many ordered pairs of positive integers $(x, y)$ satisfy $lcm(x, y) = 500$?
[b]D9 / L4.[/b] There are $50$ dogs in the local animal shelter. Each dog is enemies with at least $2$ other dogs. Steven wants to adopt as many dogs as possible, but he doesn’t want to adopt any pair of enemies, since they will cause a ruckus. Considering all possible enemy networks among the dogs, find the maximum number of dogs that Steven can possibly adopt.
[b]D10 / L7.[/b] Unit circles $a, b, c$ satisfy $d(a, b) = 1$, $d(b, c) = 2$, and $d(c, a) = 3,$ where $d(x, y)$ is defined to be the minimum distance between any two points on circles $x$ and $y$. Find the radius of the smallest circle entirely containing $a$, $b$, and $c$.
[b]D11 / L8.[/b] The numbers $1$ through $5$ are written on a chalkboard. Every second, Sara erases two numbers $a$ and $b$ such that $a \ge b$ and writes $\sqrt{a^2 - b^2}$ on the board. Let M and m be the maximum and minimum possible values on the board when there is only one number left, respectively. Find the ordered pair $(M, m)$.
[b]D12 / L9.[/b] $N$ people stand in a line. Bella says, “There exists an assignment of nonnegative numbers to the $N$ people so that the sum of all the numbers is $1$ and the sum of any three consecutive people’s numbers does not exceed $1/2019$.” If Bella is right, find the minimum value of $N$ possible.
[b]D13 / L10.[/b] In triangle $\vartriangle ABC$, $D$ is on $AC$ such that $BD$ is an altitude, and $E$ is on $AB$ such that $CE$ is an altitude. Let F be the intersection of $BD$ and $CE$. If $EF = 2FC$, $BF = 8DF$, and $DC = 3$, then find the area of $\vartriangle CDF$.
[b]D14 / L11.[/b] Consider nonnegative real numbers $a_1, ..., a_6$ such that $a_1 +... + a_6 = 20$. Find the minimum possible value of $$\sqrt{a^2_1 + 1^2} +\sqrt{a^2_2 + 2^2} +\sqrt{a^2_3 + 3^2} +\sqrt{a^2_4 + 4^2} +\sqrt{a^2_5 + 5^2} +\sqrt{a^2_6 + 6^2}.$$
[b]D15 / L13.[/b] Find an $a < 1000000$ so that both $a$ and $101a$ are triangular numbers. (A triangular number is a number that can be written as $1 + 2 +... + n$ for some $n \ge 1$.)
Note: There are multiple possible answers to this problem. You only need to find one.
[b]L6.[/b] How many ordered pairs of positive integers $(x, y)$, where $x$ is a perfect square and $y$ is a perfect cube, satisfy $lcm(x, y) = 81000000$?
[b]L12.[/b] Given two points $A$ and $B$ in the plane with $AB = 1$, define $f(C)$ to be the incenter of triangle $ABC$, if it exists. Find the area of the region of points $f(f(X))$ where $X$ is arbitrary.
[b]L14.[/b] Leptina and Zandar play a game. At the four corners of a square, the numbers $1, 2, 3$, and $4$ are written in clockwise order. On Leptina’s turn, she must swap a pair of adjacent numbers. On Zandar’s turn, he must choose two adjacent numbers $a$ and $b$ with $a \ge b$ and replace $a$ with $ a - b$. Zandar wants to reduce the sum of the numbers at the four corners of the square to $2$ in as few turns as possible, and Leptina wants to delay this as long as possible. If Leptina goes first and both players play optimally, find the minimum number of turns Zandar can take after which Zandar is guaranteed to have reduced the sum of the numbers to $2$.
[b]L15.[/b] There exist polynomials $P, Q$ and real numbers $c_0, c_1, c_2, ... , c_{10}$ so that the three polynomials $P, Q$, and $$c_0P^{10} + c_1P^9Q + c_2P^8Q^2 + ... + c_{10}Q^{10}$$ are all polynomials of degree 2019. Suppose that $c_0 = 1$, $c_1 = -7$, $c_2 = 22$. Find all possible values of $c_{10}$.
Note: The answer(s) are rational numbers. It suffices to give the prime factorization(s) of the numerator(s) and denominator(s).
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1987 Brazil National Olympiad, 1
$p(x_1, x_2, ... , x_n)$ is a polynomial with integer coefficients. For each positive integer $r, k(r)$ is the number of $n$-tuples $(a_1, a_2,... , a_n)$ such that $0 \le a_i \le r-1 $ and $p(a_1, a_2, ... , a_n)$ is prime to $r$. Show that if $u$ and $v$ are coprime then $k(u\cdot v) = k(u)\cdot k(v)$, and if p is prime then $k(p^s) = p^{n(s-1)} k(p)$.
2010 All-Russian Olympiad, 4
In each unit square of square $100*100$ write any natural number. Called rectangle with sides parallel sides of square $good$ if sum of number inside rectangle divided by $17$. We can painted all unit squares in $good$ rectangle. One unit square cannot painted twice or more.
Find maximum $d$ for which we can guaranteed paint at least $d$ points.
V Soros Olympiad 1998 - 99 (Russia), grade8
[b]p1.[/b] Given two irreducible fractions. The denominator of the first fraction is $4$, the denominator of the second fraction is $6$. What can the denominator of the product of these fractions be equal to if the product is represented as an irreducible fraction?
[b]p2.[/b] Three horses compete in the race. The player can bet a certain amount of money on each horse. Bets on the first horse are accepted in the ratio $1: 4$. This means that if the first horse wins, then the player gets back the money bet on this horse, and four more times the same amount. Bets on the second horse are accepted in the ratio $1:3$, on the third -$ 1:1$. Money bet on a losing horse is not returned. Is it possible to bet in such a way as to win whatever the outcome of the race?
[b]p3.[/b] A quadrilateral is inscribed in a circle, such that the center of the circle, point $O$, is lies inside it. Let $K$, $L$, $M$, $N$ be the midpoints of the sides of the quadrilateral, following in this order. Prove that the bisectors of angles $\angle KOM$ and $\angle LOC$ are perpendicular (Fig.).
[img]https://cdn.artofproblemsolving.com/attachments/b/8/ea4380698eba7f4cc2639ce20e3057e0294a7c.png[/img]
[b]p4.[/b] Prove that the number$$\underbrace{33...33}_{1999 \,\,\,3s}1$$ is not divisible by $7$.
[b]p5.[/b] In triangle $ABC$, the median drawn from vertex $A$ to side $BC$ is four times smaller than side $AB$ and forms an angle of $60^o$ with it. Find the greatest angle of this triangle.
[b]p6.[/b] Given a $7\times 8$ rectangle made up of 1x1 cells. Cut it into figures consisting of $1\times 1$ cells, so that each figure consists of no more than $5$ cells and the total length of the cuts is minimal (give an example and prove that this cannot be done with a smaller total length of the cuts). You can only cut along the boundaries of the cells.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
MBMT Geometry Rounds, 2023
[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide]
[b]B1.[/b] If the values of two angles in a triangle are $60$ and $75$ degrees respectively, what is the measure of the third angle?
[b]B2.[/b] Square $ABCD$ has side length $1$. What is the area of triangle $ABC$?
[b]B3 / G1.[/b] An equilateral triangle and a square have the same perimeter. If the side length of the equilateral triangle is $8$, what is the square’s side length?
[b]B4 / G2.[/b] What is the maximum possible number of sides and diagonals of equal length in a quadrilateral?
[b]B5.[/b] A square of side length $4$ is put within a circle such that all $4$ corners lie on the circle. What is the diameter of the circle?
[b]B6 / G3.[/b] Patrick is rafting directly across a river $20$ meters across at a speed of $5$ m/s. The river flows in a direction perpendicular to Patrick’s direction at a rate of $12$ m/s. When Patrick reaches the shore on the other end of the river, what is the total distance he has traveled?
[b]B7 / G4.[/b] Quadrilateral $ABCD$ has side lengths $AB = 7$, $BC = 15$, $CD = 20$, and $DA = 24$. It has a diagonal length of $BD = 25$. Find the measure, in degrees, of the sum of angles $ABC$ and $ADC$.
[b]B8 / G5.[/b] What is the largest $P$ such that any rectangle inscribed in an equilateral triangle of side length $1$ has a perimeter of at least $P$?
[b]G6.[/b] A circle is inscribed in an equilateral triangle with side length $s$. Points $A$,$B$,$C$,$D$,$E$,$F$ lie on the triangle such that line segments $AB$, $CD$, and $EF$ are parallel to a side of the triangle, and tangent to the circle. If the area of hexagon $ABCDEF = \frac{9\sqrt3}{2}$ , find $s$.
[b]G7.[/b] Let $\vartriangle ABC$ be such that $\angle A = 105^o$, $\angle B = 45^o$, $\angle C = 30^o$. Let $M$ be the midpoint of $AC$. What is $\angle MBC$?
[b]G8.[/b] Points $A$, $B$, and $C$ lie on a circle centered at $O$ with radius $10$. Let the circumcenter of $\vartriangle AOC$ be $P$. If $AB = 16$, find the minimum value of $PB$.
[i]The circumcenter of a triangle is the intersection point of the three perpendicular bisectors of the sides.
[/i]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1955 Poland - Second Round, 6
Inside the trihedral angle $ OABC $, whose plane angles $ AOB $, $ BOC $, $ COA $ are equal, a point $ S $ is chosen equidistant from the faces of this angle. Through point $ S $ a plane is drawn that intersects the edges $ OA $, $ OB $, $ OC $ at points $ M $, $ N $, $ P $, respectively. Prove that the sum
$$
\frac{1}{OM} + \frac{1}{ON} + \frac{1}{OP}$$
has a constant value, i.e. independent of the position of the plane $ MNP $.
2009 Harvard-MIT Mathematics Tournament, 5
Compute \[\lim_{h\to 0}\dfrac{\sin(\frac{\pi}{3}+4h)-4\sin(\frac{\pi}{3}+3h)+6\sin(\frac{\pi}{3}+2h)-4\sin(\frac{\pi}{3}+h)+\sin(\frac{\pi}{3})}{h^4}.\]
1991 USAMO, 2
For any nonempty set $\,S\,$ of numbers, let $\,\sigma(S)\,$ and $\,\pi(S)\,$ denote the sum and product, respectively, of the elements of $\,S\,$. Prove that
\[ \sum \frac{\sigma(S)}{\pi(S)} = (n^2 + 2n) - \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right) (n+1), \]
where ``$\Sigma$'' denotes a sum involving all nonempty subsets $S$ of $\{1,2,3, \ldots,n\}$.
2013 Romanian Masters In Mathematics, 1
For a positive integer $a$, define a sequence of integers $x_1,x_2,\ldots$ by letting $x_1=a$ and $x_{n+1}=2x_n+1$ for $n\geq 1$. Let $y_n=2^{x_n}-1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_1,\ldots,y_k$ are all prime.
2003 Germany Team Selection Test, 2
Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.
2011-2012 SDML (High School), 12
Kate multiplied all the integers from $1$ to her age and got $1,307,674,368,000$. How old is Kate?
$\text{(A) }14\qquad\text{(B) }15\qquad\text{(C) }16\qquad\text{(D) }17\qquad\text{(E) }18$
2015 ASDAN Math Tournament, 5
Laurie loves multiplying numbers in her head. One day she decides to multiply two $2$-digit numbers $x$ and $y$ such that $x\leq y$ and the two numbers collectively have at least three distinct digits. Unfortunately, she accidentally remembers the digits of each number in the opposite order (for example, instead of remembering $51$ she remembers $15$). Surprisingly, the product of the two numbers after flipping the digits is the same as the product of the two original numbers. How many possible pairs of numbers could Laurie have tried to multiply?
1960 Poland - Second Round, 5
There are three different points on the line $ A $, $ B $, $ C $ and a point $ S $ outside this line; perpendicularly drawn at points $ A $, $ B $, $ C $ to the lines $ SA $, $ SB $, $ SC $ intersect at points $ M $, $ N $, $ P $. Prove that the points $ M $, $ N $, $ P $, $ S $ lie on the circle
2001 Estonia Team Selection Test, 1
Consider on the coordinate plane all rectangles whose
(i) vertices have integer coordinates;
(ii) edges are parallel to coordinate axes;
(iii) area is $2^k$, where $k = 0,1,2....$
Is it possible to color all points with integer coordinates in two colors so that no such rectangle has all its vertices of the same color?
2019 Romanian Masters In Mathematics, 6
Find all pairs of integers $(c, d)$, both greater than 1, such that the following holds:
For any monic polynomial $Q$ of degree $d$ with integer coefficients and for any prime $p > c(2c+1)$, there exists a set $S$ of at most $\big(\tfrac{2c-1}{2c+1}\big)p$ integers, such that
\[\bigcup_{s \in S} \{s,\; Q(s),\; Q(Q(s)),\; Q(Q(Q(s))),\; \dots\}\]
contains a complete residue system modulo $p$ (i.e., intersects with every residue class modulo $p$).
2002 Denmark MO - Mohr Contest, 2
Prove that for any integer $n$ greater than $5$, a square can be divided into $n$ squares.
1976 Bundeswettbewerb Mathematik, 2
Each of the two opposite sides of a convex quadrilateral is divided into seven equal parts, and corresponding division points are connected by a segment, thus dividing the quadrilateral into seven smaller quadrilaterals. Prove that the area of at least one of the small quadrilaterals equals $1\slash 7$ slash of the area of the large quadrilateral.
1991 Denmark MO - Mohr Contest, 4
Let $a, b, c$ and $d$ be arbitrary real numbers. Prove that if $$a^2+b^2+c^2+d^2=ab+bc+cd+da,$$ then $a=b=c=d$.
2016 239 Open Mathematical Olympiad, 6
A graph is called $7-chip$ if it obtained by removing at most three edges that have no vertex in common from a complete graph with seven vertices. Consider a complete graph $G$ with $v$ vertices which each edge of its is colored blue or red. Prove that there is either a blue path with $100$ edges or a red $7-chip$.
Kvant 2024, M2811
A sequence of positive integer numbers $a_1,...,a_{100}$ such is that $a_1=1$, and for all $n=1, 2,...,100$ number $(a_1+...+a_n) \left ( \frac{1}{a_1}+...+\frac{1}{a_n} \right )$ is integer. What is the maximum value that can take $ a_{100}$?
[i] M. Turevskii [/i]
1997 Federal Competition For Advanced Students, Part 2, 3
For every natural number $n$, find all polynomials $x^2+ax+b$, where $a^2 \geq 4b$, that divide $x^{2n} + ax^n + b$.
2013 Bangladesh Mathematical Olympiad, 5
Higher Secondary P5
Let $x>1$ be an integer such that for any two positive integers $a$ and $b$, if $x$ divides $ab$ then $x$ either divides $a$ or divides $b$. Find with proof the number of positive integers that divide $x$.
2011 Puerto Rico Team Selection Test, 1
A set of ten two-digit numbers is given. Prove that one can always choose two disjoint subsets of this set such that the sum of their elements is the same.
Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )
2011 China Girls Math Olympiad, 4
A tennis tournament has $n>2$ players and any two players play one game against each other (ties are not allowed). After the game these players can be arranged in a circle, such that for any three players $A,B,C$, if $A,B$ are adjacent on the circle, then at least one of $A,B$ won against $C$. Find all possible values for $n$.