Found problems: 85335
2015 Math Hour Olympiad, 5-7
[u]Round 1[/u]
[b]p1.[/b] A party is attended by ten people (men and women). Among them is Pat, who always lies to people of the opposite gender and tells the truth to people of the same gender.
Pat tells five of the guests: “There are more men than women at the party.”
Pat tells four of the guests: “There are more women than men at the party.”
Is Pat a man or a woman?
[b]p2.[/b] Once upon a time in a land far, far away there lived $100$ knights, $99$ princesses, and $101$ dragons. Over time, knights beheaded dragons, dragons ate princesses, and princesses poisoned knights. But they always obeyed an ancient law that prohibits killing any creature who has killed an odd number of others. Eventually only one creature remained alive. Could it have been a knight? A dragon? A princess?
[b]p3.[/b] The numbers $1 \circ 2 \circ 3 \circ 4 \circ 5 \circ 6 \circ 7 \circ 8 \circ 9 \circ 10$ are written in a line. Alex and Vicky play a game, taking turns inserting either an addition or a multiplication symbol between adjacent numbers. The last player to place a symbol wins if the resulting expression is odd and loses if it is even. Alex moves first. Who wins?
(Remember that multiplication is performed before addition.)
[b]p4.[/b] A chess tournament had $8$ participants. Each participant played each other participant once. The winner of a game got $1$ point, the loser $0$ points, and in the case of a draw each got $1/2$ a point. Each participant scored a different number of points, and the person who got $2$nd prize scored the same number of points as the $5$th, $6$th, $7$th and $8$th place participants combined.
Can you determine the result of the game between the $3$rd place player and the $5$th place player?
[b]p5.[/b] One hundred gnomes sit in a circle. Each gnome gets a card with a number written on one side and a different number written on the other side. Prove that it is possible for all the gnomes to lay down their cards so that no two neighbors have the same numbers facing up.
[u]Round 2[/u]
[b]p6.[/b] A casino machine accepts tokens of $32$ different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to $\$10$ cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either $\$5$ plus a red token or $\$3$ plus a yellow token; a black token can always be exchanged either for $\$10$ (but no token) or for a brown token (but no cash). A player may keep playing as long as he has a token. Rob and Bob each have one white token. Rob watches Bob play and win $\$500$. Prove that Rob can win at least $\$1000$.
[img]https://cdn.artofproblemsolving.com/attachments/6/6/e55614bae92233c9b2e7d66f5f425a18e6475a.png[/img]
[b]p7.[/b] Each of the $100$ residents of Pleasantville has at least $30$ friends in town. A resident votes in the mayoral election only if one of her friends is a candidate. Prove that it is possible to nominate two candidates for mayor so that at least half of the residents will vote.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2020 SJMO, 2
Anthony writes the $(n+1)^2$ distinct positive integer divisors of $10^n$, each once, on a whiteboard. On a move, he may choose any two distinct numbers $a$ and $b$ on the board, erase them both, and write $\gcd(a, b)$ twice. Anthony keeps making moves until all of the numbers on the board are the same. Find the minimum possible number of moves Anthony could have made.
[i]Proposed by Andrew Wen[/i]
2014 ELMO Shortlist, 1
Let $ABC$ be a triangle with symmedian point $K$. Select a point $A_1$ on line $BC$ such that the lines $AB$, $AC$, $A_1K$ and $BC$ are the sides of a cyclic quadrilateral. Define $B_1$ and $C_1$ similarly. Prove that $A_1$, $B_1$, and $C_1$ are collinear.
[i]Proposed by Sammy Luo[/i]
MOAA Team Rounds, 2023.6
Call a set of integers [i]unpredictable[/i] if no four elements in the set form an arithmetic sequence. How many unordered [i]unpredictable[/i] sets of five distinct positive integers $\{a, b, c, d, e\}$ exist such that all elements are strictly less than $12$?
[i]Proposed by Anthony Yang[/i]
2016 Abels Math Contest (Norwegian MO) Final, 3b
Let $ABC$ be an acute triangle with $AB < AC$. The points $A_1$ and $A_2$ are located on the line $BC$ so that $AA_1$ and $AA_2$ are the inner and outer angle bisectors at $A$ for the triangle $ABC$. Let $A_3$ be the mirror image $A_2$ with respect to $C$, and let $Q$ be a point on $AA_1$ such that $\angle A_1QA_3 = 90^o$. Show that $QC // AB$.
2020 Harvard-MIT Mathematics Tournament, 3
Each unit square of a $4 \times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color? (L-trominos are made up of three unit squares sharing a corner, as shown below.)
[asy]
draw((0,0) -- (2,0) -- (2,1) -- (0,1));
draw((0,0) -- (0,2) -- (1,2) -- (1,0));
draw((4,1) -- (6,1) -- (6,2) -- (4,2));
draw((4,2) -- (4,0) -- (5,0) -- (5,2));
draw((10,0) -- (8,0) -- (8,1) -- (10,1));
draw((9,0) -- (9,2) -- (10,2) -- (10,0));
draw((14,1) -- (12,1) -- (12,2) -- (14,2));
draw((13,2) -- (13,0) -- (14,0) -- (14,2));
[/asy]
[i]Proposed by Andrew Lin.[/i]
2013 Princeton University Math Competition, 4
Suppose $a,b$ are nonzero integers such that two roots of $x^3+ax^2+bx+9a$ coincide, and all three roots are integers. Find $|ab|$.
2004 Czech and Slovak Olympiad III A, 1
Find all triples $(x,y,z)$ of real numbers such that
\[x^2+y^2+z^2\le 6+\min (x^2-\frac{8}{x^4},y^2-\frac{8}{y^4},z^2-\frac{8}{z^4}).\]
1970 IMO Longlists, 6
There is an equation $\sum_{i=1}^{n}{\frac{b_i}{x-a_i}}=c$ in $x$, where all $b_i >0$ and $\{a_i\}$ is a strictly increasing sequence. Prove that it has $n-1$ roots such that $x_{n-1}\le a_n$, and $a_i \le x_i$ for each $i\in\mathbb{N}, 1\le i\le n-1$.
2019 Simon Marais Mathematical Competition, B2
For each odd prime number $p$, prove that the integer
$$1!+2!+3!+\cdots +p!-\left\lfloor \frac{(p-1)!}{e}\right\rfloor$$is divisible by $p$
(Here, $e$ denotes the base of the natural logarithm and $\lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$.)
2019 Philippine TST, 6
Let $k$ be a positive integer. The organising commitee of a tennis tournament is to schedule the matches for $2k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay $1$ coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. Determine this minimum cost.
2012 Bundeswettbewerb Mathematik, 1
Alex writes the sixteen digits $2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9$ side by side in any order and then places a colon somewhere between two digits, so that a division task arises. Can the result of this calculation be $2$?
2009 AMC 8, 18
The diagram represents a $ 7$-foot-by-$ 7$-foot floor that is tiled with $ 1$-square-foot black tiles and white tiles. Notice that the corners have white tiles. If a $ 15$-foot-by-$ 15$-foot floor is to be tiled in the same manner, how many white tiles will be needed?
[asy]unitsize(10);
draw((0,0)--(7,0)--(7,7)--(0,7)--cycle);
draw((1,7)--(1,0));
draw((6,7)--(6,0));
draw((5,7)--(5,0));
draw((4,7)--(4,0));
draw((3,7)--(3,0));
draw((2,7)--(2,0));
draw((0,1)--(7,1));
draw((0,2)--(7,2));
draw((0,3)--(7,3));
draw((0,4)--(7,4));
draw((0,5)--(7,5));
draw((0,6)--(7,6));
fill((1,0)--(2,0)--(2,7)--(1,7)--cycle,black);
fill((3,0)--(4,0)--(4,7)--(3,7)--cycle,black);
fill((5,0)--(6,0)--(6,7)--(5,7)--cycle,black);
fill((0,5)--(0,6)--(7,6)--(7,5)--cycle,black);
fill((0,3)--(0,4)--(7,4)--(7,3)--cycle,black);
fill((0,1)--(0,2)--(7,2)--(7,1)--cycle,black);[/asy]
$ \textbf{(A)}\ 49 \qquad \textbf{(B)}\ 57 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 96 \qquad \textbf{(E)}\ 126$
2019 Moroccan TST, 2
Let $a>1$ be a real number. Prove that for all $n\in\mathbb{N}*$ that :
$\frac{a^n-1}{n}\ge \sqrt{a}^{n+1}-\sqrt{a}^{n-1}$
2025 Harvard-MIT Mathematics Tournament, 9
Let $f$ be the unique polynomial of degree at most $2026$ such that for all $n \in \{1,2, 3, \ldots, 2027\},$ $$f(n)=\begin{cases} 1 & \text{if } $n$ \text{ is a perfect square}, \\
0 & \text{otherwise.}
\end{cases}$$ Suppose that $\tfrac{a}{b}$ is the coefficient of $x^{2025}$ in $f,$ where $a$ and $b$ are integers such that $\gcd(a,b)=1.$ Compute the unique integer $r$ between $0$ and $2026$ (inclusive) such that $a-rb$ is divisible by $2027.$ (Note that $2027$ is prime.)
Novosibirsk Oral Geo Oly IX, 2016.1
In the quadrilateral $ABCD$, angles $B$ and $C$ are equal to $120^o$, $AB = CD = 1$, $CB = 4$. Find the length $AD$.
1978 IMO Longlists, 14
Let $p(x, y)$ and $q(x, y)$ be polynomials in two variables such that for $x \ge 0, y \ge 0$ the following conditions hold:
$(i) p(x, y)$ and $q(x, y)$ are increasing functions of $x$ for every fixed $y$.
$(ii) p(x, y)$ is an increasing and $q(x)$ is a decreasing function of $y$ for every fixed $x$.
$(iii) p(x, 0) = q(x, 0)$ for every $x$ and $p(0, 0) = 0$.
Show that the simultaneous equations $p(x, y) = a, q(x, y) = b$ have a unique solution in the set $x \ge 0, y \ge 0$ for all $a, b$ satisfying $0 \le b \le a$ but lack a solution in the same set if $a < b$.
2017 Caucasus Mathematical Olympiad, 4
Determine if there exist $101$ positive integers (not necessarily distinct) such that their product is equal to the sum of all their pairwise LCM.
2004 IMO Shortlist, 2
The function $f$ from the set $\mathbb{N}$ of positive integers into itself is defined by the equality \[f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}.\]
a) Prove that $f(mn)=f(m)f(n)$ for every two relatively prime ${m,n\in\mathbb{N}}$.
b) Prove that for each $a\in\mathbb{N}$ the equation $f(x)=ax$ has a solution.
c) Find all ${a\in\mathbb{N}}$ such that the equation $f(x)=ax$ has a unique solution.
2020 Brazil Cono Sur TST, 3
Let $a_0,a_1,a_2,\dots$ be a periodic sequence of real numbers(that is, there is a fixed positive integer $k$ such that $a_n=a_{n+k}$ for every integer $n\geq 0$). The following equality is true, for all $n\geq 0$:
$a_{n+2}=\frac{1}{n+2} (a_n - \frac{n+1}{a_{n+1}})$
if $a_0=2020$, determine the value of $a_1$.
2003 Italy TST, 3
Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy
\[f(f(x)+y)=2x+f(f(y)-x)\quad\text{for all real}\ x,y. \]
2009 Thailand Mathematical Olympiad, 5
Determine all functions $f : R\to R$ satisfying: $$f(xy + 2x + 2y - 1) = f(x)f(y) + f(y) + x -2$$ for all real numbers $x, y$.
2015 Switzerland Team Selection Test, 4
Find all relatively prime integers $a,b$ such that $$a^2+a=b^3+b$$
2009 Today's Calculation Of Integral, 444
Evaluate $ \int_0^{\frac {\pi}{6}} \frac {\sin x \plus{} \cos x}{1 \minus{} \sin 2x}\ln\ (2 \plus{} \sin 2x)\ dx.$
2019 AMC 8, 25
Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the people has at least 2 apples?
$\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380$