Found problems: 85335
[b]p1.[/b] Find the largest k such that the equation $x^2 - 2x + k = 0$ has at least one real root.
[b]p2.[/b] Indiana Jones needs to cross a flimsy rope bridge over a mile long gorge. It is so dark that it is impossible to cross the bridge without a flashlight. Furthermore, the bridge is so weak that it can only support the weight of two people. The party has only one flashlight, which has a weak beam so whenever two people cross, they are constrained to walk together, at the speed of the slower person. Indiana Jones can cross the bridge in $5$ minutes. His girlfriend can cross in $10$ minutes. His father needs $20$ minutes, and his father’s side kick needs $25$ minutes. They need to get everyone across safely in on hour to escape the bay guys. Can they do it?
[b]p3.[/b] There are ten big bags with coins. Nine of them contain fare coins weighing $10$ g. each, and one contains counterfeit coins weighing $9$ g. each. By one weighing on a digital scale find the bag with counterfeit coins.
[b]p4.[/b] Solve the equation: $\sqrt{x^2 + 4x + 4} = x^2 + 5x + 5$.
[b]p5.[/b] (a) In the $x - y$ plane, analytically determine the length of the path $P \to A \to C \to B \to P$ around the circle $(x - 6)^2 + (y - 8)^2 = 25$ from the point $P(12, 16)$ to itself.
[img]https://cdn.artofproblemsolving.com/attachments/f/b/24888b5b478fa6576a54d0424ce3d3c6be2855.png[/img]
(b) Determine coordinates of the points $A$ and $B$.
[b]p6.[/b] (a) Let $ABCD$ be a convex quadrilateral (it means that diagonals are inside the quadrilateral). Prove that
$$Area\,\, (ABCD) \le \frac{|AB| \cdot |AD| + |BC| \cdot |CD|}{2}$$
(b) Let $ABCD$ be an arbitrary quadrilateral (not necessary convex). Prove the same inequality as in part (a).
(c) For an arbitrary quadrilateral $ABCD$ prove that $Area\,\, (ABCD) \le \frac{|AB| \cdot |CD| + |BC| \cdot |AD|}{2}$
PS. You should use hide for answers.
What is the smallest possible sum of six distinct positive integers for which the sum of any five of them is prime?
Given a triangle $ABC$ with $\angle A = 45^\circ$. Let $A'$ be the antipode of $A$ in the circumcircle of $ABC$. Points $E$ and $F$ on segments $AB$ and $AC$ respectively are such that $A'B = BE$, $A'C = CF$. Let $K$ be the second intersection of circumcircles of triangles $AEF$ and $ABC$. Prove that $EF$ bisects $A'K$.
We call a tetrahedron divisor of a parallelepiped if the parallelepiped can be divided into $6$ copies of that tetrahedron. Does there exist a parallelepiped that it has at least two different divisor tetrahedrons?
Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral.
Let $ABC$ be a triangle with $AB=209$, $AC=243$, and $\angle BAC = 60^\circ$, and denote by $N$ the midpoint of the major arc $\widehat{BAC}$ of circle $\odot(ABC)$. Suppose the parallel to $AB$ through $N$ intersects $\overline{BC}$ at a point $X$. Compute the ratio $\tfrac{BX}{XC}$.
Let $O$ denote the origin and let $\gamma$ be the circle with center $(1,0)$ and radius $1$ in the Cartesian system of coordinates. Let $\lambda$ be a real number from the interval $(0,2)$, and let the line $x=\lambda$ intersect the circle $\gamma$ at points $P$ and $Q$. The lines $OP$ and $OQ$ intersect the line $x=2-\lambda$ at the points $P'$ and $Q'$, respectively. Let $\mathcal G$ denote the locus of such points $P'$ and $Q'$ as $\lambda$ varies over the interval $(0,2)$. Prove that there exist points $R$ and $S$ different from the origin in the plane such that for every $A\in \mathcal G$ there exists a point $A'$ on line $OA$ satisfying
\[ A'R^2=(A'S-OS)^2=A'A\cdot A'O.\]
[i]Proposed by: Áron Bán-Szabó, Budapest[/i]
Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points?
Let $a, b, c$, and $d$ be real numbers. The six sums of two numbers $x$ and $y$, different from the previous four, are $117$, $510$, $411$, $252$, in no particular order. Determine the maximum possible value of $x + y$.
Altitudes $AA_1$ and $CC_1$ of an acute-angled triangle $ABC$ intersect at point $H$. A straight line passing through point $H$ parallel to line $A_1C_1$ intersects the circumscribed circles of triangles $AHC_1$ and $CHA_1$ at points $X$ and $Y$, respectively. Prove that points $X$ and $Y$ are equidistant from the midpoint of segment $BH$.
Let $A$ be an $m\times n$ matrix with rational entries. Suppose that there are at least $m+n$ distinct prime numbers among the absolute values of the entries of $A.$ Show that the rank of $A$ is at least $2.$
Prove that the number $\binom np-\left\lfloor\frac np\right\rfloor$ is divisible by $p$ for every prime number and integer $n\ge p$.
Source: 1976 Euclid Part A Problem 1
-----
In the diagram, $ABCD$ and $EFGH$ are similar rectangles. $DK:KC=3:2$. Then rectangle $ABCD:$ rectangle $EFGH$ is equal to
[asy]draw((75,0)--(0,0)--(0,50)--(75,50)--(75,0)--(55,0)--(55,20)--(100,20)--(100,0)--cycle);
draw((55,5)--(60,5)--(60,0));
draw((75,5)--(80,5)--(80,0));
label("A",(0,50),NW);
label("B",(0,0),SW);
label("C",(75,0),SE);
label("D",(75,50),NE);
label("E",(55,20),NW);
label("F",(55,0),SW);
label("G",(100,0),SE);
label("H",(100,20),NE);
label("K",(75,20),NE);[/asy]
$\textbf{(A) } 3:2 \qquad \textbf{(B) } 9:4 \qquad \textbf{(C) } 5:2 \qquad \textbf{(D) } 25:4 \qquad \textbf{(E) } 6:2$
Triangle $ABC$ is given. Let $M$ be the midpoint of the segment $AB$ and $T$ be the midpoint of the arc $BC$ not containing $A$ of the circumcircle of $ABC.$ The point $K$ inside the triangle $ABC$ is such that $MATK$ is an isosceles trapezoid with $AT\parallel MK.$ Show that $AK = KC.$
There are $ n\geq 5$ people in a party. Assume that among any three of them some two know each other. Show that one can select at least $ \frac{n}{2}$ people and arrange them at a round table so that each person sits between two of his/her acquaintances.
Consider a matrix whose entries are integers. Adding a same integer to all entries on a same row, or on a same column, is called an operation. It is given that, for infinitely many positive integers $n$, one can obtain, through a finite number of operations, a matrix having all entries divisible by $n$. Prove that, through a finite number of operations, one can obtain the null matrix.
A circle passing through the vertices $A$ and $B$ of a cyclic quadrilateral $ABCD$ intersects diagonals $AC$ and $BD$ at $E$ and $F$, respectively. The lines $AF$ and $BC$ meet at a point $P$, and the lines $BE$ and $AD$ meet at a point $Q$. Prove that $PQ$ is parallel to $CD$.
Find all odd postive integers less than $2007$ such that the sum of all of its positive divisors is odd.
Let $ABC$ be an acute triangle, and let $M$ and $N$ be two points on the line $AC$ such that the vectors $MN$ and $AC$ are identical. Let $X$ be the orthogonal projection of $M$ on $BC$, and let $Y$ be the orthogonal projection of $N$ on $AB$. Finally, let $H$ be the orthocenter of triangle $ABC$.
Show that the points $B$, $X$, $H$, $Y$ lie on one circle.
Paul starts at $1$ and counts by threes: $1, 4, 7, 10, ... $. At the same time and at the same speed, Penny counts backwards from $2017$ by fi ves: $2017, 2012, 2007, 2002,...$ . Find the one number that both Paul and Penny count at the same time.
Northside's Drum and Bugle Corps raised money for a trip. The drummers and bugle players kept separate sales records. According to the double bar graph, in what month did one group's sales exceed the other's by the greatest percent?
[asy]
unitsize(12);
fill((2,0)--(2,9)--(3,9)--(3,0)--cycle,lightgray);
draw((3,0)--(3,9)--(2,9)--(2,0));
draw((2,7)--(1,7)--(1,0));
draw((2,8)--(3,8));
draw((2,7)--(3,7));
for (int a = 1; a <= 6; ++a)
{
draw((1,a)--(3,a));
}
fill((5,0)--(5,3)--(6,3)--(6,0)--cycle,lightgray);
draw((6,0)--(6,3)--(5,3)--(5,0));
draw((5,3)--(5,5)--(4,5)--(4,0));
draw((4,4)--(5,4));
draw((4,3)--(5,3));
draw((4,2)--(6,2));
draw((4,1)--(6,1));
fill((8,0)--(8,6)--(9,6)--(9,0)--cycle,lightgray);
draw((9,0)--(9,6)--(8,6)--(8,0));
draw((8,6)--(8,9)--(7,9)--(7,0));
draw((7,8)--(8,8));
draw((7,7)--(8,7));
draw((7,6)--(8,6));
for (int a = 1; a <= 5; ++a)
{
draw((7,a)--(9,a));
}
fill((11,0)--(11,12)--(12,12)--(12,0)--cycle,lightgray);
draw((12,0)--(12,12)--(11,12)--(11,0));
draw((11,9)--(10,9)--(10,0));
draw((11,11)--(12,11));
draw((11,10)--(12,10));
draw((11,9)--(12,9));
for (int a = 1; a <= 8; ++a)
{
draw((10,a)--(12,a));
}
fill((14,0)--(14,10)--(15,10)--(15,0)--cycle,lightgray);
draw((15,0)--(15,10)--(14,10)--(14,0));
draw((14,8)--(13,8)--(13,0));
draw((14,9)--(15,9));
draw((14,8)--(15,8));
for (int a = 1; a <= 7; ++a)
{
draw((13,a)--(15,a));
}
draw((16,0)--(0,0)--(0,13),black);
label("Jan",(2,0),S);
label("Feb",(5,0),S);
label("Mar",(8,0),S);
label("Apr",(11,0),S);
label("May",(14,0),S);
label("$\textbf{MONTHLY SALES}$",(8,14),N);
label("S",(0,8),W);
label("A",(0,7),W);
label("L",(0,6),W);
label("E",(0,5),W);
label("S",(0,4),W);
draw((4,12.5)--(4,13.5)--(5,13.5)--(5,12.5)--cycle);
label("Drums",(4,13),W);
fill((15,12.5)--(15,13.5)--(16,13.5)--(16,12.5)--cycle,lightgray);
draw((15,12.5)--(15,13.5)--(16,13.5)--(16,12.5)--cycle);
label("Bugles",(15,13),W);[/asy]
$\text{(A)}\ \text{Jan} \qquad \text{(B)}\ \text{Feb} \qquad \text{(C)}\ \text{Mar} \qquad \text{(D)}\ \text{Apr} \qquad \text{(E)}\ \text{May}$
Oscar buys 13 pencils and 3 erasers for $ \$1.00$. A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser?
$ \textbf{(A) } 10\qquad \textbf{(B) } 12\qquad \textbf{(C) } 15\qquad \textbf{(D) } 18\qquad \textbf{(E) } 20$
Let us say, that a natural number has the property $P(k)$ if it can be represented as a product of $k$ succeeding natural numbers greater than $1$.
a) Find k such that there exists n which has properties $P(k)$ and $P(k+2)$ simultaneously.
b) Prove that there is no number having properties $P(2)$ and $P(4)$ simultaneously
We have a pool table $8$ meters long and $2$ meters wide with a single ball in the center. We throw the ball in a straight line and, after traveling $29$ meters, it stops at a corner of the table. How many times did the ball hit the edges of the table?
Note: When the ball rebounds on the edge of the table, the two angles that form its trajectory with the edge of the table are the same.
Triangle $ABC$ has $AB=25$, $AC=29$, and $BC=36$. Additionally, $\Omega$ and $\omega$ are the circumcircle and incircle of $\triangle ABC$. Point $D$ is situated on $\Omega$ such that $AD$ is a diameter of $\Omega$, and line $AD$ intersects $\omega$ in two distinct points $X$ and $Y$. Compute $XY^2$.
[i]Proposed by David Altizio[/i]