Found problems: 85335
2020 LMT Fall, 18
Given that $\sqrt{x+2y}-\sqrt{x-2y}=2,$ compute the minimum value of $x+y.$
[i]Proposed by Alex Li[/i]
2016 India IMO Training Camp, 1
Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.
1996 Estonia National Olympiad, 3
An equilateral triangle of side$ 1$ is rotated around its center, yielding another equilareral triangle. Find the area of the intersection of these two triangles.
2025 Alborz Mathematical Olympiad, P1
Let \( M \) and \( N \) be the midpoints of sides \( BC \) and \( AC \), respectively, in an acute-angled triangle \( ABC \). Suppose there exists a point \( P \) on the line segment \( AM \) such that \( \angle NPC = \angle MPC \). Let \( D \) be the intersection point of the line \( NP \) and the line parallel to \( CP \) passing through \( B \). Prove that \( AD = AB \).
Proposed by Soroush Behroozifar
Gheorghe Țițeica 2025, P1
Let $G$ be a finite group and $a\in G$ a fixed element. Define the set $$S_a=\{g\in G\mid ga\neq ag, \,ga^2=a^2g\}.$$ Show that:
[list=a]
[*] if $g\in S_a$, then $ag^{-1}\in S_a$;
[*] $|S_a|$ is divisible by $4$.
2021 Latvia Baltic Way TST, P12
Five points $A,B,C,P,Q$ are chosen so that $A,B,C$ aren't collinear. The following length conditions hold: $\frac{AP}{BP}=\frac{AQ}{BQ}=\frac{21}{20}$ and $\frac{BP}{CP}=\frac{BQ}{CQ}=\frac{20}{19}$. Prove that line $PQ$ goes through the circumcentre of $\triangle ABC$.
2021 CCA Math Bonanza, TB4
For all integers $0 \leq k \leq 16$, let \[S_k = \sum_{j=0}^{k}(-1)^j {\binom{16}{j}}.\] Compute $\max(S_0,S_1, \ldots S_{16})$.
[i]2021 CCA Math Bonanza Tiebreaker Round #4[/i]
2008 Mongolia Team Selection Test, 1
How many ways to fill the board $ 4\times 4$ by nonnegative integers, such that sum of the numbers of each row and each column is 3?
2019 Teodor Topan, 4
Let $ S $ be a finite [url=https://en.wikipedia.org/wiki/Cancellation_property]cancellative semigroup.[/url]
[b]a)[/b] Prove that $ S $ contains an idempotent element.
[b]b)[/b] Prove that $ S $ is a group.
[b]c)[/b] Disprove subpoint [b]b)[/b] in the case that $ S $ would not be finite.
[i]Vlad Mihaly[/i]
2019 India IMO Training Camp, P2
Let $ABC$ be an acute-angled scalene triangle with circumcircle $\Gamma$ and circumcenter $O$. Suppose $AB < AC$. Let $H$ be the orthocenter and $I$ be the incenter of triangle $ABC$. Let $F$ be the midpoint of the arc $BC$ of the circumcircle of triangle $BHC$, containing $H$.
Let $X$ be a point on the arc $AB$ of $\Gamma$ not containing $C$, such that $\angle AXH = \angle AFH$. Let $K$ be the circumcenter of triangle $XIA$. Prove that the lines $AO$ and $KI$ meet on $\Gamma$.
[i]Proposed by Anant Mudgal[/i]
2017 AMC 12/AHSME, 13
In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?
[asy]
size(100);
pair A, B, C, D, E, F;
A = (0,0);
B = (1,0);
C = (2,0);
D = rotate(60, A)*B;
E = B + D;
F = rotate(60, A)*C;
draw(Circle(A, 0.5));
draw(Circle(B, 0.5));
draw(Circle(C, 0.5));
draw(Circle(D, 0.5));
draw(Circle(E, 0.5));
draw(Circle(F, 0.5));
[/asy]
$\textbf{(A) } 6 \qquad \textbf{(B) } 8 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 15$
2015 Postal Coaching, Problem 4
For every positive integer$ n$, let $P(n)$ be the greatest prime divisor of $n^2+1$. Show that there are infinitely many quadruples $(a, b, c, d)$ of positive integers that satisfy $a < b < c < d$ and $P(a) = P(b) = P(c) = P(d)$.
2013 NIMO Summer Contest, 3
Jacob and Aaron are playing a game in which Aaron is trying to guess the outcome of an unfair coin which shows heads $\tfrac{2}{3}$ of the time. Aaron randomly guesses ``heads'' $\tfrac{2}{3}$ of the time, and guesses ``tails'' the other $\tfrac{1}{3}$ of the time. If the probability that Aaron guesses correctly is $p$, compute $9000p$.
[i]Proposed by Aaron Lin[/i]
Kettering MO, 2008
[b]p1.[/b] The case of Mr. Brown, Mr. Potter, and Mr. Smith is investigated. One of them has committed a crime. Everyone of them made two statements.
Mr. Brown: I have not done it. Mr. Potter has not done it.
Mr. Potter: Mr. Brown has not done it. Mr. Smith has done it.
Mr. Smith: I have not done it. Mr. Brown has done it.
It is known that one of them told the truth both times, one lied both times, and one told the truth one time and lied one time. Who has committed the crime?
[b]p2.[/b] Is it possible to draw in a plane $1000001$ circles of the radius $1$ such that every circle touches exactly three other circles?
[b]p3.[/b] Consider a circle of radius $R$ centered at the origin. A particle is “launched” from the $x$-axis at a distance, $d$, from the origin with $0 < d < R$, and at an angle, $\alpha$, with the $x$-axis. The particle is reflected from the boundary of the circle so that the [b]angle of incidence[/b] equals the [b]angle of reflection[/b]. Determine the angle $\alpha$ so that the path of the particle contacts the circle’s interior at exactly eight points. Please note that $\alpha$ should be determined in terms of the qunatities $R$ and $d$.
[img]https://cdn.artofproblemsolving.com/attachments/e/3/b8ef9bb8d1b54c263bf2b68d3de60be5b41ad0.png[/img]
[b]p4.[/b] Is it possible to find four different real numbers such that the cube of every number equals the square of the sum of the three others?
[b]p5. [/b]The Fibonacci sequence $(1, 2, 3, 5, 8, 13, 21, . . .)$ is defined by the following formula:
$f_n = f_{n-2} + f_{n-1}$, where $f_1 = 1$, $f_2 = 2$. Prove that any positive integer can be represented as a sum of different members of the Fibonacci sequence.
[b]p6.[/b] $10,000$ points are arbitrary chosen inside a square of area $1$ m$^2$ . Does there exist a broken line connecting all these points, the length of which is less than $201$ m$^2?
PS. You should use hide for answers.
2007 National Olympiad First Round, 26
Let $c$ be the least common multiple of positive integers $a$ and $b$, and $d$ be the greatest common divisor of $a$ and $b$. How many pairs of positive integers $(a,b)$ are there such that
\[
\dfrac {1}{a} + \dfrac {1}{b} + \dfrac {1}{c} + \dfrac {1}{d} = 1?
\]
$
\textbf{(A)}\ 6
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ 2
$
2015 AIME Problems, 8
For positive integer $n$, let $s(n)$ denote the sum of the digits of $n$. Find the smallest positive integer $n$ satisfying $s(n)=s(n+864)=20$.
2020 Thailand TST, 3
Let $\mathbb Z$ be the set of integers. We consider functions $f :\mathbb Z\to\mathbb Z$ satisfying
\[f\left(f(x+y)+y\right)=f\left(f(x)+y\right)\]
for all integers $x$ and $y$. For such a function, we say that an integer $v$ is [i]f-rare[/i] if the set
\[X_v=\{x\in\mathbb Z:f(x)=v\}\]
is finite and nonempty.
(a) Prove that there exists such a function $f$ for which there is an $f$-rare integer.
(b) Prove that no such function $f$ can have more than one $f$-rare integer.
[i]Netherlands[/i]
2016 SDMO (Middle School), 2
Let $AB$ be a diameter of a circle and let $C$ be a point on $AB$ with $2\cdot AC=BC$. Let $D$ and $E$ be points on the circle such that $DC\perp AB$ and $DE$ is a second diameter. What is the ratio of the area of $\triangle{DCE}$ to the area of $\triangle{ABD}$?
2024-25 IOQM India, 18
Let $p,q$ be two-digit number neither of which are divisible by $10$. Let $r$ be the four-digit number by putting the digits of $p$ followed by the digits of $q$ (in order). As $p,q$ very, a computer prints $r$ on the screen if $\gcd(p,q) = 1$ and $p+q$ divides $r$. Suppose that the largest number that is printed by the computer is $N$. Determine the number formed by the last two digits of $N$ (in the same order).
1981 USAMO, 4
The sum of the measures of all the face angles of a given complex polyhedral angle is equal to the sum of all its dihedral angles. Prove that the polyhedral angle is a trihedral angle.
$\mathbf{Note:}$ A convex polyhedral angle may be formed by drawing rays from an exterior point to all points of a convex polygon.
2011 IFYM, Sozopol, 6
In a group of $n$ people each one has an Easter Egg. They exchange their eggs in the following way: On each exchange two people exchange the eggs they currently have. Each two exchange eggs between themselves at least once. After a certain amount of such exchanges it turned out that each one of the $n$ people had the same egg he had from the beginning. Determine the least amount of exchanges needed, if:
a) $n=5$;
b) $n=6$.
2012 CIIM, Problem 5
Let $D=\{0,1,\dots,9\}$. A direction function for $D$ is a function $f:D \times D \to \{0,1\}.$
A real $r\in [0,1]$ is compatible with $f$ if it can be written in the form $$r = \sum_{j=1}^{\infty} \frac{d_j}{10^j}$$ with $d_j \in D$ and $f(d_j,d_{j+1})=1$ for every positive integer $j$.
Determine the least integer $k$ such that for any direction fuction $f$, if there are $k$ compatible reals with $f$ then there are infinite reals compatible with $f$.
2014 Romania National Olympiad, 3
Let $ P,Q $ be the midpoints of the diagonals $ BD, $ respectively, $ AC, $ of the quadrilateral $ ABCD, $ and points $ M,N,R,S $ on the segments $ BC,CD,PQ, $ respectively $ AC, $ except their extremities, such that
$$ \frac{BM}{MC}=\frac{DN}{NC}=\frac{PR}{RQ}=\frac{AS}{SC} . $$
Show that the center of mass of the triangle $ AMN $ is situated on the segment $ RS. $
May Olympiad L1 - geometry, 2005.4
There are two paper figures: an equilateral triangle and a rectangle. The height of rectangle is equal to the height of the triangle and the base of the rectangle is equal to the base of the triangle. Divide the triangle into three parts and the rectangle into two, using straight cuts, so that with the five pieces can be assembled, without gaps or overlays, a equilateral triangle. To assemble the figure, each part can be rotated and / or turned around.
1982 Vietnam National Olympiad, 2
For a given parameter $m$, solve the equation
\[x(x + 1)(x + 2)(x + 3) + 1 - m = 0.\]