Found problems: 85335
2023 Assara - South Russian Girl's MO, 2
In the convex quadrilateral $ABCD$, point $X$ is selected on side $AD$, and the diagonals intersect at point $E$. It is known that $AC = BD$, $\angle ABX = \angle AX B = 50^o$, $\angle CAD = 51^o$, $\angle AED = 80^o$. Find the value of angle $\angle AXC$.
2012 India PRMO, 20
$PS$ is a line segment of length $4$ and $O$ is the midpoint of $PS$. A semicircular arc is drawn with $PS$ as diameter. Let $X$ be the midpoint of this arc. $Q$ and $R$ are points on the arc $PXS$ such that $QR$ is parallel to $PS$ and the semicircular arc drawn with $QR$ as diameter is tangent to $PS$. What is the area of the region $QXROQ$ bounded by the two semicircular arcs?
2009 IMO Shortlist, 7
Let $ a_1, a_2, \ldots , a_n$ be distinct positive integers and let $ M$ be a set of $ n \minus{} 1$ positive integers not containing $ s \equal{} a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n.$ A grasshopper is to jump along the real axis, starting at the point $ 0$ and making $ n$ jumps to the right with lengths $ a_1, a_2, \ldots , a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $ M.$
[i]Proposed by Dmitry Khramtsov, Russia[/i]
1963 Polish MO Finals, 5
Prove that a fifth-degree polynomial $$ P(x) = x^5 - 3x^4 + 6x^3 - 3x^2 + 9x - 6$$ is not the product of two lower-degree polynomials with integer coefficients.
2019 LIMIT Category C, Problem 7
The value of
$$\left(1+\frac26+\frac{2\cdot5}{6\cdot12}+\frac{2\cdot5\cdot8}{6\cdot12\cdot18}+\ldots\right)^3$$
KoMaL A Problems 2019/2020, A. 757
For every $n$ non-negative integer let $S(n)$ denote a subset of the positive integers, for which $i$ is an element of $S(n)$ if and only if the $i$-th digit (from the right) in the base two representation of $n$ is a digit $1$.
Two players, $A$ and $B$ play the following game: first, $A$ chooses a positive integer $k$, then $B$ chooses a positive integer $n$ for which $2^n\geqslant k$. Let $X$ denote the set of integers $\{ 0,1,\dotsc ,2^n-1\}$, let $Y$ denote the set of integers $\{ 0,1,\dotsc ,2^{n+1}-1\}$. The game consists of $k$ rounds, and in each round player $A$ chooses an element of set $X$ or $Y$, then player $B$ chooses an element from the other set. For $1\leqslant i\leqslant k$ let $x_i$ denote the element chosen from set $X$, let $y_i$ denote the element chosen from set $Y$.
Player $B$ wins the game, if for every $1\leqslant i\leqslant k$ and $1\leqslant j\leqslant k$, $x_i<x_j$ if and only if $y_i<y_j$ and $S(x_i)\subset S(x_j)$ if and only if $S(y_i)\subset S(y_j)$. Which player has a winning strategy?
[i]Proposed by Levente Bodnár, Cambridge[/i]
2007 China Second Round Olympiad, 1
In an acute triangle $ABC$, $AB<AC$. $AD$ is the altitude dropped onto $BC$ and $P$ is a point on $AD$. Let $PE\perp AC$ at $E$, $PF\perp AB$ at $F$ and let $J,K$ be the circumcentres of triangles $BDF, CDE$ respectively. Prove that $J,K,E,F$ are concyclic if and only if $P$ is the orthocentre of triangle $ABC$.
2008 Peru IMO TST, 4
Let $\mathcal{S}_1$ and $\mathcal{S}_2$ be two non-concentric circumferences such that $\mathcal{S}_1$ is inside $\mathcal{S}_2$. Let $K$ be a variable point on $\mathcal{S}_1$. The line tangent to $\mathcal{S}_1$ at point $K$ intersects $\mathcal{S}_2$ at points $A$ and $B$. Let $M$ be the midpoint of arc $AB$ that is in the semiplane determined by $AB$ that does not contain $\mathcal{S}_1$. Determine the locus of the point symmetric to $M$ with respect to $K.$
2021/2022 Tournament of Towns, P2
Peter picked an arbitrary positive integer, multiplied it by 5, multiplied the result by 5, then multiplied the result by 5 again and so on. Is it true that from some moment all the numbers that Peter obtains contain 5 in their decimal representation?
2019 Dutch IMO TST, 4
Find all functions $f : Z \to Z$ satisfying
$\bullet$ $ f(p) > 0$ for all prime numbers $p$,
$\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.
2000 Bulgaria National Olympiad, 3
Let $A$ be the set of all binary sequences of length $n$ and denote $o =(0, 0, \ldots , 0) \in A$. Define the addition on $A$ as $(a_1, \ldots , a_n)+(b_1, \ldots , b_n) =(c_1, \ldots , c_n)$, where $c_i = 0$ when $a_i = b_i$ and $c_i = 1$ otherwise. Suppose that $f\colon A \to A$ is a function such that $f(0) = 0$, and for each $a, b \in A$, the sequences $f(a)$ and $f(b)$ differ in exactly as many places as $a$ and $b$ do. Prove that if $a$ , $b$, $c \in A$ satisfy $a+ b + c = 0$, then $f(a)+ f(b) + f(c) = 0$.
2019 Harvard-MIT Mathematics Tournament, 1
What is the smallest positive integer that cannot be written as the sum of two nonnegative palindromic integers? (An integer is [i]palindromic[/i] if the sequence of decimal digits are the same when read backwards.)
2015 Postal Coaching, Problem 1
Find all positive integer $n$ such that
$$\frac{\sin{n\theta}}{\sin{\theta}} - \frac{\cos{n\theta}}{\cos{\theta}} = n-1$$
holds for all $\theta$ which are not integral multiples of $\frac{\pi}{2}$
2019 Purple Comet Problems, 18
A container contains five red balls. On each turn, one of the balls is selected at random, painted blue, and returned to the container. The expected number of turns it will take before all fi ve balls are colored blue is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2013 BmMT, Ind. Round
[b]p1.[/b] Ten math students take a test, and the average score on the test is $28$. If five students had an average of $15$, what was the average of the other five students' scores?
[b]p2.[/b] If $a\otimes b = a^2 + b^2 + 2ab$, find $(-5\otimes 7) \otimes 4$.
[b]p3.[/b] Below is a $3 \times 4$ grid. Fill each square with either $1$, $2$ or $3$. No two squares that share an edge can have the same number. After filling the grid, what is the $4$-digit number formed by the bottom row?
[img]https://cdn.artofproblemsolving.com/attachments/9/6/7ef25fc1220d1342be66abc9485c4667db11c3.png[/img]
[b]p4.[/b] What is the angle in degrees between the hour hand and the minute hand when the time is $6:30$?
[b]p5.[/b] In a small town, there are some cars, tricycles, and spaceships. (Cars have $4$ wheels, tricycles have $3$ wheels, and spaceships have $6$ wheels.) Among the vehicles, there are $24$ total wheels. There are more cars than tricycles and more tricycles than spaceships. How many cars are there in the town?
[b]p6.[/b] You toss five coins one after another. What is the probability that you never get two consecutive heads or two consecutive tails?
[b]p7.[/b] In the below diagram, $\angle ABC$ and $\angle BCD$ are right angles. If $\overline{AB} = 9$, $\overline{BD} = 13$, and $\overline{CD} = 5$, calculate $\overline{AC}$.
[img]https://cdn.artofproblemsolving.com/attachments/7/c/8869144e3ea528116e2d93e14a7896e5c62229.png[/img]
[b]p8.[/b] Out of $100$ customers at a market, $80$ purchased oranges, $60$ purchased apples, and $70$ purchased bananas. What is the least possible number of customers who bought all three items?
[b]p9.[/b] Francis, Ted and Fred planned to eat cookies after dinner. But one of them sneaked o earlier and ate the cookies all by himself. The three say the following:
Francis: Fred ate the cookies.
Fred: Ted did not eat the cookies.
Ted: Francis is lying.
If exactly one of them is telling the truth, who ate all the cookies?
[b]p11.[/b] Let $ABC$ be a triangle with a right angle at $A$. Suppose $\overline{AB} = 6$ and $\overline{AC} = 8$. If $AD$ is the perpendicular from $A$ to $BC$, what is the length of $AD$?
[b]p12.[/b] How many three digit even numbers are there with an even number of even digits?
[b]p13.[/b] Three boys, Bob, Charles and Derek, and three girls, Alice, Elizabeth and Felicia are all standing in one line. Bob and Derek are each adjacent to precisely one girl, while Felicia is next to two boys. If Alice stands before Charles, who stands before Elizabeth, determine the number of possible ways they can stand in a line.
[b]p14.[/b] A man $5$ foot, $10$ inches tall casts a $14$ foot shadow. $20$ feet behind the man, a flagpole casts ashadow that has a $9$ foot overlap with the man's shadow. How tall (in inches) is the flagpole?
[b]p15.[/b] Alvin has a large bag of balls. He starts throwing away balls as follows: At each step, if he has $n$ balls and 3 divides $n$, then he throws away a third of the balls. If $3$ does not divide $n$ but $2$ divides $n$, then he throws away half of them. If neither $3$ nor $2$ divides $n$, he stops throwing away the balls. If he began with $1458$ balls, after how many steps does he stop throwing away balls?
[b]p16.[/b] Oski has $50$ coins that total to a value of $82$ cents. You randomly steal one coin and find out that you took a quarter. As to not enrage Oski, you quickly put the coin back into the collection. However, you are both bored and curious and decide to randomly take another coin. What is the probability that this next coin is a penny? (Every coin is either a penny, nickel, dime or quarter).
[b]p17.[/b] Let $ABC$ be a triangle. Let $M$ be the midpoint of $BC$. Suppose $\overline{MA} = \overline{MB} = \overline{MC} = 2$ and $\angle ACB = 30^o$. Find the area of the triangle.
[b]p18.[/b] A spirited integer is a positive number representable in the form $20^n + 13k$ for some positive integer $n$ and any integer $k$. Determine how many spirited integers are less than $2013$.
[b]p19. [/b]Circles of radii $20$ and $13$ are externally tangent at $T$. The common external tangent touches the circles at $A$, and $B$, respectively where $A \ne B$. The common internal tangent of the circles at $T$ intersects segment $AB$ at $X$. Find the length of $AX$.
[b]p20.[/b] A finite set of distinct, nonnegative integers $\{a_1, ... , a_k\}$ is called admissible if the integer function $f(n) = (n + a_1) ... (n + a_k)$ has no common divisor over all terms; that is, $gcd \left(f(1), f(2),... f(n)\right) = 1$ for any integer$ n$. How many admissible sets only have members of value less than $10$? $\{4\}$ and $\{0, 2, 6\}$ are such sets, but $\{4, 9\}$ and $\{1, 3, 5\}$ are not.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1997 Federal Competition For Advanced Students, P2, 1
Let $ a$ be a fixed integer. Find all integer solutions $ x,y,z$ of the system:
$ 5x\plus{}(a\plus{}2)y\plus{}(a\plus{}2)z\equal{}a,$
$ (2a\plus{}4)x\plus{}(a^2\plus{}3)y\plus{}(2a\plus{}2)z\equal{}3a\minus{}1,$
$ (2a\plus{}4)x\plus{}(2a\plus{}2)y\plus{}(a^2\plus{}3)z\equal{}a\plus{}1.$
2011 Belarus Team Selection Test, 1
Given natural number $a>1$ and different odd prime numbers $p_1,p_2,...,p_n$ with
$a^{p_1}\equiv 1$ (mod $p_2$), $a^{p_2}\equiv 1$ (mod $p_3$), ..., $a^{p_n}\equiv 1$(mod $p_1$).
Prove that
a) $(a-1)\vdots p_i$ for some $i=1,..,n$
b) Can $(a-1)$ be divisible by $p_i $for exactly one $i$ of $i=1,...,n$?
I. Bliznets
2024 Junior Balkan Team Selection Tests - Romania, P2
Let $ABC$ be a triangle with $AB<AC$ and $\omega$ be its circumcircle. The tangent line to $\omega$ at $A$ intersects line $BC$ at $D$ and let $E$ be a point on $\omega$ such that $BE$ is parallel to $AD$. $DE$ intersects segment $AB$ and $\omega$ at $F$ and $G$, respectively. The circumcircle of $BGF$ intersects $BE$ at $N$. The line $NF$ intersects lines $AD$ and $EA$ at $S$ and $T$, respectively. Prove that $DGST$ is cyclic.
2008 Oral Moscow Geometry Olympiad, 1
Each of two similar triangles was cut into two triangles so that one of the resulting parts of one triangle is similar to one of the parts of the other triangle. Is it true that the remaining parts are also similar?
(D. Shnol)
2007 India IMO Training Camp, 1
Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.
2012 Philippine MO, 4
Let $\star$ be an operation defined in the set of nonnegative integers with the following properties: for any nonnegative integers $x$ and $y$,
(i) $(x + 1)\star 0 = (0\star x) + 1$
(ii) $0\star (y + 1) = (y\star 0) + 1$
(iii) $(x + 1)\star (y + 1) = (x\star y) + 1$.
If $123\star 456 = 789$, find $246\star 135$.
2023 AMC 12/AHSME, 7
A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in $2023$ will each digit appear an even number of times in the 8-digital display for that date?
$\textbf{(A)}~5\qquad\textbf{(B)}~6\qquad\textbf{(C)}~7\qquad\textbf{(D)}~8\qquad\textbf{(E)}~9$
2015 QEDMO 14th, 12
Steve stands in the middle of a field of an infinitely large chessboard, all of which are fields square and one square meter. Every second it randomly wanders into the middle one of the four neighboring fields, each of which has the same probability. How high is the probability that after $2015$ steps, he will have taken exactly five meters way from his starting square?
2015 Thailand TSTST, 3
Let $a, b, c$ be positive real numbers. Prove that
$$\frac {3(ab + bc + ca)}{2(a^2b^2+b^2c^2+c^2a^2)}\leq \frac1{a^2 + bc} + \frac1{b^2 + ca} + \frac1{c^2 + ab}\leq\frac{a+b+c}{2abc}.$$
2011 Macedonia National Olympiad, 1
Let $~$ $ a,\,b,\,c,\,d\, >\, 0$ $~$ and $~$ $a+b+c+d\, =\, 1\, .$ $~$ Prove the inequality
\[ \frac{1}{4a+3b+c}+\frac{1}{3a+b+4d}+\frac{1}{a+4c+3d}+\frac{1}{4b+3c+d}\; \ge\; 2\, . \]