Found problems: 85335
2019 Saudi Arabia Pre-TST + Training Tests, 5.2
Let the bisector of the outside angle of $A$ of triangle $ABC$ and the circumcircle of triangle $ABC$ meet at point $P$. The circle passing through points $A$ and $P$ intersects segments $BP$ and $CP$ at points $E$ and $F$ respectively. Let $AD$ is the angle bisector of triangle $ABC$. Prove that $\angle PED = \angle PFD$.
[img]https://cdn.artofproblemsolving.com/attachments/0/3/0638429a220f07227703a682479ed150302aae.png[/img]
V Soros Olympiad 1998 - 99 (Russia), 8.1 - 8.4
[b]p1.[/b] Is it possible to write $5$ different fractions that add up to $1$, such that their numerators are equal to one and their denominators are natural numbers?
[b]p2.[/b] The following is known about two numbers $x$ and $y$:
if $x\ge 0$, then $y = 1 -x$;
if $y\le 1$, then $x = 1 + y$;
if $x\le 1$, then $x = |1 + y|$.
Find $x$ and $y$.
[b]p3.[/b] Five people living in different cities received a salary, some more, others less ($143$, $233$, $313$, $410$ and $413$ rubles). Each of them can send money to the other by mail. In this case, the post office takes $10\%$ of the amount of money sent for the transfer (in order to receive $100$ rubles, you need to send $10\%$ more, that is, $110$ rubles). They want to send money so that everyone has the same amount of money, and the post office receives as little money as possible. How much money will each person have using the most economical shipping method?
[b]p4.[/b] a) List three different natural numbers $m$, $n$ and $k$ for which $m! = n! \cdot k!$ .
b) Is it possible to come up with $1999$ such triplets?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
1997 Vietnam Team Selection Test, 3
Find the greatest real number $ \alpha$ for which there exists a sequence of infinitive integers $ (a_n)$, ($ n \equal{} 1, 2, 3, \ldots$) satisfying the following conditions:
1) $ a_n > 1997n$ for every $ n \in\mathbb{N}^{*}$;
2) For every $ n\ge 2$, $ U_n\ge a^{\alpha}_n$, where $ U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}$.
1990 Romania Team Selection Test, 10
Let $p,q$ be positive prime numbers and suppose $q>5$. Prove that if $q \mid 2^{p}+3^{p}$, then $q>p$.
[i]Laurentiu Panaitopol[/i]
2010 ELMO Shortlist, 5
Given a prime $p$, let $d(a,b)$ be the number of integers $c$ such that $1 \leq c < p$, and the remainders when $ac$ and $bc$ are divided by $p$ are both at most $\frac{p}{3}$. Determine the maximum value of \[\sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)(x_a + 1)(x_b + 1)} - \sqrt{\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}d(a,b)x_ax_b}\] over all $(p-1)$-tuples $(x_1,x_2,\ldots,x_{p-1})$ of real numbers.
[i]Brian Hamrick.[/i]
2011 Belarus Team Selection Test, 2
Positive real $a,b,c$ satisfy the condition $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=1+\frac{1}{6}\left( \frac{a}{c}+\frac{b}{a}+\frac{c}{b} \right)$$ Prove that $$\frac{a^3bc}{b+c}+\frac{b^3ca}{a+c}+\frac{c^3ab}{a+b}\ge \frac{1}{6}(ab+bc+ca)^2$$
I.Voronovich
LMT Team Rounds 2021+, A11 B17
In $\triangle ABC$ with $\angle BAC = 60^{\circ}$ and circumcircle $\omega$, the angle bisector of $\angle BAC$ intersects side $\overline{BC}$ at point $D$, and line $AD$ is extended past $D$ to a point $A'$. Let points $E$ and $F$ be the feet of the perpendiculars of $A'$ onto lines $AB$ and $AC$, respectively. Suppose that $\omega$ is tangent to line $EF$ at a point $P$ between $E$ and $F$ such that $\tfrac{EP}{FP} = \tfrac{1}{2}$. Given that $EF=6$, the area of $\triangle ABC$ can be written as $\tfrac{m\sqrt{n}}{p}$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.
[i]Proposed by Taiki Aiba[/i]
2020/2021 Tournament of Towns, P2
Maria has a balance scale that can indicate which of its pans is heavier or whether they have equal weight. She also has 4 weights that look the same but have masses of 1001, 1002, 1004 and 1005g. Can Maria determine the mass of each weight in 4 weightings? The weights for a new weighing may be picked when the result of the previous ones is known.
[i]The Jury[/i]
(For the senior paper) The same question when the left pan of the scale is lighter by 1g than the right one, so the scale indicates equality when the mass on the left pan is heavier by 1g than the mass on the right pan.
[i]Alexey Tolpygo[/i]
2009 VJIMC, Problem 1
A positive integer $m$ is called self-descriptive in base $b$, where $b\ge2$ is an integer, if
i) The representation of $m$ in base $b$ is of the form $(a_0a_1\ldots a_{b-1})_b$ (that is $m=a_0b^{b-1}+a_1b^{b-2}+\ldots+a_{b-2}b+a_{b-1}$, where $0\le a_i\le b-1$ are integers).
ii) $a_i$ is equal to the number of occurences of the number $i$ in the sequence $(a_0a_1\ldots a_{b-1})$.
For example, $(1210)_4$ is self-descriptive in base $4$, because it has four digits and contains one $0$, two $1$s, one $2$ and no $3$s.
2019 Balkan MO Shortlist, C1
100 couples are invited to a traditional Modolvan dance. The $200$ people stand in a line, and then in a $\textit{step}$, (not necessarily adjacent) many swap positions. Find the least $C$ such that whatever the initial order, they can arrive at an ordering where everyone is dancing next to their partner in at most $C$ steps.
1977 IMO Longlists, 3
In a company of $n$ persons, each person has no more than $d$ acquaintances, and in that company there exists a group of $k$ persons, $k\ge d$, who are not acquainted with each other. Prove that the number of acquainted pairs is not greater than $\left[ \frac{n^2}{4}\right]$.
2019 PUMaC Geometry A, 2
Let $\triangle ABC$ be a triangle with circumcenter $O$ and orthocenter $H$. Let $D$ be a point on the circumcircle of $ABC$ such that $AD \perp BC$. Suppose that $AB = 6, DB = 2$, and the ratio $\tfrac{\text{area}(\triangle ABC)}{\text{area}(\triangle HBC)}=5.$ Then, if $OA$ is the length of the circumradius, then $OA^2$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
2002 India National Olympiad, 5
Do there exist distinct positive integers $a$, $b$, $c$ such that $a$, $b$, $c$, $-a+b+c$, $a-b+c$, $a+b-c$, $a+b+c$ form an arithmetic progression (in some order).
2013 District Olympiad, 1
Let $a,b\in \mathbb{R}$ and $z\in \mathbb{C}\backslash \mathbb{R}$ so that $\left| a-b \right|=\left| a+b-2z \right|$.
a) Prove that the equation ${{\left| z-a \right|}^{x}}+{{\left| \bar{z}-b \right|}^{x}}={{\left| a-b \right|}^{x}}$, with the unknown number $x\in \mathbb{R}$, has a unique solution.
b) Solve the following inequation ${{\left| z-a \right|}^{x}}+{{\left| \bar{z}-b \right|}^{x}}\le {{\left| a-b \right|}^{x}}$, with the unknown number $x\in \mathbb{R}$.
The Mathematical Gazette
2008 Balkan MO, 1
Given a scalene acute triangle $ ABC$ with $ AC>BC$ let $ F$ be the foot of the altitude from $ C$. Let $ P$ be a point on $ AB$, different from $ A$ so that $ AF\equal{}PF$. Let $ H,O,M$ be the orthocenter, circumcenter and midpoint of $ [AC]$. Let $ X$ be the intersection point of $ BC$ and $ HP$. Let $ Y$ be the intersection point of $ OM$ and $ FX$ and let $ OF$ intersect $ AC$ at $ Z$. Prove that $ F,M,Y,Z$ are concyclic.
2000 AIME Problems, 15
Find the least positive integer $n$ such that \[ \frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}. \]
1977 IMO Shortlist, 5
There are $2^n$ words of length $n$ over the alphabet $\{0, 1\}$. Prove that the following algorithm generates the sequence $w_0, w_1, \ldots, w_{2^n-1}$ of all these words such that any two consecutive words differ in exactly one digit.
(1) $w_0 = 00 \ldots 0$ ($n$ zeros).
(2) Suppose $w_{m-1} = a_1a_2 \ldots a_n,\quad a_i \in \{0, 1\}$. Let $e(m)$ be the exponent of $2$ in the representation of $n$ as a product of primes, and let $j = 1 + e(m)$. Replace the digit $a_j$ in the word $w_{m-1}$ by $1 - a_j$. The obtained word is $w_m$.
MBMT Guts Rounds, 2019
[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide]
[u]Set 4[/u]
[b]D.16 / L.6[/b] Alex has $100$ Bluffy Funnies in some order, which he wants to sort in order of height. They’re already almost in order: each Bluffy Funny is at most $1$ spot off from where it should be. Alex can only swap pairs of adjacent Bluffy Funnies. What is the maximum possible number of swaps necessary for Alex to sort them?
[b]D.17[/b] I start with the number $1$ in my pocket. On each round, I flip a coin. If the coin lands heads heads, I double the number in my pocket. If it lands tails, I divide it by two. After five rounds, what is the expected value of the number in my pocket?
[b]D.18 / L.12[/b] Point $P$ inside square $ABCD$ is connected to each corner of the square, splitting the square into four triangles. If three of these triangles have area $25$, $25$, and $15$, what are all the possible values for the area of the fourth triangle?
[b]D.19[/b] Mr. Stein and Mr. Schwartz are playing a yelling game. The teachers alternate yelling. Each yell is louder than the previous and is also relatively prime to the previous. If any teacher yells at $100$ or more decibels, then they lose the game. Mr. Stein yells first, at $88$ decibels. What volume, in decibels, should Mr. Schwartz yell at to guarantee that he will win?
[b]D.20 / L.15[/b] A semicircle of radius $1$ has line $\ell$ along its base and is tangent to line $m$. Let $r$ be the radius of the largest circle tangent to $\ell$, $m$, and the semicircle. As the point of tangency on the semicircle varies, the range of possible values of $r$ is the interval $[a, b]$. Find $b - a$.
[u]Set 5[/u]
[b]D.21 / L.14[/b] Hungryman starts at the tile labeled “$S$”. On each move, he moves $1$ unit horizontally or vertically and eats the tile he arrives at. He cannot move to a tile he already ate, and he stops when the sum of the numbers on all eaten tiles is a multiple of nine. Find the minimum number of tiles that Hungryman eats.
[img]https://cdn.artofproblemsolving.com/attachments/e/7/c2ecc2a872af6c4a07907613c412d3b86cd7bc.png
[/img]
[b]D.22 / L.11[/b] How many triples of nonnegative integers $(x, y, z)$ satisfy the equation $6x + 10y +15z = 300$?
[b]D.23 / L.16[/b] Anson, Billiam, and Connor are looking at a $3D$ figure. The figure is made of unit cubes and is sitting on the ground. No cubes are floating; in other words, each unit cube must either have another unit cube or the ground directly under it. Anson looks from the left side and says, “I see a $5 \times 5$ square.” Billiam looks from the front and says the same thing. Connor looks from the top and says the same thing. Find the absolute difference between the minimum and maximum volume of the figure.
[b]D.24 / L.13[/b] Tse and Cho are playing a game. Cho chooses a number $x \in [0, 1]$ uniformly at random, and Tse guesses the value of $x(1 - x)$. Tse wins if his guess is at most $\frac{1}{50}$ away from the correct value. Given that Tse plays optimally, what is the probability that Tse wins?
[b]D.25 / L.20[/b] Find the largest solution to the equation $$2019(x^{2019x^{2019}-2019^2+2019})^{2019}) = 2019^{x^{2019}+1}.$$
[u]Set 6[/u]
[i]This round is an estimation round. No one is expected to get an exact answer to any of these questions, but unlike other rounds, you will get points for being close. In the interest of transparency, the formulas for determining the number of points you will receive are located on the answer sheet, but they aren’t very important when solving these problems.[/i]
[b]D.26 / L.26[/b] What is the sum over all MBMT volunteers of the number of times that volunteer has attended MBMT (as a contestant or as a volunteer, including this year)? Last year there were $47$ volunteers; this is the fifth MBMT.
[b]D.27 / L.27[/b] William is sharing a chocolate bar with Naveen and Kevin. He first randomly picks a point along the bar and splits the bar at that point. He then takes the smaller piece, randomly picks a point along it, splits the piece at that point, and gives the smaller resulting piece to Kevin. Estimate the probability that Kevin gets less than $10\%$ of the entire chocolate bar.
[b]D.28 / L.28[/b] Let $x$ be the positive solution to the equation $x^{x^{x^x}}= 1.1$. Estimate $\frac{1}{x-1}$.
[b]D.29 / L.29[/b] Estimate the number of dots in the following box:
[img]https://cdn.artofproblemsolving.com/attachments/8/6/416ba6379d7dfe0b6302b42eff7de61b3ec0f1.png[/img]
It may be useful to know that this image was produced by plotting $(4\sqrt{x}, y)$ some number of times, where x, y are random numbers chosen uniformly randomly and independently from the interval $[0, 1]$.
[b]D.30 / L.30[/b] For a positive integer $n$, let $f(n)$ be the smallest prime greater than or equal to $n$. Estimate $$(f(1) - 1) + (f(2) - 2) + (f(3) - 3) + ...+ (f(10000) - 10000).$$
For $26 \le i \le 30$, let $E_i$ be your team’s answer to problem $i$ and let $A_i$ be the actual answer to problem $i$. Your score $S_i$ for problem $i$ is given by
$S_{26} = \max(0, 12 - |E_{26} - A_{26}|/5)$
$S_{27} = \max(0, 12 - 100|E_{27} - A_{27}|)$
$S_{28} = \max(0, 12 - 5|E_{28} - A_{28}|))$
$S_{29} = 12 \max \left(0, 1 - 3 \frac{|E_{29} - A_{29}|}{A_{29}} \right)$
$S_{30} = \max (0, 12 - |E_{30} - A_{30}|/2000)$
PS. You should use hide for answers. D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here [/url] and L10,16-30 [url=https://artofproblemsolving.com/community/c3h2790825p24541816]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2025 Malaysian IMO Training Camp, 3
Minivan and Megavan play a game. For a positive integer $n$, Minivan selects a sequence of integers $a_1,a_2,\ldots,a_n$. An operation on $a_1,a_2,\ldots,a_n$ means selecting an $a_i$ and increasing it by $1$. Minivan and Megavan take turns, with Minivan going first. On Minivan's turn, he performs at most $2025$ operations, and he may choose the same integer repeatedly. On Megavan's turn, he performs exactly $1$ operation instead. Megavan wins if at any point in the game, including in the middle of Minivan's operations, two numbers in the sequence are equal.
[i](Proposed by Ho Janson)[/i]
2014 Junior Balkan Team Selection Tests - Romania, 5
Let $D$ and $E$ be the midpoints of sides $[AB]$ and $[AC]$ of the triangle $ABC$. The circle of diameter $[AB]$ intersects the line $DE$ on the opposite side of $AB$ than $C$, in $X$. The circle of diameter $[AC]$ intersects $DE$ on the opposite side of $AC$ than $B$ in $Y$ . Let $T$ be the intersection of $BX$ and $CY$.
Prove that the orthocenter of triangle $XY T$ lies on $BC$.
2001 Moldova Team Selection Test, 12
Let $n{}$ $(n\geq 1)$ be an integer and a set $A=\{1,2,\ldots,n\}$. The set $A{}$ is $k-partitionable$ if it can be partitioned in $k{}$ disjoint sets with the same sum of elements. Show that $A{}$ is $k-partitionable$ if and only if $2k$ divides $n(n+1)$ and $2k\leq n+1$.
JBMO Geometry Collection, 2005
Let $ABC$ be an acute-angled triangle inscribed in a circle $k$. It is given that the tangent from $A$ to the circle meets the line $BC$ at point $P$. Let $M$ be the midpoint of the line segment $AP$ and $R$ be the second intersection point of the circle $k$ with the line $BM$. The line $PR$ meets again the circle $k$ at point $S$ different from $R$.
Prove that the lines $AP$ and $CS$ are parallel.
2021 Polish Junior MO First Round, 2
A triangle $ABC$ is given with $AC = BC = 5$. The altitude of this triangle drawn from vertex $A$ has length $4$. Calculate the length of the altitude of $ABC$ drawn from vertex $C$.
Kyiv City MO Seniors 2003+ geometry, 2009.10.4
In the triangle $ABC$ the angle bisectors $AL$ and $BT$ are drawn, which intersect at the point $I$, and their extensions intersect the circle circumscribed around the triangle $ABC$ at the points $E$ and $D$ respectively. The segment $DE$ intersects the sides $AC$ and $BC$ at the points $F$ and $K$, respectively. Prove that:
a) quadrilateral $IKCF$ is rhombus;
b) the side of this rhombus is $\sqrt {DF \cdot EK}$.
(Rozhkova Maria)
1983 Vietnam National Olympiad, 2
Decide whether $S_n$ or $T_n$ is larger, where
\[S_n =\displaystyle\sum_{k=1}^n \frac{k}{(2n - 2k + 1)(2n - k + 1)}, T_n =\displaystyle\sum_{k=1}^n\frac{1}{k}\]