Found problems: 85335
It is known that, having $6$ weighs, it is possible to balance the scales with loads, which weights are successing natural numbers from $1$ to $63$. Find all such sets of weighs.
Let $p$ be the largest prime less than $2013$ for which \[ N = 20 + p^{p^{p+1}-13} \] is also prime. Find the remainder when $N$ is divided by $10^4$.
[i]Proposed by Evan Chen and Lewis Chen[/i]
Let $a, b, c, d$ and $u$ be integers such that each of the numbers
$$ac\ \ , \ \ bc + ad \ \ , \ \ bd$$
is a multiple of $u$. Show that $bc$ and $ad$ are multiples of $u$.
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.
Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$ , $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$. The diagonals and intersect at point $P$ . Determine the measure of $\angle{APD}$.
Ward and Gabrielle are playing a game on a large sheet of paper. At the start of the game, there are $999$ ones on the sheet of paper. Ward and Gabrielle each take turns alternatingly, and Ward has the first turn.
During their turn, a player must pick two numbers a and b on the sheet such that $gcd(a, b) = 1$, erase these numbers from the sheet, and write the number $a + b$ on the sheet. The first player who is not able to do so, loses.
Determine which player can always win this game.
Let $I$ be the center of inscribed circle of right triangle $\Delta ABC$ with $\angle A = 90$ and point $M$ is the midpoint of $(BC)$.The bisector of $\angle BAC$ intersects the circumcircle of $\Delta ABC $ in point $W$.Point $U$ is situated on the line $AB$ such that the lines $AB$ and $WU$ are perpendiculars.Point $P$ is situated on the line $WU$ such that the lines $PI$ and $WU$ are perpendiculars.Prove that the line $MP$ bisects the segment $CI$.
Let $x,y,z$ be positive real numbers such that
$x^4+y^4+z^4=1$ .
Determine with proof the minimum value of
$\frac{x^3}{1-x^8}+\frac{y^3}{1-y^8}+\frac{z^3}{1-z^8}$
Maria have $14$ days to train for an olympiad. The only conditions are that she cannot train by $3$ consecutive days and she cannot rest by $3$ consecutive days. Determine how many configurations of days(in training) she can reach her goal.
Given some square carpets with the total area $4$. Prove that they can fully cover the unit square.
Given four points $O,\ A,\ B,\ C$ on a plane such that $OA=4,\ OB=3,\ OC=2,\ \overrightarrow{OB}\cdot \overrightarrow{OC}=3.$
Find the maximum area of $\triangle{ABC}$.
Let $S$ be a string of $99$ characters, $66$ of which are $A$ and $33$ are $B$. We call $S$ [i]good[/i] if, for each $n$ such that $1\le n \le 99$, the sub-string made from the first $n$ characters of $S$ has an odd number of distinct permutations. How many good strings are there? Which strings are good?
Determine all pairs of positive integers $(a,b)$ such that the following fraction is an integer: \[\frac{(a+b)^2}{4+4a(a-b)^2}.\]
Determine whether or nor there exist two disjoint infinite sets $ A$ and $ B$ of points in the plane satisfying the following conditions:
a.) No three points in $ A \cup B$ are collinear, and the distance between any two points in $ A \cup B$ is at least 1.
b.) There is a point of $ A$ in any triangle whose vertices are in $ B,$ and there is a point of $ B$ in any triangle whose vertices are in $ A.$
Solve the system of equations in integers:
\[\begin{cases}z^x=y^{2x}\\ 2^z=4^x\\ x+y+z=20.\end{cases}\]
We have $n{}$ coins, one of which is fake, which differs in weight from the real ones and a two-pan scale which works correctly if the weights on the pans are different, but can show any outcome if the weights on the pans are equal. For what $n{}$ can we determine which coin is fake and whether it is lighter or heavier than the real coins, in at most $k{}$ weightings?
[i]Proposed by A. Zaslavsky[/i]
Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the figure. She plants one flower per square foot in each region. Asters cost $ \$$1 each, begonias $ \$$1.50 each, cannas $ \$$2 each, dahlias $ \$$2.50 each, and Easter lilies $ \$$3 each. What is the least possible cost, in dollars, for her garden?
[asy]unitsize(5mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
draw((6,0)--(0,0)--(0,1)--(6,1));
draw((0,1)--(0,6)--(4,6)--(4,1));
draw((4,6)--(11,6)--(11,3)--(4,3));
draw((11,3)--(11,0)--(6,0)--(6,3));
label("1",(0,0.5),W);
label("5",(0,3.5),W);
label("3",(11,1.5),E);
label("3",(11,4.5),E);
label("4",(2,6),N);
label("7",(7.5,6),N);
label("6",(3,0),S);
label("5",(8.5,0),S);[/asy]$ \textbf{(A)}\ 108 \qquad
\textbf{(B)}\ 115 \qquad
\textbf{(C)}\ 132 \qquad
\textbf{(D)}\ 144 \qquad
\textbf{(E)}\ 156$
Let $x,y,z,w $ be real numbers such that $x+2y+3z+4w=1$. Find the minimum of $x^2+y^2+z^2+w^2+(x+y+z+w)^2$.
Let $\Delta ABC$ be an acute-angled triangle and let $H$ be its orthocentre. Let $G_1, G_2$ and $G_3$ be the centroids of the triangles $\Delta HBC , \Delta HCA$ and $\Delta HAB$ respectively. If the area of $\Delta G_1G_2G_3$ is $7$ units, what is the area of $\Delta ABC $?
Let $u_n$ denote the least common multiple of the first $n$ terms of a strictly increasing sequence of positive integers.
Prove that the series
$$\sum_{n=1}^{\infty} \frac{1}{ u_n }$$
is convergent
Compute $1+2\cdot3^4$.
[i]Proposed by Evan Chen[/i]
A $\text{5.0 kg}$ block with a speed of $\text{8.0 m/s}$ travels $\text{2.0 m}$ along a horizontal surface where it makes a head-on, perfectly elastic collision with a $\text{15.0 kg}$ block which is at rest. The coefficient of kinetic friction between both blocks and the surface is $0.35$. How far does the $\text{15.0 kg}$ block travel before coming to rest?
(A) $\text{0.76 m}$
(B) $\text{1.79 m}$
(C) $\text{2.29 m}$
(D) $\text{3.04 m}$
(E) $\text{9.14 m}$
Define $ x\otimes y \equal{} x^3 \minus{} y$. What is $ h\otimes (h\otimes h)$?
$ \textbf{(A) } \minus{} h\qquad \textbf{(B) } 0\qquad \textbf{(C) } h\qquad \textbf{(D) } 2h\qquad \textbf{(E) } h^3$
Find the number of values of $x$ such that the number of square units in the area of the isosceles triangle with sides $x$, $65$, and $65$ is a positive integer.
$T_1$ consists of a single branch from 0 to 1 in the complex plane. This branch then splits into two branches at the endpoint which each form a $135^\circ$ angle with the branch in $T_1$, and each branch has length $\frac{1}{\sqrt{2}}$. This process is repeated so that from each terminal branch in $T_n$, we form two more branches at angles $135^\circ$ with $\frac{1}{\sqrt{2}}$ the length. Let $L_n$ be the collection of the $2^{n-1}$ endpoints of the tree $T_n$. If multiple terminal branches end at the same point, then that point is counted multiple times in $L_n$. Shown below is $T_k$ for $k=1, 2, 3$ with dots at the points in $L_k$. Find the sum of $\ell^2$ over all points $\ell$ in $L_{10}$.