Found problems: 85335
2021 AMC 12/AHSME Fall, 2
Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch?
$\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }19\qquad\textbf{(E) }20$
1950 Miklós Schweitzer, 3
For any system $ x_1,x_2,...,x_n$ of positive real numbers, let
$ f_1(x_1,x_2,...,x_n) \equal{} x_1$,
and
$ f_{\nu} \equal{} \frac {x_1 \plus{} 2x_2 \plus{} \cdots \plus{} \nu x_{\nu}}{\nu \plus{} (\nu \minus{} 1)x_1 \plus{} (\nu \minus{} 2)x_2 \plus{} \cdots \plus{} 1\cdot x_{\nu \minus{} 1}}$
for $ \nu \equal{} 2,3,...,n$. Show that for any $ \epsilon > 0$, a positive integer $ n_0 < n_0(\epsilon)$ can be found such that for every $ n > n_0$ there exists a system $ x_1',x_2',...,x_n'$ of positive real numbers
with $ x_1' \plus{} x_2' \plus{} \cdots \plus{} x_n' \equal{} 1$ and $ f_{\nu}(x_1',x_2',...,x_n')\le \epsilon$ for $ \nu \equal{} 1,2,...,n$ .
2001 Junior Balkan Team Selection Tests - Romania, 1
Let $ABC$ be an arbitrary triangle. A circle passes through $B$ and $C$ and intersects the lines $AB$ and $AC$ at $D$ and $E$, respectively. The projections of the points $B$ and $E$ on $CD$ are denoted by $B'$ and $E'$, respectively. The projections of the points $D$ and $C$ on $BE$ are denoted by $D'$ and $C'$, respectively. Prove that the points $B',D',E'$ and $C'$ lie on the same circle.
OMMC POTM, 2022 2
Find all functions $f:\mathbb R \to \mathbb R$ (from the set of real numbers to itself) where$$f(x-y)+xf(x-1)+f(y)=x^2$$for all reals $x,y.$
Proposed by [b]cj13609517288[/b]
2022 Stanford Mathematics Tournament, 2
The straight line $y=ax+16$ intersects the graph of $y=x^3$ at $2$ distinct points. What is the value of $a$?
2002 National Olympiad First Round, 6
The thousands digit of a five-digit number which is divisible by $37$ and $173$ is $3$. What is the hundreds digit of this number?
$
\textbf{a)}\ 0
\qquad\textbf{b)}\ 2
\qquad\textbf{c)}\ 4
\qquad\textbf{d)}\ 6
\qquad\textbf{e)}\ 8
$
2010 National Olympiad First Round, 16
$11$ different books are on a $3$-shelf bookcase. In how many different ways can the books be arranged such that at most one shelf is empty?
$ \textbf{(A)}\ 75\cdot 11!
\qquad\textbf{(B)}\ 62\cdot 11!
\qquad\textbf{(C)}\ 68\cdot 12!
\qquad\textbf{(D)}\ 12\cdot 13!
\qquad\textbf{(E)}\ 6 \cdot 13!
$
2023 VIASM Summer Challenge, Problem 3
Given an $8 \times 8$ chess board. Each knight is allowed to move between two squares located at opposite vertices of $2 \times 3$ or $3 \times 2$ rectangles. There are four knights that move on the board, evenly start from the same cell $X$ and return to $X$ and then stop. Assume that every square on the chessboard has at least one of these four roosters moving through.
Prove that there exists a square $Y$ that is different from $X$ such that it is moved over no less than twice by the same knight or by different knights.
1988 AMC 12/AHSME, 27
In the figure, $AB \perp BC$, $BC \perp CD$, and $BC$ is tangent to the circle with center $O$ and diameter $AD$. In which one of the following cases is the area of $ABCD$ an integer?
[asy]
size(170);
defaultpen(fontsize(10pt)+linewidth(.8pt));
pair O=origin, A=(-1/sqrt(2),1/sqrt(2)), B=(-1/sqrt(2),-1), C=(1/sqrt(2),-1), D=(1/sqrt(2),-1/sqrt(2));
draw(unitcircle);
dot(O);
draw(A--B--C--D--A);
label("$A$",A,dir(A));
label("$B$",B,dir(B));
label("$C$",C,dir(C));
label("$D$",D,dir(D));
label("$O$",O,N);
[/asy]
$ \textbf{(A)}\ AB=3, CD=1\qquad\textbf{(B)}\ AB=5, CD=2\qquad\textbf{(C)}\ AB=7, CD=3\qquad\textbf{(D)}\ AB=9, CD=4\qquad\textbf{(E)}\ AB=11, CD=5 $
1984 Miklós Schweitzer, 3
[b]3.[/b] Let $a$ and $b$ be positive integers such that when dividing them by any prime $p$, the remainder of $a$ is always less than or equal to the remainder of $b$. Prove that $a=b$. ([b]N.16[/b])
[P. Erdos, P. P. Pálify]
2017 JBMO Shortlist, A2
Let $a$ and $b$ be positive real numbers such that $3a^2 + 2b^2 = 3a + 2b$. Find the minimum value of $A =\sqrt{\frac{a}{b(3a+2)}} + \sqrt{\frac{b}{a(2b+3)}} $
2023 Sharygin Geometry Olympiad, 20
Let a point $D$ lie on the median $AM$ of a triangle $ABC$. The tangents to the circumcircle of triangle $BDC$ at points $B$ and $C$ meet at point $K$. Prove that $DD'$ is parallel to $AK$, where $D'$ is isogonally conjugated to $D$ with respect to $ABC$.
2011 IMO Shortlist, 2
Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that
\[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\]
[i]Proposed by Alexey Gladkich, Israel[/i]
1990 Swedish Mathematical Competition, 6
Find all positive integers $m, n$ such that $\frac{117}{158} > \frac{m}{n} > \frac{97}{131}$ and $n \le 500$.
2013 Iran Team Selection Test, 12
Let $ABCD$ be a cyclic quadrilateral that inscribed in the circle $\omega$.Let $I_{1},I_{2}$ and $r_{1},r_{2}$ be incenters and radii of incircles of triangles $ACD$ and $ABC$,respectively.assume that $r_{1}=r_{2}$. let $\omega'$ be a circle that touches $AB,AD$ and touches $\omega$ at $T$. tangents from $A,T$ to $\omega$ meet at the point $K$.prove that $I_{1},I_{2},K$ lie on a line.
2008 National Olympiad First Round, 21
Let $ABC$ be a right triangle with $m(\widehat{A})=90^\circ$. Let $APQR$ be a square with area $9$ such that $P\in [AC]$, $Q\in [BC]$, $R\in [AB]$. Let $KLMN$ be a square with area $8$ such that $N,K\in [BC]$, $M\in [AB]$, and $L\in [AC]$. What is $|AB|+|AC|$?
$
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 10
\qquad\textbf{(C)}\ 12
\qquad\textbf{(D)}\ 14
\qquad\textbf{(E)}\ 16
$
2012 AMC 10, 19
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at $\text{8:00 AM}$, and all three always take the same amount of time to eat lunch. On Monday the three of them painted $50\%$ of a house, quitting at $\text{4:00 PM}$. On Tuesday, when Paula wasn't there, the two helpers painted only $24\%$ of the house and quit at $\text{2:12 PM}$. On Wednesday Paula worked by herself and finished the house by working until $\text{7:12 PM}$. How long, in minutes, was each day's lunch break?
$ \textbf{(A)}\ 30
\qquad\textbf{(B)}\ 36
\qquad\textbf{(C)}\ 42
\qquad\textbf{(D)}\ 48
\qquad\textbf{(E)}\ 60
$
2017 Caucasus Mathematical Olympiad, 2
On Mars a basketball team consists of 6 players. The coach of the team Mars can select any line-up of 6 players among 100 candidates. The coach considers some line-ups as [i]appropriate[/i] while the other line-ups are not (there exists at least one appropriate line-up). A set of 5 candidates is called [i]perspective[/i] if one more candidate could be added to it to obtain an appropriate line-up. A candidate is called [i]universal[/i] if he completes each perspective set of 5 candidates (not containing him) upto an appropriate line-up. The coach has selected a line-up of 6 universal candidates. Determine if it follows that this line-up is appropriate.
2010 Federal Competition For Advanced Students, P2, 3
On a circular billiard table a ball rebounds from the rails as if the rail was the tangent to the circle at the point of impact.
A regular hexagon with its vertices on the circle is drawn on a circular billiard table.
A (point-shaped) ball is placed somewhere on the circumference of the hexagon, but not on one of its edges.
Describe a periodical track of this ball with exactly four points at the rails.
With how many different directions of impact can the ball be brought onto such a track?
2002 Iran Team Selection Test, 4
$O$ is a point in triangle $ABC$. We draw perpendicular from $O$ to $BC,AC,AB$ which intersect $BC,AC,AB$ at $A_{1},B_{1},C_{1}$. Prove that $O$ is circumcenter of triangle $ABC$ iff perimeter of $ABC$ is not less than perimeter of triangles $AB_{1}C_{1},BC_{1}A_{1},CB_{1}A_{1}$.
1992 Canada National Olympiad, 3
In the diagram, $ ABCD$ is a square, with $ U$ and $ V$ interior points of the sides $ AB$ and $ CD$ respectively. Determine all the possible ways of selecting $ U$ and $ V$ so as to maximize the area of the quadrilateral $ PUQV$.
[img]http://i250.photobucket.com/albums/gg265/geometry101/CMO1992Number3.jpg[/img]
2023 Federal Competition For Advanced Students, P1, 4
Find all pairs of positive integers $(n, k)$ satisfying the equation $$n!+n=n^k.$$
2012 Belarus Team Selection Test, 1
For any point $X$ inside an acute-angled triangle $ABC$ we define $$f(X)=\frac{AX}{A_1X}\cdot \frac{BX}{B_1X}\cdot \frac{CX}{C_1X}$$ where $A_1, B_1$, and $C_1$ are the intersection points of the lines $AX, BX,$ and $CX$ with the sides $BC, AC$, and $AB$, respectively. Let $H, I$, and $G$ be the orthocenter, the incenter, and the centroid of the triangle $ABC$, respectively. Prove that $f(H) \ge f(I) \ge f(G)$ .
(D. Bazylev)
2015 Poland - Second Round, 1
Points $E, F, G$ lie, and on the sides $BC, CA, AB$, respectively of a triangle $ABC$, with $2AG=GB, 2BE=EC$ and $2CF=FA$. Points $P$ and $Q$ lie on segments $EG$ and $FG$, respectively such that $2EP = PG$ and $2GQ=QF$. Prove that the quadrilateral $AGPQ$ is a parallelogram.
1994 IMO, 6
Show that there exists a set $ A$ of positive integers with the following property: for any infinite set $ S$ of primes, there exist [i]two[/i] positive integers $ m$ in $ A$ and $ n$ not in $ A$, each of which is a product of $ k$ distinct elements of $ S$ for some $ k \geq 2$.