Found problems: 85335
2015 AMC 12/AHSME, 8
What is the value of $(625^{\log_{5}{2015}})^{\frac{1}{4}}$?
$\textbf{(A) }5\qquad\textbf{(B) }\sqrt[4]{2015}\qquad\textbf{(C) }625\qquad\textbf{(D) }2015\qquad\textbf{(E) }\sqrt[4]{5^{2015}}$
2007 Cono Sur Olympiad, 3
Show that for each positive integer $n$, there is a positive integer $k$ such that the decimal representation of each of the numbers $k, 2k,\ldots, nk$ contains all of the digits $0, 1, 2,\ldots, 9$.
2007 Mid-Michigan MO, 7-9
[b]p1.[/b] The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were on each bus?
[b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $99$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a $6$ cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $500$ coins?
[b]p3.[/b] Find all solutions $a, b, c, d, e, f, g, h$ if these letters represent distinct digits and the following multiplication is correct:
$\begin{tabular}{ccccc}
& & a & b & c \\
+ & & & d & e \\
\hline
& f & a & g & c \\
x & b & b & h & \\
\hline
f & f & e & g & c \\
\end{tabular}$
[b]p4.[/b] Is it possible to find a rectangle of perimeter $10$ m and cut it in rectangles (as many as you want) so that the sum of the perimeters is $500$ m?
[b]p5.[/b] The picture shows a maze with chambers (shown as circles) and passageways (shown as segments). A cat located in chamber $C$ tries to catch a mouse that was originally in the chamber $M$. The cat makes the first move, moving from chamber $C$ to one of the neighboring chambers. Then the mouse moves, then the cat, and so forth. At each step, the cat and the mouse can move to any neighboring chamber or not move at all. The cat catches the mouse by moving into the chamber currently occupied by the mouse. Can the cat get the mouse?
[img]https://cdn.artofproblemsolving.com/attachments/9/9/25f61e1499ff1cfeea591cb436d33eb2cdd682.png[/img]
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2022 Bulgarian Spring Math Competition, Problem 9.2
Let $\triangle ABC$ have median $CM$ ($M\in AB$) and circumcenter $O$. The circumcircle of $\triangle AMO$ bisects $CM$. Determine the least possible perimeter of $\triangle ABC$ if it has integer side lengths.
1986 Bundeswettbewerb Mathematik, 3
Let $d_n$ be the last digit, distinct from 0, in the decimal expansion of $n!$. Prove that the sequence $d_1,d_2,d_3, \ldots$ is not periodic.
1970 Spain Mathematical Olympiad, 2
A plant grows in the way we describe below. has a trunk which forks into two branches; each branch of the plant can, in turn, branch off into other two branches, or end in a bud. We will call the [i]load [/i] of a branch the total number of buds it bears, that is, the number of buds fed by the sap that passes by that branch; and we will call the [i]distance [/i] of a bud the number of bifurcations that it sap has to go through to get from the trunk to that bud.
If n is the number of bifurcations that a certain plant of that type has, it is asks
a) the number of branches of the plant,
b) the number of buds,
c) show that the sum of the charges of all the branches is equal to the sum of the clearances of all buds.
Hint: You can proceed by induction, showing that if some results are correct for a given plant, they remain correct for the plant that is obtained substituting a bud in it for a pair of branches ending in individual buds.
2003 German National Olympiad, 6
Prove that there are infinitely many coprime, positive integers $a,b$ such that $a$ divides $b^2 -5$ and $b$ divides $a^2 -5.$
2019 Romanian Master of Mathematics, 2
Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$. Let $E$ be the midpoint of $AC$. Denote by $\omega$ and $\Omega$ the circumcircles of the triangles $ABE$ and $CDE$, respectively. Let $P$ be the crossing point of the tangent to $\omega$ at $A$ with the tangent to $\Omega$ at $D$. Prove that $PE$ is tangent to $\Omega$.
[i]Jakob Jurij Snoj, Slovenia[/i]
2002 Flanders Junior Olympiad, 1
Prove that for all $a,b,c \in \mathbb{R}^+_0$ we have \[\frac{a}{bc}+\frac{b}{ac}+\frac{c}{ab} \ge \frac2a+\frac2b-\frac2c\] and determine when equality occurs.
2022 Francophone Mathematical Olympiad, 4
find the smallest integer $n\geq1$ such that the equation :
$$a^2+b^2+c^2-nd^2=0 $$
has $(0,0,0,0)$ as unique solution .
1997 Baltic Way, 7
Let $P$ and $Q$ be polynomials with integer coefficients. Suppose that the integers $a$ and $a+1997$ are roots of $P$, and that $Q(1998)=2000$. Prove that the equation $Q(P(x))=1$ has no integer solutions.
2013 Iran Team Selection Test, 18
A special kind of parallelogram tile is made up by attaching the legs of two right isosceles triangles of side length $1$. We want to put a number of these tiles on the floor of an $n\times n$ room such that the distance from each vertex of each tile to the sides of the room is an integer and also no two tiles overlap. Prove that at least an area $n$ of the room will not be covered by the tiles.
[i]Proposed by Ali Khezeli[/i]
2020 New Zealand MO, 3
There are $13$ marked points on the circumference of a circle with radius $13$. Prove that we can choose three of the marked points which form a triangle with area less than $13$.
2014 Harvard-MIT Mathematics Tournament, 6
We have a calculator with two buttons that displays and integer $x$. Pressing the first button replaces $x$ by $\lfloor \frac{x}{2} \rfloor$, and pressing the second button replaces $x$ by $4x+1$. Initially, the calculator displays $0$. How many integers less than or equal to $2014$ can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here, $\lfloor y \rfloor$ denotes the greatest integer less than or equal to the real number $y$).
2012 Princeton University Math Competition, A2 / B4
If $x, y$, and $z$ are real numbers with $\frac{x - y}{z}+\frac{y - z}{x}+\frac{z - x}{y}= 36$, find $2012 +\frac{x - y}{z}\cdot \frac{y - z}{x}\cdot\frac{z - x}{y}$ .
2013 Indonesia Juniors, day 1
p1. It is known that $f$ is a function such that $f(x)+2f\left(\frac{1}{x}\right)=3x$ for every $x\ne 0$. Find the value of $x$ that satisfies $f(x) = f(-x)$.
p2. It is known that ABC is an acute triangle whose vertices lie at circle centered at point $O$. Point $P$ lies on side $BC$ so that $AP$ is the altitude of triangle ABC. If $\angle ABC + 30^o \le \angle ACB$, prove that $\angle COP + \angle CAB < 90^o$.
p3. Find all natural numbers $a, b$, and $c$ that are greater than $1$ and different, and fulfills the property that $abc$ divides evenly $bc + ac + ab + 2$.
p4. Let $A, B$, and $ P$ be the nails planted on the board $ABP$ . The length of $AP = a$ units and $BP = b$ units. The board $ABP$ is placed on the paths $x_1x_2$ and $y_1y_2$ so that $A$ only moves freely along path $x_1x_2$ and only moves freely along the path $y_1y_2$ as in following image. Let $x$ be the distance from point $P$ to the path $y_1y_2$ and y is with respect to the path $x_1x_2$ . Show that the equation for the path of the point $P$ is $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$.
[img]https://cdn.artofproblemsolving.com/attachments/4/6/d88c337370e8c3bc5a1833bc9588d3fb047bd0.png[/img]
p5. There are three boxes $A, B$, and $C$ each containing $3$ colored white balls and $2$ red balls. Next, take three
ball with the following rules:
1. Step 1
Take one ball from box $A$.
2. Step 2
$\bullet$ If the ball drawn from box $A$ in step 1 is white, then the ball is put into box $B$. Next from box $B$ one ball is drawn, if it is a white ball, then the ball is put into box $C$, whereas if the one drawn is red ball, then the ball is put in box $A$.
$\bullet$ If the ball drawn from box $A$ in step 1 is red, then the ball is put into box $C$. Next from box $C$ one ball is taken. If what is drawn is a white ball then the ball is put into box $A$, whereas if the ball drawn is red, the ball is placed in box $B$.
3. Step 3
Take one ball each from squares $A, B$, and $C$.
What is the probability that all the balls drawn in step 3 are colored red?
2018 AMC 12/AHSME, 20
Let $ABCDEF$ be a regular hexagon with side length $1$. Denote by $X$, $Y$, and $Z$ the midpoints of sides $\overline{AB},\overline{CD},\overline{EF}$, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of $\triangle{ACE}$ and $\triangle{XYZ}$?
$\textbf{(A) }\dfrac{3}{8}\sqrt{3}\qquad\textbf{(B) }\dfrac{7}{16}\sqrt{3}\qquad\textbf{(C) }\dfrac{15}{32}\sqrt{3}\qquad\textbf{(D) }\dfrac{1}{2}\sqrt{3}\qquad\textbf{(E) }\dfrac{9}{16}\sqrt{3}$
1981 Yugoslav Team Selection Test, Problem 2
Suppose that there is a point $S$ inside a quadrilateral $ABCD$ such that segments $SA,SB,SC,SD$ divide the quadrilateral into four triangles of equal areas. Prove that one of the diagonals of the quadrilateral bisects the other one.
2008 District Olympiad, 1
Let $ \{a_n\}_{n\geq 1}$ be a sequence of real numbers such that $ |a_{n\plus{}1}\minus{}a_n|\leq 1$, for all positive integers $ n$. Let $ \{b_n\}_{n\geq 1}$ be the sequence defined by \[ b_n \equal{} \frac { a_1\plus{} a_2 \plus{} \cdots \plus{}a_n} {n}.\] Prove that $ |b_{n\plus{}1}\minus{}b_n | \leq \frac 12$, for all positive integers $ n$.
2005 Thailand Mathematical Olympiad, 4
Triangle $\vartriangle ABC$ is inscribed in the circle with diameter $BC$. If $AB = 3$, $AC = 4$, and $O$ is the incenter of $\vartriangle ABC$, then find $BO \cdot OC$.
2011 USAMTS Problems, 3
You have $14$ coins, dated $1901$ through $1914$. Seven of these coins are real and weigh $1.000$ ounce each. The other seven are counterfeit and weigh $0.999$ ounces each. You do not know which coins are real or counterfeit. You also cannot tell which coins are real by look or feel.
Fortunately for you, Zoltar the Fortune-Weighing Robot is capable of making very precise measurements. You may place any number of coins in each of Zoltar's two hands and Zoltar will do the following:
[list][*] If the weights in each hand are equal, Zoltar tells you so and returns all of the coins.
[*] If the weight in one hand is heavier than the weight in the other, then Zoltar takes one coin, at random, from the heavier hand as tribute. Then Zoltar tells you which hand was heavier, and returns the remaining coins to you.[/list]
Your objective is to identify a single real coin that Zoltar has not taken as tribute. Is there a strategy that guarantees this? If so, then describe the strategy and why it works. If not, then prove that no such strategy exists.
2021 Czech and Slovak Olympiad III A, 1
A fraction with $1010$ squares in the numerator and $1011$ squares in the denominator serves as a game board for a two player game. $$\frac{\square + \square +...+ \square}{\square + \square +...+ \square+ \square}$$ Players take turns in moves. In each turn, the player chooses one of the numbers $1, 2,. . . , 2021$ and inserts it in any empty field. Each number can only be used once. The starting player wins if the value of the fraction after all the fields is filled differs from number $1$ by less than $10^{-6}$. Otherwise, the other player wins. Decide which of the players has a winning strategy.
(Pavel Šalom)
2010 Nordic, 2
Three circles $\Gamma_A$, $\Gamma_B$ and $\Gamma_C$ share a common point of intersection $O$. The other common point of $\Gamma_A$ and $\Gamma_B$ is $C$, that of $\Gamma_A$ and $\Gamma_C$ is $B$, and that of $\Gamma_C$ and $\Gamma_B$ is $A$. The line $AO$ intersects the circle $\Gamma_A$ in the point $X \ne O$. Similarly, the line $BO$ intersects the circle $\Gamma_B$ in the point $Y \ne O$, and the line $CO$ intersects the circle $\Gamma_C$ in the point $Z \ne O$. Show that
\[\frac{|AY |\cdot|BZ|\cdot|CX|}{|AZ|\cdot|BX|\cdot|CY |}= 1.\]
1977 Czech and Slovak Olympiad III A, 2
The numbers $p,q>0$ are given. Construct a rectangle $ABCD$ with $AE=p,AF=q$ where $E,F$ are midpoints of $BC,CD,$ respectively. Discuss conditions of solvability.
2012 JBMO TST - Turkey, 1
Let $a, b, c$ be the side-lengths of a triangle, $r$ be the inradius and $r_a, r_b, r_c$ be the corresponding exradius. Show that
\[ \frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \leq 2 \cdot \frac{\sqrt{{r_a}^2+{r_b}^2+{r_c}^2}}{r_a+r_b+r_c-3r} \]