Found problems: 85335
2018 Dutch IMO TST, 3
Determine all pairs $(a,b)$ of positive integers such that $(a+b)^3-2a^3-2b^3$ is a power of two.
2017 Pan-African Shortlist, G3
Let $ABCDE$ be a regular pentagon, and $F$ some point on the arc $AB$ of the circumcircle of $ABCDE$. Show that
\[
\frac{FD}{FE + FC} = \frac{FB + FA}{FD} = \frac{-1 + \sqrt{5}}{2},
\]
and that $FD + FB + FA = FE + FC$.
2021 Sharygin Geometry Olympiad, 6
Three circles $\Gamma_1,\Gamma_2,\Gamma_3$ are inscribed into an angle(the radius of $\Gamma_1$ is the minimal, and the radius of $\Gamma_3$ is the maximal) in such a way that $\Gamma_2$ touches $\Gamma_1$ and $\Gamma_3$ at points $A$ and $B$ respectively. Let $\ell$ be a tangent to $A$ to $\Gamma_1$. Consider circles $\omega$ touching $\Gamma_1$ and $\ell$. Find the locus of meeting points of common internal tangents to $\omega$ and $\Gamma_3$.
2000 Czech And Slovak Olympiad IIIA, 4
For which quadratic polynomials $f(x)$ does there exist a quadratic polynomial $g(x)$ such that the equations $g(f(x)) = 0$ and $f(x)g(x) = 0$ have the same roots, which are mutually distinct and form an arithmetic progression?
2015 Regional Olympiad of Mexico Center Zone, 4
Find all natural integers $m, n$ such that $m, 2+m, 2^n+m, 2+2^n+m$ are all prime numbers
1999 Greece National Olympiad, 1
Let $f(x)=ax^2+bx+c$, where $a,b,c$ are nonnegative real numbers, not all equal to zero. Prove that $f(xy)^2\le f(x^2)f(y^2)$ for all real numbers $x,y$.
2004 Regional Olympiad - Republic of Srpska, 4
Set $S=\{1,2,...,n\}$ is firstly divided on $m$ disjoint nonempty subsets, and then on $m^2$ disjoint nonempty subsets. Prove that some $m$ elements of set $S$ were after first division in same set, and after the second division were in $m$ different sets
2017 Korea Winter Program Practice Test, 2
Alice and Bob play a game. There are $100$ gold coins, $100$ silver coins, and $100$ bronze coins. Players take turns to take at least one coin, but they cannot take two or more coins of same kind at once. Alice goes first. The player who cannot take any coin loses. Who has a winning strategy?
2011 Finnish National High School Mathematics Competition, 3
Points $D$ and $E$ divides the base $BC$ of an isosceles triangle $ABC$ into three equal parts and $D$ is between $B$ and $E.$ Show that $\angle BAD<\angle DAE.$
2022 Mexico National Olympiad, 1
A number $x$ is "Tlahuica" if there exist prime numbers $p_1,\ p_2,\ \dots,\ p_k$ such that
\[x=\frac{1}{p_1}+\frac{1}{p_2}+\dots+\frac{1}{p_k}.\]
Find the largest Tlahuica number $x$ such that $0<x<1$ and there exists a positive integer $m\leq 2022$ such that $mx$ is an integer.
2020 JHMT, MS Team
Use the following description of a machine to solve the first 4 problems in the round.
A machine displays four digits: $0000$. There are two buttons: button $A$ moves all digits one position to the left and fills the rightmost position with $0$ (for example, it changes $1234$ to $2340$), and button $B$ adds $11$ to the current number, displaying only the last four digits if the sum is greater than $9999$ (for example, it changes $1234$ to $1245$, and changes $9998$ to $0009$). We can denote a sequence of moves by writing down the buttons pushed from left to right. A sequence of moves that outputs $2100$, for example, is $BABAA$.
[b]p1[/b]. Give a sequence of $17$ or less moves so that the machine displays $2020$.
[b]p2.[/b] Using the same machine, how many outputs are possible if you make at most three moves?
[b]p3.[/b] Button $ B$ now adds n to the four digit display, while button $ A$ remains the same. For how many positive integers $n \le 20$ (including $11$) can every possible four-digit output be reached?
[b]p4.[/b] Suppose the function of button $ A$ changes to: move all digits one position to the right and fill the leftmost position with $2$. Then, what is the minimum number of moves required for the machine to display $2020$, if it initially displays $0000$?
[b]p5.[/b] In the figure below, every inscribed triangle has vertices that are on the midpoints of its circumscribed triangle’s sides. If the area of the largest triangle is $64$, what is the area of the shaded region?
[img]https://cdn.artofproblemsolving.com/attachments/6/f/fe17b6a6d0037163f0980a5a5297c1493cc5bb.png[/img]
[b]p6.[/b] A bee flies $10\sqrt2$ meters in the direction $45^o$ clockwise of North (that is, in the NE direction). Then, the bee turns $135^o$ clockwise, and flies $20$ forward meters. It continues by turning $60^o$ counterclockwise, and flies forward $14$ meters. Finally, the bee turns $120^o$ clockwise and flies another $14$ meters forward before finally finding a flower to pollinate. How far is the bee from its starting location in meters?
[b]p7.[/b] All the digits of a $15$-digit number are either $p$ or $c$. $p$ shows up $3$ more times than $c$ does, and the average of the digits is $c - p$. What is $p + c$?
[b]p8.[/b] Let $m$ be the sum of the factors of $75$ (including $1$ and $75$ itself). What is the ones digit of $m^{75}$ ?
[b]p9.[/b] John flips a coin twice. For each flip, if it lands tails, he does nothing. If it lands heads, he rolls a fair $4$-sided die with sides labeled 1 through $4$. Let $a/b$ be the probability of never rolling a $3$, in simplest terms. What is $a + b$?
[b]p10.[/b] Let $\vartriangle ABC$ have coordinates $(0, 0)$, $(0, 3)$,$(18, 0)$. Find the number of integer coordinates interior (excluding the vertices and edges) of the triangle.
[b]p11.[/b] What is the greatest integer $k$ such that $2^k$ divides the value $20! \times 20^{20}$?
[b]p12.[/b] David has $n$ pennies, where $n$ is a natural number. One apple costs $3$ pennies, one banana costs $5$ pennies, and one cranberry costs $7$ pennies. If David spends all his money on apples, he will have $2$ pennies left; if David spends all his money on bananas, he will have $4$ pennies left; is David spends all his money on cranberries, he will have $6$ pennies left. What is the second least possible amount of pennies that David can have?
[b]p13.[/b] Elvin is currently at Hopperville which is $40$ miles from Waltimore and $50$ miles from Boshington DC. He takes a taxi back to Waltimore, but unfortunately the taxi gets lost. Elvin now finds himself at Kinsville, but he notices that he is still $40$ miles from Waltimore and $50$ miles from Boshington $DC$. If Waltimore and Boshington DC are $30$ miles apart, What is the maximum possible distance between Hopperville and Kinsville?
[b]p14.[/b] After dinner, Rick asks his father for $1000$ scoops of ice cream as dessert. Rick’s father responds, “I will give you $2$ scoops of ice cream, plus $ 1$ additional scoop for every ordered pair $(a, b)$ of real numbers satisfying $\frac{1}{a + b}= \frac{1}{a}+ \frac{1}{b}$ you can find.” If Rick finds every solution to the equation, how many scoops of ice cream will he receive?
[b]p15.[/b] Esther decides to hold a rock-paper-scissors tournament for the $56$ students at her school. As a rule, competitors must lose twice before they are eliminated. Each round, all remaining competitors are matched together in best-of-1 rock-paper-scissors duels. If there is an odd number of competitors in a round, one random competitor will not compete that round. What is the maximum number of matches needed to determine the rock-paper-scissors champion?
[b]p16.[/b] $ABCD$ is a rectangle. $X$ is a point on $\overline{AD}$, $Y$ is a point on $\overline{AB}$, and $N$ is a point outside $ABCD$ such that $XYNC$ is also a rectangle and $YN$ intersects $\overline{BC}$ at its midpoint $M$. $ \angle BYM = 45^o$. If $MN = 5$, what is the sum of the areas of $ABCD$ and $XYNC$?
[b]p17. [/b] Mr. Brown has $10$ identical chocolate donuts and $15$ identical glazed donuts. He knows that Amar wants $6$ donuts, Benny wants $9$ donuts, and Callie wants $9$ donuts. How many ways can he distribute out his $25$ donuts?
[b]p18.[/b] When Eric gets on the bus home, he notices his $ 12$-hour watch reads $03: 30$, but it isn’t working as expected. The second hand makes a full rotation in $4$ seconds, then makes another in $8$ seconds, then another in $ 12$ seconds, and so on until it makes a full rotation in $60$ seconds. Then it repeats this process, and again makes a full rotation in $4$ second, then $8$ seconds, etc. Meanwhile, the minute hand and hour hand continue to function as if every full rotation of the second hand represents $60$ seconds. When Eric gets off the bus $75$ minutes later, his watch reads $AB: CD$. What is $A + B + C + D$?
[b]p19.[/b] Alex and Betty want to meet each other at the airport. Alex will arrive at the airport between $12: 00$ and $13: 15$, and will wait for Betty for $15$ minutes before he leaves. Betty will arrive at the airport between $12: 30$ and $13: 10$, and will wait for Alex for $10$ minutes before she leaves. The chance that they arrive at any time in their respective time intervals is equally likely. The probability that they will meet at the airport can be expressed as $a/b$ where $a/b$ is a fraction written in simplest form. What is $a + b$?
[b]p20.[/b] Let there be $\vartriangle ABC$ such that $A = (0, 0)$, $B = (23, 0)$, $C = (a, b)$. Furthermore, $D$, the center of the circle that circumscribes $\vartriangle ABC$, lies on $\overline{AB}$. Let $\angle CDB = 150^o$. If the area of $\vartriangle ABC$ is $m/n$ where $m, n$ are in simplest integer form, find the value of $m \,\, \mod \,\,n$ (The remainder of $m$ divided by $n$).
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2023 Austrian MO Regional Competition, 1
Let $a$, $b$ and $c$ be real numbers with $0 \le a, b, c \le 2$. Prove that
$$(a - b)(b - c)(a- c) \le 2.$$
When does equality hold?
[i](Karl Czakler)[/i]
2021 Macedonian Balkan MO TST, Problem 3
Suppose that $a_1, a_2, \dots a_{2021}$ are non-negative numbers such that $\sum_{k=1}^{2021} a_k=1$. Prove that
$$ \sum_{k=1}^{2021}\sqrt[k]{a_1 a_2\dots a_k} \leq 3. $$
IV Soros Olympiad 1997 - 98 (Russia), 10.3
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 $r(A, B) = max\{| x_1- x_2 | , | y_1 - y_2 |\}$.
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 . Find the locus of points $C$ for which
a) $r(A, C) + r(C, B) = r(A, B)$
b) $r(A, C) = r(C, B).$
2013 Saudi Arabia BMO TST, 3
Find all positive integers $x, y, z$ such that $2^x + 21^y = z^2$
2004 Junior Tuymaada Olympiad, 2
For which natural $ n \geq 3 $ numbers from 1 to $ n $ can be arranged by a circle so that each number does not exceed $60$ % of the sum of its two neighbors?
2019 USAMTS Problems, 5
Post your solutions below! :D
[b]Also, I think it is beneficial to everyone if you all attempt to comment on each other's solutions.[/b]
5/1/31. Let $n$ be a positive integer. For integers a, b with $0 \leq a b \leq n - 1$, let $r_n(a, b)$ denote
the remainder when $ab$ is divided by $n$. If $S_n$ denotes the sum of all $n^2$ remainders $r_n(a, b)$,
prove that
$\frac{1}{2}-\frac{1}{\sqrt{n}}\leq \frac{S_n}{n^3} \leq \frac{1}{2}$
2022 Benelux, 3
Let $ABC$ be a scalene acute triangle. Let $B_1$ be the point on ray $[AC$ such that $|AB_1|=|BB_1|$. Let $C_1$ be the point on ray $[AB$ such that $|AC_1|=|CC_1|$. Let $B_2$ and $C_2$ be the points on line $BC$ such that $|AB_2|=|CB_2|$ and $|BC_2|=|AC_2|$. Prove that $B_1$, $C_1$, $B_2$, $C_2$ are concyclic.
1998 AIME Problems, 12
Let $ABC$ be equilateral, and $D, E,$ and $F$ be the midpoints of $\overline{BC}, \overline{CA},$ and $\overline{AB},$ respectively. There exist points $P, Q,$ and $R$ on $\overline{DE}, \overline{EF},$ and $\overline{FD},$ respectively, with the property that $P$ is on $\overline{CQ}, Q$ is on $\overline{AR},$ and $R$ is on $\overline{BP}.$ The ratio of the area of triangle $ABC$ to the area of triangle $PQR$ is $a+b\sqrt{c},$ where $a, b$ and $c$ are integers, and $c$ is not divisible by the square of any prime. What is $a^{2}+b^{2}+c^{2}$?
2016 AIME Problems, 3
Let $x,y$ and $z$ be real numbers satisfying the system \begin{align*}
\log_2(xyz-3+\log_5 x) &= 5 \\
\log_3(xyz-3+\log_5 y) &= 4 \\
\log_4(xyz-3+\log_5 z) &= 4.
\end{align*} Find the value of $|\log_5 x|+|\log_5 y|+|\log_5 z|$.
2013 Stars Of Mathematics, 4
Given a (fixed) positive integer $N$, solve the functional equation
\[f \colon \mathbb{Z} \to \mathbb{R}, \ f(2k) = 2f(k) \textrm{ and } f(N-k) = f(k), \ \textrm{for all } k \in \mathbb{Z}.\]
[i](Dan Schwarz)[/i]
2023 Novosibirsk Oral Olympiad in Geometry, 6
Let's call a convex figure, the boundary of which consists of two segments and an arc of a circle, a mushroom-gon (see fig.). An arbitrary mushroom-gon is given. Use a compass and straightedge to draw a straight line dividing its area in half.
[img]https://cdn.artofproblemsolving.com/attachments/d/e/e541a83a7bb31ba14b3637f82e6a6d1ea51e22.png[/img]
2008 Harvard-MIT Mathematics Tournament, 23
Two mathematicians, Kelly and Jason, play a cooperative game. The computer selects some secret positive integer $ n < 60$ (both Kelly and Jason know that $ n < 60$, but that they don't know what the value of $ n$ is). The computer tells Kelly the unit digit of $ n$, and it tells Jason the number of divisors of $ n$. Then, Kelly and Jason have the following dialogue:
Kelly: I don't know what $ n$ is, and I'm sure that you don't know either. However, I know that $ n$ is divisible by at least two different primes.
Jason: Oh, then I know what the value of $ n$ is.
Kelly: Now I also know what $ n$ is.
Assuming that both Kelly and Jason speak truthfully and to the best of their knowledge, what are all the possible values of $ n$?
2019 Durer Math Competition Finals, 13
There are $12$ chairs arranged in a circle, numbered from $ 1$ to $ 12$. How many ways are there to select some of the chairs in such a way that our selection includes $3$ consecutive chairs somewhere?
2016 USAMO, 5
An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\in\overline{AB}$, $Q\in\overline{AC}$, and $N,P\in\overline{BC}$. Let $S$ be the intersection of $\overleftrightarrow{MN}$ and $\overleftrightarrow{PQ}$. Denote by $\ell$ the angle bisector of $\angle MSQ$.
Prove that $\overline{OI}$ is parallel to $\ell$, where $O$ is the circumcenter of triangle $ABC$, and $I$ is the incenter of triangle $ABC$.