Found problems: 85335
2004 Czech and Slovak Olympiad III A, 5
Let $L$ be an arbitrary point on the minor arc $CD$ of the circumcircle of square $ABCD$. Let $K,M,N$ be the intersection points of $AL,CD$; $CL,AD$; and $MK,BC$ respectively. Prove that $B,M,L,N$ are concyclic.
1956 AMC 12/AHSME, 49
Triangle $ PAB$ is formed by three tangents to circle $ O$ and $ < APB \equal{} 40^{\circ}$; then angle $ AOB$ equals:
$ \textbf{(A)}\ 45^{\circ} \qquad\textbf{(B)}\ 50^{\circ} \qquad\textbf{(C)}\ 55^{\circ} \qquad\textbf{(D)}\ 60^{\circ} \qquad\textbf{(E)}\ 70^{\circ}$
2014 ASDAN Math Tournament, 14
Consider a round table on which $2014$ people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon receiving the plate, will serve himself if necessary and similarly pass the plate either left or right with equal probability. Compute the probability that you are served last if you are seated $2$ seats away from the person at the head of the table.
2019 Purple Comet Problems, 15
Let $a, b, c$, and $d$ be prime numbers with $a \le b \le c \le d > 0$.
Suppose $a^2 + 2b^2 + c^2 + 2d^2 = 2(ab + bc - cd + da)$. Find $4a + 3b + 2c + d$.
2008 Czech and Slovak Olympiad III A, 1
Find all pairs of real numbers $(x,y)$ satisfying:
\[x+y^2=y^3,\]\[y+x^2=x^3.\]
1969 AMC 12/AHSME, 22
Let $K$ be the measure of the area bounded by the $x$-axis, the line $x=8$, and the curve defined by \[f=\{(x,y)\,|\, y=x\text{ when }0\leq x\leq 5,\,y=2x-5\text{ when }5\leq x\leq 8\}.\] Then $K$ is:
$\textbf{(A) }21.5\qquad
\textbf{(B) }36.4\qquad
\textbf{(C) }36.5\qquad
\textbf{(D) }44\qquad$
$\textbf{ (E) }\text{less than 44 but arbitrarily close to it.}$
1993 Bulgaria National Olympiad, 4
Find all natural numbers $n > 1$ for which there exists such natural numbers $a_1,a_2,...,a_n$ for which the numbers
$\{a_i +a_j | 1 \le i \le j \le n \}$ form a full system modulo $\frac{n(n+1)}{2}$.
2023 VIASM Summer Challenge, Problem 4
Let $ABCD$ be a parallelogram and $P$ be an arbitrary point in the plane. Let $O$ be the intersection of two diagonals $AC$ and $BD.$ The circumcircles of triangles $POB$ and $POC$ intersect the circumcircles of triangle $OAD$ at $Q$ and $R,$ respectively $(Q,R \ne O).$ Construct the parallelograms $PQAM$ and $PRDN.$
Prove that: the circumcircle of triangle $MNP$ passes through $O.$
[i]Proposed by Tran Quang Hung ([url=https://artofproblemsolving.com/community/user/68918]buratinogigle[/url])[/i]
1967 IMO Shortlist, 3
Suppose $\tan \alpha = \dfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove that the number $\tan \beta$ for which $\tan {2 \beta} = \tan {3 \alpha}$ is rational only when $p^2 + q^2$ is the square of an integer.
MOAA Accuracy Rounds, 2023.3
Ms. Raina's math class has 6 students, including the troublemakers Andy and Harry. For a group project, Ms. Raina randomly divides the students into three groups containing 1, 2, and 3 people. The probability that Andy and Harry unfortunately end up in the same group can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by Andy Xu[/i]
2004 Switzerland - Final Round, 1
Let $\Gamma$ be a circle and $P$ a point outside of $\Gamma$ . A tangent from $P$ to the circle intersects it in $A$. Another line through $P$ intersects $\Gamma$ at the points $B$ and $C$. The bisector of $\angle APB$ intersects $AB$ at $D$ and $AC$ at $E$. Prove that the triangle $ADE$ is isosceles.
1953 Putnam, A4
From the identity
$$ \int_{0}^{\pi \slash 2} \log \sin 2x \, dx = \int_{0}^{\pi \slash 2} \log \sin x \, dx + \int_{0}^{\pi \slash 2} \log \cos x \, dx +\int_{0}^{\pi \slash 2} \log 2 \, dx, $$
deduce the value of $\int_{0}^{\pi \slash 2} \log \sin x \, dx.$
2022 USAJMO, 4
Let $ABCD$ be a rhombus, and let $K$ and $L$ be points such that $K$ lies inside the rhombus, $L$ lies outside the rhombus, and $KA = KB = LC = LD$. Prove that there exist points $X$ and $Y$ on lines $AC$ and $BD$ such that $KXLY$ is also a rhombus.
[i]Proposed by Ankan Bhattacharya[/i]
1968 Bulgaria National Olympiad, Problem 5
The point $M$ is inside the tetrahedron $ABCD$ and the intersection points of the lines $AM,BM,CM$ and $DM$ with the opposite walls are denoted with $A_1,B_1,C_1,D_1$ respectively. It is given also that the ratios $\frac{MA}{MA_1}$, $\frac{MB}{MB_1}$, $\frac{MC}{MC_1}$, and $\frac{MD}{MD_1}$ are equal to the same number $k$. Find all possible values of $k$.
[i]K. Petrov[/i]
1990 IMO Longlists, 85
Let $A_1, A_2, \ldots, A_n (n \geq 4)$ be $n$ convex sets in plane. Knowing that every three convex sets have a common point. Prove that there exists a point belonging to all the sets.
2018 Romania National Olympiad, 3
Let $f:[a,b] \to \mathbb{R}$ be an integrable function and $(a_n) \subset \mathbb{R}$ such that $a_n \to 0.$
$\textbf{a) }$ If $A= \{m \cdot a_n \mid m,n \in \mathbb{N}^* \},$ prove that every open interval of strictly positive real numbers contains elements from $A.$
$\textbf{b) }$ If, for any $n \in \mathbb{N}^*$ and for any $x,y \in [a,b]$ with $|x-y|=a_n,$ the inequality $\left| \int_x^yf(t)dt \right| \leq |x-y|$ is true, prove that $$\left| \int_x^y f(t)dt \right| \leq |x-y|, \: \forall x,y \in [a,b]$$
[i]Nicolae Bourbacut[/i]
2019 IFYM, Sozopol, 2
$\Delta ABC$ is a triangle with center $I$ of its inscribed circle and $B_1$ and $C_1$ are feet of its angle bisectors through $B$ and $C$. Let $S$ be the middle point on the arc $\widehat{BAC}$ of the circumscribed circle of $\Delta ABC$ (denoted with $\Omega$) and let $\omega_a$ be the excircle of $\Delta ABC$ opposite to $A$. Let $\omega_a (I_a)$ be tangent to $AB$ and $AC$ in points $D$ and $E$ respectively and $SI\cap \Omega=\{S,P\}$. Let $M$ be the middle point of $DE$ and $N$ be the middle point of $SI$. If $MN\cap AP=K$, prove that $KI_a\perp B_1 C_1$.
2017-2018 SDPC, 6
Let $ABC$ be an acute triangle with circumcenter $O$. Let the parallel to $BC$ through $A$ intersect line $BO$ at $B_A$ and $CO$ at $C_A$. Lines $B_AC$ and $BC_A$ intersect at $A'$. Define $B'$ and $C'$ similarly.
(a) Prove that the the perpendicular from $A'$ to $BC$, the perpendicular from $B'$ to $AC$, and $C'$ to $AB$ are concurrent.
(b) Prove that likes $AA'$, $BB'$, and $CC'$ are concurrent.
2024 India Iran Friendly Math Competition, 2
Let $ABCD$ be a cyclic quadrilateral with circumcentre $O_1$. The diagonals $AC$ and $BD$ meet at point $P$. Suppose the four incentres of triangles $PAB, PBC, PCD, PDA$ lie on a circle with centre $O_2$. Prove that $P, O_1, O_2$ are collinear.
[i]Proposed by Shantanu Nene[/i]
2016 Indonesia TST, 2
Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]
Kyiv City MO Juniors 2003+ geometry, 2003.9.4
The diagonals of a convex quadrilateral divide it into four triangles. The radii of the circles circumscribed around these triangles are equal. Can such a property have a quadrilateral other than:
a) parallelogram,
b) rhombus?
(Sharygin Igor)
2006 South africa National Olympiad, 6
Consider the function $f$ defined by
\[f(n)=\frac{1}{n}\left (\left \lfloor\frac{n}{1}\right \rfloor+\left \lfloor\frac{n}{2}\right \rfloor+\cdots+\left \lfloor\frac{n}{n}\right \rfloor \right )\]
for all positive integers $n$. (Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.) Prove that
(a) $f(n+1)>f(n)$ for infinitely many $n$.
(b) $f(n+1)<f(n)$ for infinitely many $n$.
2020 Australian Maths Olympiad, 3
Let $ABC$ be a triangle with $\angle ACB=90^{\circ}$. Suppose that the tangent line at $C$ to the circle passing through $A,B,C$ intersects the line $AB$ at $D$. Let $E$ be the midpoint of $CD$ and let $F$ be a point on $EB$ such that $AF$ is parallel to $CD$.
Prove that the lines $AB$ and $CF$ are perpendicular.
2023 4th Memorial "Aleksandar Blazhevski-Cane", P4
Let $ABCD$ be a cyclic quadrilateral such that $AB = AD + BC$ and $CD < AB$. The diagonals $AC$ and $BD$ intersect at $P$, while the lines $AD$ and $BC$ intersect at $Q$. The angle bisector of $\angle APB$ meets $AB$ at $T$. Show that the circumcenter of the triangle $CTD$ lies on the circumcircle of the triangle $CQD$.
[i]Proposed by Nikola Velov[/i]
2000 Junior Balkan Team Selection Tests - Romania, 4
Let be a triangle $ ABC, $ and three points $ A',B',C' $ on the segments $ BC,CA, $ respectively, $ AB, $ such that the lines $ AA',BB',CC' $ are concurent at $ M. $ Name $ a,b,c,x,y,z $ the areas of the triangles $ AB'M,BC'M,CA'M,AC'M,BA'M, $ respectively, $ CB'M. $ Show that:
[b]a)[/b] $ abc=xyz $
[b]b)[/b] $ ab+bc+ca=xy+yz+zx $
[i]Bogdan Enescu[/i] and [i]Marcel Chiriță[/i]