Found problems: 85335
Let $ABC$ be an acute-angled triangle, $O$ its circumcenter and $H$ its orthocenter. The orthogonal projection of the vertex $A$ to the line $BC$ lies on the perpendicular bisector of the segment $AC$. Compute $\frac{CH}{BO}$ .
(J. Willemson)
Squares $ BCA_{1}A_{2}$ , $ CAB_{1}B_{2}$ , $ ABC_{1}C_{2}$ are outwardly drawn on sides of triangle $ \triangle ABC$. If $ AB_{1}A'C_{2}$ , $ BC_{1}B'A_{2}$ , $ CA_{1}C'B_{2}$ are parallelograms then prove that:
(i) Lines $ BC$ and $ AA'$ are orthogonal.
(ii)Triangles $ \triangle ABC$ and $ \triangle A'B'C'$ have common centroid
What is the number of terms with rational coefficients among the $1001$ terms of the expression $( x \sqrt[3]{2} + y \sqrt{3})^{1000}$?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 166 \qquad\textbf{(C)}\ 167 \qquad\textbf{(D)}\
500 \qquad\textbf{(E)}\ 501$
To control her blood pressure, Jill's grandmother takes one half of a pill every other day. If one supply of medicine contains $60$ pills, then the supply of medicine would last approximately
$\text{(A)}\ 1\text{ month} \qquad \text{(B)}\ 4\text{ months} \qquad \text{(C)}\ 6\text{ months} \qquad \text{(D)}\ 8\text{ months} \qquad \text{(E)}\ 1\text{ year}$
Let $a_1, a_2, . . . , a_n$ be real numbers. Prove
$\sqrt[3]{a_1^3+ a_2^3+ . . . + a_n^3} \le \sqrt{a_1^2+ a_2^2+ . . . + a_n^2} $ (1)
When does equality hold in (1)?
Find all non-negative integers $ m$ and $ n$ that satisfy the equation:
\[ 107^{56}(m^2\minus{}1)\minus{}2m\plus{}5\equal{}3\binom{113^{114}}{n}\]
(If $ n$ and $ r$ are non-negative integers satisfying $ r\le n$, then $ \binom{n}{r}\equal{}\frac{n}{r!(n\minus{}r)!}$ and $ \binom{n}{r}\equal{}0$ if $ r>n$.)
Monica and Bogdan are playing a game, depending on given integers $n, k$. First, Monica writes some $k$ positive numbers. Bogdan wins, if he is able to find $n$ points on the plane with the following property: for any number $m$ written by Monica, there are some two points chosen by Bogdan with distance exactly $m$ between them. Otherwise, Monica wins.
Determine who has a winning strategy depending on $n, k$.
[i](Proposed by Fedir Yudin)[/i]
A "Hishgad" lottery ticket contains the numbers $1$ to $mn$, arranged in some order in a table with $n$ rows and $m$ columns. It is known that the numbers in each row increase from left to right and the numbers in each column increase from top to bottom. An example for $n=3$ and $m=4$:
[asy]
size(3cm);
Label[][] numbers = {{"$1$", "$2$", "$3$", "$9$"}, {"$4$", "$6$", "$7$", "$10$"}, {"$5$", "$8$", "$11$", "$12$"}};
for (int i=0; i<5;++i) {
draw((i,0)--(i,3));
}
for (int i=0; i<4;++i) {
draw((0,i)--(4,i));
}
for (int i=0; i<4;++i){
for (int j=0; j<3;++j){
label(numbers[2-j][i], (i+0.5, j+0.5));
}}
[/asy]
When the ticket is bought the numbers are hidden, and one must "scratch" the ticket to reveal them. How many cells does it always suffice to reveal in order to determine the whole table with certainty?
For any two points $A (x_1 , y_1)$ and $B (x_2, y_2)$, the distance $r (A, B)$ between them is determined by the equality $r(A, B) = | x_1- x_2 | + | y_1 - y_2 |$.
Prove that the triangle inequality $r(A, C) + r(C, B) \ge r(A, B)$. holds for the distance introduced in this way .
Let $A$ and $B$ be two points of the plane (you can take $A(1, 3)$, $B(3, 7)$). Find the locus of points $C$ for which
a) $r(A, C) + r(C, B) = r(A, B)$
b) $r(A, C) = r(C, B).$
Alberto wants to invite Ximena to his house. Since Alberto knows that Ximena is amateur to mathematics, instead of pointing out exactly which Transantiago buses serve him, he tells him: [i]the numbers of the buses that take me to my house have three digits, where the leftmost digit is not null, furthermore, these numbers are multiples of $13$, and the second digit of them is the average of the other two.[/i]
What are the bus lines that go to Alberto's house?
Oh no! While playing Mario Party, Theo has landed inside the Bowser Zone. If his next roll is between $1$ and $5$ inclusive, Bowser will shoot his ``Zero Flame" that sets a player's coin and star counts to zero. Fortunately, Theo has a double dice block, which lets him roll two fair $10$-sided dice labeled $1$-$10$ and take the sum of the rolls as his "roll". If he uses his double dice block, what is the probability he escapes the Bowser zone without losing his coins and stars?
[i]Proposed by Connor Gordon[/i]
Prove that from any sequence of $1996$ real numbers $a_1$, $a_2$,$...$, $a_{1996}$ one can choose one or several numbers standing successively one after another so that their sum differs from an integer by less than $0.001$.
(A Kanel)
Let points $ A = (0,0) , \ B = (1,2), \ C = (3,3), $ and $ D = (4,0) $. Quadrilateral $ ABCD $ is cut into equal area pieces by a line passing through $ A $. This line intersects $ \overline{CD} $ at point $ \left (\frac{p}{q}, \frac{r}{s} \right ) $, where these fractions are in lowest terms. What is $ p + q + r + s $?
$ \textbf{(A)} \ 54 \qquad \textbf{(B)} \ 58 \qquad \textbf{(C)} \ 62 \qquad \textbf{(D)} \ 70 \qquad \textbf{(E)} \ 75 $
A wine glass with a cross section as shown has the property of an orange in shape as a sphere with a radius of $3$ cm just can be placed in the glass without protruding above glass. Determine the height $h$ of the glass.
[img]https://1.bp.blogspot.com/-IuLm_IPTvTs/XzcH4FAjq5I/AAAAAAAAMYY/qMi4ng91us8XsFUtnwS-hb6PqLwAON_jwCLcBGAsYHQ/s0/1994%2BMohr%2Bp1.png[/img]
How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$
$\textbf{(A) } 80 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 125 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 500$
Find all ordered pairs of positive integers $(m,n)$ for which there exists a set $C=\{c_1,\ldots,c_k\}$ ($k\ge1$) of colors and an assignment of colors to each of the $mn$ unit squares of a $m\times n$ grid such that for every color $c_i\in C$ and unit square $S$ of color $c_i$, exactly two direct (non-diagonal) neighbors of $S$ have color $c_i$.
[i]David Yang.[/i]
[b]p1.[/b] Twenty five horses participate in a competition. The competition consists of seven runs, five horse compete in each run. Each horse shows the same result in any run it takes part. No two horses will give the same result. After each run you can decide what horses participate in the next run. Could you determine the three fastest horses? (You don’t have stopwatch. You can only remember the order of the horses.)
[b]p2.[/b] Prove that the equation $x^6-143x^5-917x^4+51x^3+77x^2+291x+1575=0$ does not have solutions in integer numbers.
[b]p3.[/b] Show how we can cut the figure shown in the picture into two parts for us to be able to assemble a square out of these two parts. Show how we can assemble a square.
[img]https://cdn.artofproblemsolving.com/attachments/7/b/b0b1bb2a5a99195688638425cf10fe4f7b065b.png[/img]
[b]p4.[/b] The city of Vyatka in Russia produces local drink, called “Vyatka Cola”. «Vyatka Cola» is sold in $1$, $3/4$, and $1/2$-gallon bottles. Ivan and John bought $4$ gallons of “Vyatka Cola”. Can we say for sure, that they can split the Cola evenly between them without opening the bottles?
[b]p5.[/b] Positive numbers a, b and c satisfy the condition $a + bc = (a + b)(a + c)$. Prove that $b + ac = (b + a)(b + c)$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Consider the system \[x+y=z+u,\] \[2xy=zu.\] Find the greatest value of the real constant $m$ such that $m \le \frac{x}{y}$ for any positive integer solution $(x, y, z, u)$ of the system, with $x \ge y$.
Is it true that from an arbitrary triangle you can cut three equal figures, the area of each of which is more than a quarter of the area triangle?
A triangle $ABC$ is given together with an arbitrary circle $\omega$. Let $\alpha$ be the reflection of $\omega$ with respect to $A$, $\beta$ the reflection of $\omega$ with respect to $B$, and $\gamma$ the reflection of $\omega$ with respect to $C$. It is known that the circles $\alpha, \beta, \gamma$ don't intersect each other.
Let $P$ be the meeting point of the two internal common tangents to $\beta, \gamma$ (see picture). Similarly, $Q$ is the meeting point of the internal common tangents of $\alpha, \gamma$, and $R$ is the meeting point of the internal common tangents of $\alpha, \beta$.
Prove that the triangles $PQR, ABC$ are congruent.
How many integers $n$ are there such that $n^4 + 2n^3 + 2n^2 + 2n + 1$ is a prime number?
Let $\mathfrak F=\{A_1,A_2,\ldots,A_n\}$ be a collection of subsets of the set $S=\{1,2,\ldots,n\}$ satisfying the following conditions:
(a) Any two distinct sets from $\mathfrak F$ have exactly one element in common;
(b) each element of $S$ is contained in exactly $k$ of the sets in $\mathfrak F$.
Can $n$ be equal to $1996$?
Prove that on a coordinate plane it is impossible to draw a closed broken line such that [list][*] coordinates of each vertex are rational, [*] the length of its every edge is equal to $1$, [*] the line has an odd number of vertices.[/list]
Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions:
(i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$;
(ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$.
Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list]
[i]Proposed by Hans Zantema, Netherlands[/i]
Find the value of real parameter $ a$ such that $ 2$ is the smallest integer solution of \[ \frac{x\plus{}\log_2 (2^x\minus{}3a)}{1\plus{}\log_2 a} >2.\]