Found problems: 85335
Geometry Mathley 2011-12, 3.1
$AB,AC$ are tangent to a circle $(O)$, $B,C$ are the points of tangency. $Q$ is a point iside the angle $BAC$, on the ray $AQ$, take a point $P$ suc that $OP$ is perpendicular to $AQ$. The line $OP$ meets the circumcircles triangles $BPQ$ and $CPQ$ at $I, J$. Prove that $OI = OJ$.
Hồ Quang Vinh
Kvant 2024, M2819
Ten children have several bags of candies. The children begin to divide these candies among them. They take turns picking their shares of candies from each bag, and leave just after that. The size of the share is determined as follows: the current number of candies in the bag is divided by the number of remaining children (including the one taking the turn). If the remainder is nonzero than the quotient is rounded to the lesser integer. Is it possible that all the children receive different numbers of candies if the total number of bags is:
a) 8 ;
6) 99 ?
Alexey Glebov
1999 All-Russian Olympiad Regional Round, 9.2
In triangle $ABC$, on side $AC$ there are points $D$ and $E$, that $AB = AD$ and $BE = EC$ ($E$ between $A$ and $D$). Point $F$ is midpoint of arc $BC$ of circumcircle of triangle $ABC$. Prove that the points $B, E, D, F$ lie on the same circle.
2018 Azerbaijan Junior NMO, 1
First $20$ positive integers are written on a board. It is known that, after you erase a number from the board, there exists a number that is equal to the arithmetic mean of the rest of the numbers left on the board. Find all the numbers that could've been erased.
2011 Serbia JBMO TST, 3
Let $\triangle ABC$ be a right-angled triangle and $BC > AC$. $M$ is a point on $BC$ such that $BM = AC$ and $N$ is a point on $AC$ such that $AN = CM$. Find the angle between $BN$ and $AM$.
2018 Costa Rica - Final Round, LRP4
On a $30\times 30$ board both rows $ 1$ to $30$ and columns are numbered, in addition, to each box is assigned the number $ij$, where the box is in row $i$ and column $j$.
$N$ columns and $m$ rows are chosen, where $1 <n$ and $m <30$, and the cells that are simultaneously in any of the rows and in any of the selected columns are painted blue. They paint the others red .
(a) Prove that the sum of the numbers in the blue boxes cannot be prime.
(b) Can the sum of the numbers in the red cells be prime?
2007 Korea - Final Round, 3
Find all triples of $ (x, y, z)$ of positive intergers satisfying $ 1\plus{}{4}^{x}\plus{}{4}^{y}\equal{}z^2$.
2012 Junior Balkan Team Selection Tests - Moldova, 2
Let $ a,b,c,d$ be positive real numbers and $cd=1$. Prove that there exists a positive integer $n$ such that
$ab\leq n^2\leq (a+c)(b+d)$
1985 AMC 8, 18
Nine copies of a certain pamphlet cost less than $ \$10.00$ while ten copies of the same pamphlet (at the same price) cost more than $ \$11.00$. How much does one copy of this pamphlet cost?
\[ \textbf{(A)}\ \$1.07 \qquad
\textbf{(B)}\ \$1.08 \qquad
\textbf{(C)}\ \$1.09 \qquad
\textbf{(D)}\ \$1.10 \qquad
\textbf{(E)}\ \$1.11
\]
1979 Dutch Mathematical Olympiad, 2
Solve in $N$:
$$\begin{cases} a^3=b^3+c^3+12a \\ a^2=5(b+c) \end{cases}$$
2017 Saudi Arabia JBMO TST, 6
Find all pairs of prime numbers $(p, q)$ such that $p^2 + 5pq + 4q^2$ is a perfect square.
1953 Moscow Mathematical Olympiad, 246
a) On a plane, $11$ gears are arranged so that the teeth of the first gear mesh with the teeth of the second gear, the teeth of the second gear with those of the third gear, etc., and the teeth of the last gear mesh with those of the first gear. Can the gears rotate?
b) On a plane, $n$ gears are arranged so that the teeth of the first gear mesh with the teeth of the second gear, the teeth of the second gear with those of the third gear, etc., and the teeth of the last gear mesh with those of the first gear. Can the gears rotate?
1982 USAMO, 5
$A,B$, and $C$ are three interior points of a sphere $S$ such that $AB$ and $AC$ are perpendicular to the diameter of $S$ through $A$, and so that two spheres can be constructed through $A$, $B$, and $C$ which are both tangent to $S$. Prove that the sum of their radii is equal to the radius of $S$.
PEN O Problems, 3
Prove that the set of integers of the form $2^{k}-3$ ($k=2,3,\cdots$) contains an infinite subset in which every two members are relatively prime.
2016 CHMMC (Fall), 3
For a positive integer $m$, let $f(m)$ be the number of positive integers $q \le m$ such that $\frac{q^2-4}{m}$ is an integer. How many positive square-free integers $m < 2016$ satisfy $f(m) \ge 16$?
1998 Greece Junior Math Olympiad, 3
Let $k$ be a prime, such as $k\neq 2, 5$, prove that between the first $k$ terms of the sequens $1, 11, 111, 1111,....,1111....1$, where the last term have $k$ ones, is divisible by $k$.
2016 JBMO Shortlist, 3
A $5 \times 5$ table is called regular f each of its cells contains one of four pairwise distinct real numbers,such that each of them occurs exactly one in every $2 \times 2$ subtable.The sum of all numbers of a regular table is called the total sum of the table.With any four numbers,one constructs all possible regular tables,computes their total sums and counts the distinct outcomes.Determine the maximum possible count.
1999 Denmark MO - Mohr Contest, 3
A function $f$ satisfies $$f(x)+xf(1-x)=x$$ for all real numbers $x$. Determine the number $f (2)$. Find $f$ .
2007 ITAMO, 2
We define two polynomials with integer coefficients P,Q to be similar if the coefficients of P are a permutation of the coefficients of Q.
a) if P,Q are similar, then $P(2007)-Q(2007)$ is even
b) does there exist an integer $k > 2$ such that $k \mid P(2007)-Q(2007)$ for all similar polynomials P,Q?
2007 F = Ma, 32
A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. Let the rotational inertia of the rod about its center be $md^2$.
The rod is suspended from a distance $kd$ from the center, and undergoes small oscillations with an angular frequency $\beta \sqrt{\frac{g}{d}}$.
Find an expression for $\beta$ in terms of $k$.
$ \textbf{(A)}\ 1+k^2$
$ \textbf{(B)}\ \sqrt{1+k^2}$
$ \textbf{(C)}\ \sqrt{\frac{k}{1+k}}$
$ \textbf{(D)}\ \sqrt{\frac{k^2}{1+k}}$
$ \textbf{(E)}\ \text{none of the above}$
2014 Contests, 2
Paul owes Paula $35$ cents and has a pocket full of $5$-cent coins, $10$-cent coins, and $25$-cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5$
Kyiv City MO 1984-93 - geometry, 1991.10.5
Diagonal sections of a regular 8-gon pyramid, which are drawn through the smallest and largest diagonals of the base, are equal. At what angle is the plane passing through the vertex, the pyramids and the smallest diagonal of the base inclined to the base?
[hide=original wording]Діагональні перерізи правильної 8-кутної піраміди, які Проведені через найменшу і найбільшу діагоналі основи, рівновеликі. Під яким кутом до основи нахилена площина, що проходить через вершину, піраміди і найменшу діагональ основи?[/hide]
2009 AMC 10, 10
Triangle $ ABC$ has a right angle at $ B$. Point $ D$ is the foot of the altitude from $ B$, $ AD\equal{}3$, and $ DC\equal{}4$. What is the area of $ \triangle{ABC}$?
[asy]unitsize(5mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;
pair B=(0,0), C=(sqrt(28),0), A=(0,sqrt(21));
pair D=foot(B,A,C);
pair[] ps={B,C,A,D};
draw(A--B--C--cycle);
draw(B--D);
draw(rightanglemark(B,D,C));
dot(ps);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,NE);
label("$3$",midpoint(A--D),NE);
label("$4$",midpoint(D--C),NE);[/asy]$ \textbf{(A)}\ 4\sqrt3 \qquad
\textbf{(B)}\ 7\sqrt3 \qquad
\textbf{(C)}\ 21 \qquad
\textbf{(D)}\ 14\sqrt3 \qquad
\textbf{(E)}\ 42$
2013 Korea Junior Math Olympiad, 4
Prove that there exists a prime number $p$ such that the minimum positive integer $n$ such that $p|2^n -1$ is $3^{2013}$.
2010 USAJMO, 3
Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.