Found problems: 85335
2023 Malaysia IMONST 2, 6
Suppose $ABCD$ is a cyclic quadrilateral with $\angle ABC = \angle ADC = 90^{\circ}$. Let $E$ and $F$ be the feet of perpendiculars from $A$ and $C$ to $BD$ respectively. Prove that $BE = DF$.
1985 Miklós Schweitzer, 2
[b]2.[/b] Let $S$ be a given finite set of hyperplanes in $\mathbb{R}^n$, and let $O$ be a point.
Show that there exists a compact set $K \subseteq \mathbb{R}^n$ containing $O$ such that the orthogonal projection of any point of $K$ onto any hyperplane in $S$ is also in $K$. ([b]G.37[/b])
[Gy. Pap]
1984 AMC 12/AHSME, 8
Figure $ABCD$ is a trapezoid with $AB || DC, AB = 5, BC = 3 \sqrt 2, \measuredangle BCD = 45^\circ$, and $\measuredangle CDA = 60^\circ$. The length of $DC$ is
$\textbf{(A) }7 + \frac{2}{3} \sqrt{3}\qquad
\textbf{(B) }8\qquad
\textbf{(C) }9 \frac{1}{2}\qquad
\textbf{(D) }8 + \sqrt 3\qquad
\textbf{(E) }8 + 3 \sqrt 3$
1985 Vietnam Team Selection Test, 3
Suppose a function $ f: \mathbb R\to \mathbb R$ satisfies $ f(f(x)) \equal{} \minus{} x$ for all $ x\in \mathbb R$. Prove that $ f$ has infinitely many points of discontinuity.
2016 AMC 12/AHSME, 22
How many ordered triples $(x, y, z)$ of positive integers satisfy $\text{lcm}(x, y) = 72$, $\text{lcm}(x, z)= 600$, and $\text{lcm}(y, z) = 900$?
$\textbf{(A) } 15 \qquad\textbf{(B) } 16 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 27 \qquad\textbf{(E) } 64$
2018 Junior Regional Olympiad - FBH, 4
It is given $4$ circles in a plane and every one of them touches the other three as shown:
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC82L2FkYWQ5NThhMWRiMjAwZjYxOWFhYmE1M2YzZDU5YWI2N2IyYzk2LnBuZw==&rn=a3J1Z292aS5wbmc=[/img]
Biggest circle has radius $2$, and every one of the medium has $1$. Find out the radius of fourth circle.
2020 Yasinsky Geometry Olympiad, 3
There is a ruler and a "rusty" compass, with which you can construct a circle of radius $R$. The point $K$ is from the line $\ell$ at a distance greater than $R$. How to use this ruler and this compass to draw a line passing through the point $K$ and perpendicular to line $\ell$?
(Misha Sidorenko, Katya Sidorenko, Rodion Osokin)
2024 IMAR Test, P3
Let $ABC$ be a triangle . A circle through $B$ and $C$ crosses sides $AB$ and $AC$ at $P$ and $Q$, respectively. Points $X$ and $Y$ on segments $BQ$ and $CP$, respectively, satisfy $\angle ABY=\angle AXP$ and $ACX=\angle AYQ$. Prove that $XY$ and $BC$ are parallel.
2015 CHMMC (Fall), 4
The following number is the product of the divisors of $n$.
$$46, 656, 000, 000$$
What is $n$?
2008 National Olympiad First Round, 31
If the inequality
\[
((x+y)^2+4)((x+y)^2-2)\geq A\cdot (x-y)^2
\]
is hold for every real numbers $x,y$ such that $xy=1$, what is the largest value of $A$?
$
\textbf{(A)}\ 12
\qquad\textbf{(B)}\ 14
\qquad\textbf{(C)}\ 16
\qquad\textbf{(D)}\ 18
\qquad\textbf{(E)}\ 20
$
2010 IMO Shortlist, 1
In a concert, 20 singers will perform. For each singer, there is a (possibly empty) set of other singers such that he wishes to perform later than all the singers from that set. Can it happen that there are exactly 2010 orders of the singers such that all their wishes are satisfied?
[i]Proposed by Gerhard Wöginger, Austria[/i]
2024 LMT Fall, 15
Amy has a six-sided die which always rolls values greater than or equal to the previous roll. She rolls the die repeatedly until she rolls a $6$. Find the expected value of the sum of all distinct values she has rolled when she finishes.
2000 AMC 12/AHSME, 16
A checkerboard of $ 13$ rows and $ 17$ columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered $ 1, 2, \ldots , 17$, the second row $ 18, 19, \ldots , 34$, and so on down the board. If the board is renumbered so that the left column, top to bottom, is $ 1, 2, \ldots , 13$, the second column $ 14, 15, \ldots , 26$ and so on across the board, some squares have the same numbers in both numbering systems. Find the sum of the numbers in these squares (under either system).
$ \textbf{(A)}\ 222 \qquad \textbf{(B)}\ 333 \qquad \textbf{(C)}\ 444 \qquad \textbf{(D)}\ 555 \qquad \textbf{(E)}\ 666$
2016 IFYM, Sozopol, 8
Let $a_i$, $i=1,2,…2016$, be fixed natural numbers. Prove that there exist infinitely many 2016-tuples $x_1,x_2…x_{2016}$ of natural numbers, for which the sum
$\sum_{i=1}^{2016}{a_i x_i^i}$
is a 2017-th power of a natural number.
1965 Putnam, A4
At a party, assume that no boy dances with every girl but each girl dances with at least one boy. Prove that there are two couples $gb$ and $g'b'$ which dance whereas $b$ does not dance with $g'$ nor does $g$ dance with $b'$.
2008 Baltic Way, 16
Let $ABCD$ be a parallelogram. The circle with diameter $AC$ intersects the line $BD$ at points $P$ and $Q$. The perpendicular to the line $AC$ passing through the point $C$ intersects the lines $AB$ and $AD$ at points $X$ and $Y$, respectively. Prove that the points $P,Q,X$ and $Y$ lie on the same circle.
2024 Belarusian National Olympiad, 8.7
On the diagonal $AC$ of the convex quadrilateral $ABCD$ points $P$,$Q$ are chosen such that triangles $ABD$,$PCD$ and $QBD$ are similar to each other in this order.
Prove that $AQ=PC$
[i]M. Zorka[/i]
2011 Sharygin Geometry Olympiad, 23
Given are triangle $ABC$ and line $\ell$ intersecting $BC, CA$ and $AB$ at points $A_1, B_1$ and $C_1$ respectively. Point $A'$ is the midpoint of the segment between the projections of $A_1$ to $AB$ and $AC$. Points $B'$ and $C'$ are defined similarly.
(a) Prove that $A', B'$ and $C'$ lie on some line $\ell'$.
(b) Suppose $\ell$ passes through the circumcenter of $\triangle ABC$. Prove that in this case $\ell'$ passes through the center of its nine-points circle.
[i]M. Marinov and N. Beluhov[/i]
2017 China Girls Math Olympiad, 2
Given quadrilateral $ABCD$ such that $\angle BAD+2 \angle BCD=180 ^ \circ .$
Let $E$ be the intersection of $BD$ and the internal bisector of $\angle BAD$.
The perpendicular bisector of $AE$ intersects $CB,CD$ at $X,Y,$ respectively.
Prove that $A,C,X,Y$ are concyclic.
2018 APMO, 1
Let $H$ be the orthocenter of the triangle $ABC$. Let $M$ and $N$ be the midpoints of the sides $AB$ and $AC$, respectively. Assume that $H$ lies inside the quadrilateral $BMNC$ and that the circumcircles of triangles $BMH$ and $CNH$ are tangent to each other. The line through $H$ parallel to $BC$ intersects the circumcircles of the triangles $BMH$ and $CNH$ in the points $K$ and $L$, respectively. Let $F$ be the intersection point of $MK$ and $NL$ and let $J$ be the incenter of triangle $MHN$. Prove that $F J = F A$.
1999 May Olympiad, 2
In a unit circle where $O$ is your circumcenter, let $A$ and $B$ points in the circle with $\angle BOA = 90$. In the arc $AB$(minor arc) we have the points $P$ and $Q$ such that $PQ$ is parallel to $AB$. Let $X$ and $Y$ be the points of intersections of the line $PQ$ with $OA$ and $OB$ respectively. Find the value of $PX^2 + PY^2$
1986 Kurschak Competition, 2
Let $n>2$ be a positive integer. Find the largest value $h$ and the smallest value $H$ for which
\[h<{a_1\over a_1+a_2}+{a_2\over a_2+a_3}+\cdots+{a_n\over a_n+a_1}<H\]
holds for any positive reals $a_1,\dots,a_n$.
2024 Brazil EGMO TST, 2
Let \( n, k \geq 1 \). In a school, there are \( n \) students and \( k \) clubs. Each student participates in at least one of the clubs. One day, a school uniform was established, which could be either blue or red. Each student chose only one of these colors. Every day, the principal visited one of the clubs, forcing all the students in it to switch the colors of the uniforms they wore.
Assuming that the students are distributed in clubs in such a way that any initial choice of uniforms they make, after a certain number of days, it is possible to have at most one student with one of the colors. Show that
\[
n \geq 2^{n-k-1} - 1.
\]
2017 CMIMC Algebra, 9
Define a sequence $\{a_{n}\}_{n=1}^{\infty}$ via $a_{1} = 1$ and $a_{n+1} = a_{n} + \lfloor \sqrt{a_{n}} \rfloor$ for all $n \geq 1$. What is the smallest $N$ such that $a_{N} > 2017$?
2001 AMC 12/AHSME, 25
Consider sequences of positive real numbers of the form $ x,2000,y,...,$ in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of $ x$ does the term 2001 appear somewhere in the sequence?
$ \textbf{(A)} \ 1 \qquad \textbf{(B)} \ 2 \qquad \textbf{(C)} \ 3 \qquad \textbf{(D)} \ 4 \qquad \textbf{(E)} \ \text{more than 4}$