Found problems: 85335
1969 IMO Longlists, 7
$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.
2019 CMIMC, 2
Determine the number of ordered pairs of positive integers $(m,n)$ with $1\leq m\leq 100$ and $1\leq n\leq 100$ such that
\[
\gcd(m+1,n+1) = 10\gcd(m,n).
\]
1997 Estonia National Olympiad, 5
In the creation of the world there is a lonely island inhabited by dragons, snakes and crocodiles. Every inhabitant eats once a day: every snake eats one dragon for breakfast, every dragon eats one crocodile for lunch and every crocodile eats a snake for dinner. Find the total number of dragons, snakes and crocodiles on the island immediately after the creation of the world (at the beginning of the first day), when, at the end of the sixth day, there is only one inhabitant alive on the island, only one crocodile and during these six days none of the inhabitants of the island considered any to give up their meals due to lack of food.
2018 PUMaC Algebra B, 2
For what value of $n$ is $\frac{1}{2\cdot5}+\frac{1}{5\cdot8}+\frac{1}{8\cdot 11}+\frac{1}{n(n+3)}=\frac{25}{154}$?
2005 Alexandru Myller, 2
Let be a point $ P $ inside a triangle $ ABC. $ Prove that the following relations are equivalent:
$ \text{(i)} $ Any collinear triple of points $ (E,P,F) $ with $ E,F $ on $ AB,AC, $ respectively, verifies the equality
$$ \frac{1}{AE} +\frac{1}{AF} =\frac{AB+BC+CA}{AB\cdot AC} $$
$ \text{(ii)} P $ is the incircle of $ ABC $
1964 IMO, 2
Suppose $a,b,c$ are the sides of a triangle. Prove that \[ a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc \]
2002 China Team Selection Test, 3
Seventeen football fans were planning to go to Korea to watch the World Cup football match. They selected 17 matches. The conditions of the admission tickets they booked were such that
- One person should book at most one admission ticket for one match;
- At most one match was same in the tickets booked by every two persons;
- There was one person who booked six tickets.
How many tickets did those football fans book at most?
2013 National Olympiad First Round, 20
The numbers $1,2,\dots, 2013$ are written on $2013$ stones weighing $1,2,\dots, 2013$ grams such that each number is used exactly once. We have a two-pan balance that shows the difference between the weights at the left and the right pans. No matter how the numbers are written, if it is possible to determine in $k$ weighings whether the weight of each stone is equal to the number that is written on the stone, what is the least possible value of $k$?
$
\textbf{(A)}\ 15
\qquad\textbf{(B)}\ 12
\qquad\textbf{(C)}\ 10
\qquad\textbf{(D)}\ 7
\qquad\textbf{(E)}\ \text{None of above}
$
2020 Purple Comet Problems, 18
In isosceles $\vartriangle ABC, AB = AC, \angle BAC$ is obtuse, and points $E$ and $F$ lie on sides $AB$ and $AC$, respectively, so that $AE = 10, AF = 15$. The area of $\vartriangle AEF$ is $60$, and the area of quadrilateral $BEFC$ is $102$. Find $BC$.
2023 Malaysian Squad Selection Test, 2
Let $ABC$ be a triangle with orthocenter $H$. Let $\ell_b, \ell_c$ be the reflection of lines $AB$ and $AC$ about $AH$ respectively. Suppose $\ell_b$ intersect $CH$ at $P$, and $\ell_c$ intersect $BH$ at $Q$. Prove that $AH, PQ, BC$ are concurrent.
[i]Proposed by Ivan Chan Kai Chin[/i]
ABMC Online Contests, 2019 Nov
[b]p1.[/b] The remainder of a number when divided by $7$ is $5$. If I multiply the number by $32$ and add $18$ to the product, what is the new remainder when divided by $7$?
[b]p2.[/b] If a fair coin is flipped $15$ times, what is the probability that there are more heads than tails?
[b]p3.[/b] Let $-\frac{\sqrt{p}}{q}$ be the smallest nonzero real number such that the reciprocal of the number is equal to the number minus the square root of the square of the number, where $p$ and $q$ are positive integers and $p$ is not divisible the square of any prime. Find $p + q$.
[b]p4.[/b] Rachel likes to put fertilizers on her grass to help her grass grow. However, she has cows there as well, and they eat $3$ little fertilizer balls on average. If each ball is spherical with a radius of $4$, then the total volume that each cow consumes can be expressed in the form $a\pi$ where $a$ is an integer. What is $a$?
[b]p5.[/b] One day, all $30$ students in Precalc class are bored, so they decide to play a game. Everyone enters into their calculators the expression $9 \diamondsuit 9 \diamondsuit 9 ... \diamondsuit 9$, where $9$ appears $2020$ times, and each $\diamondsuit$ is either a multiplication or division sign. Each student chooses the signs randomly, but they each choose one more multiplication sign than division sign. Then all $30$ students calculate their expression and take the class average. Find the expected value of the class average.
[b]p6.[/b] NaNoWriMo, or National Novel Writing Month, is an event in November during which aspiring writers attempt to produce novel-length work - formally defined as $50,000$ words or more - within the span of $30$ days. Justin wants to participate in NaNoWriMo, but he's a busy high school student: after accounting for school, meals, showering, and other necessities, Justin only has six hours to do his homework and perhaps participate in NaNoWriMo on weekdays. On weekends, he has twelve hours on Saturday and only nine hours on Sunday, because he goes to church. Suppose Justin spends two hours on homework every single day, including the weekends. On Wednesdays, he has science team, which takes up another hour and a half of his time. On Fridays, he spends three hours in orchestra rehearsal. Assume that he spends all other time on writing. Then, if November $1$st is a Friday, let $w$ be the minimum number of words per minute that Justin must type to finish the novel. Round $w$ to the nearest whole number.
[b]p7.[/b] Let positive reals $a$, $b$, $c$ be the side lengths of a triangle with area $2030$. Given $ab + bc + ca = 15000$ and $abc = 350000$, find the sum of the lengths of the altitudes of the triangle.
[b]p8.[/b] Find the minimum possible area of a rectangle with integer sides such that a triangle with side lengths $3$, $4$, $5$, a triangle with side lengths $4$, $5$, $6$, and a triangle with side lengths $\frac94$, $4$, $4$ all fit inside the rectangle without overlapping.
[b]p9.[/b] The base $16$ number $10111213...99_{16}$, which is a concatenation of all of the (base $10$) $2$-digit numbers, is written on the board. Then, the last $2n$ digits are erased such that the base $10$ value of remaining number is divisible by $51$. Find the smallest possible integer value of $n$.
[b]p10.[/b] Consider sequences that consist entirely of $X$'s, $Y$ 's and $Z$'s where runs of consecutive $X$'s, $Y$ 's, and $Z$'s are at most length $3$. How many sequences with these properties of length $8$ are there?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2010 Today's Calculation Of Integral, 606
Find the area of the part bounded by two curves $y=\sqrt{x},\ \sqrt{x}+\sqrt{y}=1$ and the $x$-axis.
1956 Tokyo Institute of Technology entrance exam
2023 Putnam, A6
Alice and Bob play a game in which they take turns choosing integers from 1 to $n$. Before any integers are chosen, Bob selects a goal of "odd" or "even". On the first turn, Alice chooses one of the $n$ integers. On the second turn, Bob chooses one of the remaining integers. They continue alternately choosing one of the integers that has not yet been chosen, until the $n$th turn, which is forced and ends the game. Bob wins if the parity of $\{k$ : the number $k$ was chosen on the $k$th turn $\}$ matches his goal. For which values of $n$ does Bob have a winning strategy?
2005 Romania Team Selection Test, 2
On the edges of a convex polyhedra we draw arrows such that from each vertex at least an arrow is pointing in and at least one is pointing out.
Prove that there exists a face of the polyhedra such that the arrows on its edges form a circuit.
[i]Dan Schwartz[/i]
2010 Belarus Team Selection Test, 7.1
Find the smallest value of the expression $|3 \cdot 5^m - 11 \cdot 13^n|$ for all $m,n \in N$.
(Folklore)
1977 All Soviet Union Mathematical Olympiad, 235
Given a closed broken line without self-intersections in a plane. Not a triple of its vertices belongs to one straight line. Let us call "special" a couple of line's segments if the one's extension intersects another. Prove that there is even number of special pairs.
2022 Stanford Mathematics Tournament, 2
Call a three-digit number $\overline{ABC}$ $\textit{spicy}$ if it satisfies $\overline{ABC}=A^3+B^3+C^3$. Compute the unique $n$ for which both $n$ and $n+1$ are $\textit{spicy}$.
2017-2018 SDML (Middle School), 5
If $(x + 1) + (x + 2) + ... + (x + 20) = 174 + 176 + 178 + ... + 192$, then what is the value of $x$?
$\mathrm{(A) \ } 80 \qquad \mathrm{(B) \ } 81 \qquad \mathrm {(C) \ } 82 \qquad \mathrm{(D) \ } 83 \qquad \mathrm{(E) \ } 84$
2023 CMIMC Algebra/NT, 10
For a given $n$, consider the points $(x,y)\in \mathbb{N}^2$ such that $x\leq y\leq n$. An ant starts from $(0,1)$ and, every move, it goes from $(a,b)$ to point $(c,d)$ if $bc-ad=1$ and $d$ is maximized over all such points. Let $g_n$ be the number of moves made by the ant until no more moves can be made. Find $g_{2023} - g_{2022}$.
[i]Proposed by David Tang[/i]
2018 Chile National Olympiad, 5
Consider the set $\Omega$ formed by the first twenty natural numbers, $\Omega = \{1, 2, . . . , 20\}$ . A nonempty subset $A$ of $\Omega$ is said to be [i]sumfree [/i ] if for all pair of elements$ x, y \in A$, the sum $(x + y)$ is not in $A$, ( $x$ can be equal to $y$). Prove that $\Omega$ has at least $2018$ sumfree subsets.
2022 Moldova Team Selection Test, 6
Let $A$ be a point outside of the circle $\Omega$. Tangents from $A$ touch $\Omega$ in points $B$ and $C$. Point $C$, collinear with $A$ and $P$, is between $A$ and $P$, such that the circumcircle of triangle $ABP$ intersects $\Omega$ again in point $E$. Point $Q$ is on the segment $BP$, such that $\angle PEQ=2 \cdot \angle APB$. Prove that the lines $BP$ and $CQ$ are perpendicular.
2016 Online Math Open Problems, 22
Let $ABC$ be a triangle with $AB=5$, $BC=7$, $CA=8$, and circumcircle $\omega$. Let $P$ be a point inside $ABC$ such that $PA:PB:PC=2:3:6$. Let rays $\overrightarrow{AP}$, $\overrightarrow{BP}$, and $\overrightarrow{CP}$ intersect $\omega$ again at $X$, $Y$, and $Z$, respectively. The area of $XYZ$ can be expressed in the form $\dfrac{p\sqrt q}{r}$ where $p$ and $r$ are relatively prime positive integers and $q$ is a positive integer not divisible by the square of any prime. What is $p+q+r$?
[i]Proposed by James Lin[/i]
2023 Ukraine National Mathematical Olympiad, 11.8
There are $2024$ cities in a country, every two of which are bidirectionally connected by exactly one of three modes of transportation - rail, air, or road. A tourist has arrived in this country and has the entire transportation scheme. He chooses a travel ticket for one of the modes of transportation and the city from which he starts his trip. He wants to visit as many cities as possible, but using only the ticket for the specified type of transportation. What is the largest $k$ for which the tourist will always be able to visit at least $k$ cities? During the route, he can return to the cities he has already visited.
[i]Proposed by Bogdan Rublov[/i]
2016 PUMaC Team, 7
In triangle $ABC$, let $S$ be on $BC$ and $T$ be on $AC$ so that $AS \perp BC$ and $BT \perp AC$, and let $AS$ and $BT$ intersect at $H$. Let $O$ be the center of the circumcircle of $\vartriangle AHT, P$ be the center of the circumcircle of $\vartriangle BHS$, and $G$ be the other point of intersection (besides $H$) of the two circles. Let $GH$ and $OP$ intersect at $X$. If $AB = 14, BH = 6$, and HA = 11, then $XO - XP$ can be written in simplest form as $\frac{m}{n}$ . Find $m + n$.
1964 AMC 12/AHSME, 31
Let \[f(n)=\dfrac{5+3\sqrt{5}}{10}\left(\dfrac{1+\sqrt{5}}{2}\right)^n+\dfrac{5-3\sqrt{5}}{10}\left(\dfrac{1-\sqrt{5}}{2}\right)^n.\] Then $f(n+1)-f(n-1)$, expressed in terms of $f(n)$, equals:
$\textbf{(A)}\ \dfrac{1}{2}f(n) \qquad
\textbf{(B)}\ f(n)\qquad
\textbf{(C)}\ 2f(n)+1 \qquad
\textbf{(D)}\ f^2(n) \qquad
\textbf{(E)}\ \dfrac{1}{2}(f^2(n)-1)$