Found problems: 85335
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$.
2002 Iran MO (3rd Round), 24
$A,B,C$ are on circle $\mathcal C$. $I$ is incenter of $ABC$ , $D$ is midpoint of arc $BAC$. $W$ is a circle that is tangent to $AB$ and $AC$ and tangent to $\mathcal C$ at $P$. ($W$ is in $\mathcal C$)
Prove that $P$ and $I$ and $D$ are on a line.
2022 Canada National Olympiad, 3
Vishal starts with $n$ copies of the number $1$ written on the board. Every minute, he takes two numbers $a, b$ and replaces them with either $a+b$ or $\min(a^2, b^2)$. After $n-1$ there is $1$ number on the board. Let the maximal possible value of this number be $f(n)$. Prove $2^{n/3}<f(n)\leq 3^{n/3}$.
2021 MOAA, 16
Let $\triangle ABC$ have $\angle ABC=67^{\circ}$. Point $X$ is chosen such that $AB = XC$, $\angle{XAC}=32^\circ$, and $\angle{XCA}=35^\circ$. Compute $\angle{BAC}$ in degrees.
[i]Proposed by Raina Yang[/i]
1976 Yugoslav Team Selection Test, Problem 3
Find the minimum and maximum values of the function
$$f(x,y,z,t)=\frac{ax^2+by^2}{ax+by}+\frac{az^2+bt^2}{az+bt},~(a>0,b>0),$$given that $x+z=y+t=1$, and $x,y,z,t\ge0$.
2022 Vietnam TST, 2
Given a convex polyhedron with 2022 faces. In 3 arbitary faces, there are already number $26; 4$ and $2022$ (each face contains 1 number). They want to fill in each other face a real number that is an arithmetic mean of every numbers in faces that have a common edge with that face. Prove that there is only one way to fill all the numbers in that polyhedron.
2016 Switzerland Team Selection Test, Problem 7
Find all positive integers $n$ such that $$\sum_{d|n, 1\leq d <n}d^2=5(n+1)$$
1997 Romania National Olympiad, 2
Let $a \ne 0$ be a natural number. Prove that $a$ is a perfect square if and only if for every $b \in N^*$ there exists $c \in N^*$ such that $a + bc$ is a perfect square.
2008 India Regional Mathematical Olympiad, 2
Prove that there exist two infinite sequences $ \{a_n\}_{n\ge 1}$ and $ \{b_n\}_{n\ge 1}$ of positive integers such that the following conditions hold simultaneously:
$ (i)$ $ 0 < a_1 < a_2 < a_3 < \cdots$;
$ (ii)$ $ a_n < b_n < a_n^2$, for all $ n\ge 1$;
$ (iii)$ $ a_n \minus{} 1$ divides $ b_n \minus{} 1$, for all $ n\ge 1$
$ (iv)$ $ a_n^2 \minus{} 1$ divides $ b_n^2 \minus{} 1$, for all $ n\ge 1$
[19 points out of 100 for the 6 problems]