Found problems: 85335
2014 Saudi Arabia Pre-TST, 4.3
Fatima and Asma are playing the following game. First, Fatima chooses $2013$ pairwise different numbers, called $a_1, a_2, ..., a_{2013}$. Then, Asma tries to know the value of each number $a_1, a_2, ..., a_{2013}$.. At each time, Asma chooses $1 \le i < j \le 2013$ and asks Fatima ''[i]What is the set $\{a_i,a_j\}$?[/i]'' (For example, if Asma asks what is the set $\{a_i,a_j\}$, and $a_1 = 17$ and $a_2 = 13$, Fatima will answer $\{13. 17\}$). Find the least number of questions Asma needs to ask, to know the value of all the numbers $a_1, a_2, ..., a_{2013}$.
2024 Mathematical Talent Reward Programme, 1
Hari the milkman delivers milk to his customers everyday by travelling on his cycle. Each litre of milk costs him Rs. $20$, and he sells it at Rs. $24$. One day while riding his cycle with $20$L, Hari trips and loses $5$L of it, and he decides to mix some water with the rest of the milk. His customers can detect if the milk is more than $10$% impure ($1$L water in $10$L misture). Given that he doesn't wish to make his customers angry, what is his maximum profit for the day?
$(A)$ Rs $12$ profit
$(B)$ Rs $24$ profit
$(C)$ No profit
$(D)$ Rs $12$ loss
2008 Iran MO (3rd Round), 1
Let $ ABC$ be a triangle with $ BC > AC > AB$. Let $ A',B',C'$ be feet of perpendiculars from $ A,B,C$ to $ BC,AC,AB$, such that $ AA' \equal{} BB' \equal{} CC' \equal{} x$. Prove that:
a) If $ ABC\sim A'B'C'$ then $ x \equal{} 2r$
b) Prove that if $ A',B'$ and $ C'$ are collinear, then $ x \equal{} R \plus{} d$ or $ x \equal{} R \minus{} d$.
(In this problem $ R$ is the radius of circumcircle, $ r$ is radius of incircle and $ d \equal{} OI$)
2011 Romania Team Selection Test, 1
Suppose a square of sidelengh $l$ is inside an unit square and does not contain its centre. Show that $l\le 1/2.$
[i]Marius Cavachi[/i]
2019 ELMO Shortlist, C1
Elmo and Elmo's clone are playing a game. Initially, $n\geq 3$ points are given on a circle. On a player's turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo's clone goes first and players alternate turns, who wins? (Your answer may be in terms of $n$.)
[i]Proposed by Milan Haiman[/i]
1995 Singapore MO Open, 3
Let $P$ be a point inside $\vartriangle ABC$. Let $D, E, F$ be the feet of the perpendiculars from $P$ to the lines $BC, CA$ and $AB$, respectively (see Fig. ). Show that
(i) $EF = AP \sin A$,
(ii) $PA+ PB + PC \ge 2(PE+ PD+ PF)$
[img]https://cdn.artofproblemsolving.com/attachments/d/f/f37d8764fc7d99c2c3f4d16f66223ef39dfd09.png[/img]
2007 Singapore Junior Math Olympiad, 5
For any positive integer $n$, let $f(n)$ denote the $n$- th positive nonsquare integer, i.e., $f(1) = 2, f(2) = 3, f(3) = 5, f(4) = 6$, etc. Prove that $f(n)=n +\{\sqrt{n}\}$ where $\{x\}$ denotes the integer closest to $x$.
(For example, $\{\sqrt{1}\} = 1, \{\sqrt{2}\} = 1, \{\sqrt{3}\} = 2, \{\sqrt{4}\} = 2$.)
2009 Princeton University Math Competition, 4
How many strings of ones and zeroes of length 10 are there such that there is an even number of ones, and no zero follows another zero?
2020 Germany Team Selection Test, 1
Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that
\[
a b \leqslant-\frac{1}{2019}.
\]
2004 Junior Balkan MO, 3
If the positive integers $x$ and $y$ are such that $3x + 4y$ and $4x + 3y$ are both perfect squares, prove that both $x$ and $y$ are both divisible with $7$.
2020 LIMIT Category 1, 5
Let $P(x),Q(x)$ be monic polynomials with integer coeeficients. Let $a_n=n!+n$ for all natural numbers $n$. Show that if $\frac{P(a_n)}{Q(a_n)}$ is an integer for all positive integer $n$ then $\frac{P(n)}{Q(n)}$ is an integer for every integer $n\neq0$.
\\
[i]Hint (given in question): Try applying division algorithm for polynomials [/i]
2022 IMO Shortlist, N4
Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]
1962 Czech and Slovak Olympiad III A, 2
Determine the set of all points $(x,y)$ in two-dimensional cartesian coordinate system such that \begin{align*}0\le &\,x\le\frac{\pi}{2}, \\ \sqrt{1-\sin 2x}-\sqrt{1+\sin 2x}\le &\,y\le\sqrt{1-\cos2x}-\sqrt{1+\cos2x}.\end{align*}
Draw a picture of the set.
2017 Harvard-MIT Mathematics Tournament, 35
Welcome to the [b]USAYNO[/b], where each question has a yes/no answer. Choose any subset of the following six problems to answer. If you answer $n$ problems and get them [b]all[/b] correct, you will receive $\max(0, (n-1)(n-2))$ points. If any of them are wrong (or you leave them all blank), you will receive $0$ points.
Your answer should be a six-character string containing 'Y' (for yes), 'N' (for no), or 'B' (for blank). For instance if you think 1, 2, and 6 are 'yes' and 3 and 4 are 'no', you should answer YYNNBY (and receive $12$ points if all five answers are correct, 0 points if any are wrong).
(a) Does there exist a finite set of points, not all collinear, such that a line between any two points in the set passes through a third point in the set?
(b) Let $ABC$ be a triangle and $P$ be a point. The [i]isogonal conjugate[/i] of $P$ is the intersection of the reflection of line $AP$ over the $A$-angle bisector, the reflection of line $BP$ over the $B$-angle bisector, and the reflection of line $CP$ over the $C$-angle bisector. Clearly the incenter is its own isogonal conjugate. Does there exist another point that is its own isogonal conjugate?
(c) Let $F$ be a convex figure in a plane, and let $P$ be the largest pentagon that can be inscribed in $F$. Is it necessarily true that the area of $P$ is at least $\frac{3}{4}$ the area of $F$?
(d) Is it possible to cut an equilateral triangle into $2017$ pieces, and rearrange the pieces into a square?
(e) Let $ABC$ be an acute triangle and $P$ be a point in its interior. Let $D,E,F$ lie on $BC, CA, AB$ respectively so that $PD$ bisects $\angle{BPC}$, $PE$ bisects $\angle{CPA}$, and $PF$ bisects $\angle{APB}$. Is it necessarily true that $AP+BP+CP\ge 2(PD+PE+PF)$?
(f) Let $P_{2018}$ be the surface area of the $2018$-dimensional unit sphere, and let $P_{2017}$ be the surface area of the $2017$-dimensional unit sphere. Is $P_{2018}>P_{2017}$?
[color = red]The USAYNO disclaimer is only included in problem 33. I have included it here for convenience.[/color]
2015 Caucasus Mathematical Olympiad, 5
Let's call a natural number a palindrome, the decimal notation of which is equally readable from left to right and right to left (decimal notation cannot start from zero; for example, the number $1221$ is a palindrome, but the numbers $1231, 1212$ and $1010$ are not). Which palindromes among the numbers from $10,000$ to $999,999$ have an odd sum of digits, which have an one even, and how many times are the ones with odd sum more than the ones with the even sum?
1993 Hungary-Israel Binational, 7
In the questions below: $G$ is a finite group; $H \leq G$ a subgroup of $G; |G : H |$ the index of $H$ in $G; |X |$ the number of elements of $X \subseteq G; Z (G)$ the center of $G; G'$ the commutator subgroup of $G; N_{G}(H )$ the normalizer of $H$ in $G; C_{G}(H )$ the centralizer of $H$ in $G$; and $S_{n}$ the $n$-th symmetric group.
Assume $|G'| = 2$. Prove that $|G : G'|$ is even.
1951 AMC 12/AHSME, 45
If you are given $ \log 8 \approx .9031$ and $ \log 9 \approx .9542$, then the only logarithm that cannot be found without the use of tables is:
$ \textbf{(A)}\ \log 17 \qquad\textbf{(B)}\ \log \frac {5}{4} \qquad\textbf{(C)}\ \log 15 \qquad\textbf{(D)}\ \log 600 \qquad\textbf{(E)}\ \log .4$
2013 Princeton University Math Competition, 7
The Miami Heat and the San Antonio Spurs are playing a best-of-five series basketball championship, in which the team that first wins three games wins the whole series. Assume that the probability that the Heat wins a given game is $x$ (there are no ties). The expected value for the total number of games played can be written as $f(x)$, with $f$ a polynomial. Find $f(-1)$.
2011 Canadian Open Math Challenge, 7
In the figure, BC is a diameter of the circle, where $BC=\sqrt{901}, BD=1$, and $DA=16$. If $EC=x$, what is the value of x?
[asy]size(2inch);
pair O,A,B,C,D,E;
B=(0,0);
O=(2,0);
C=(4,0);
D=(.333,1.333);
A=(.75,2.67);
E=(1.8,2);
draw(Arc(O,2,0,360));
draw(B--C--A--B);
label("$A$",A,N);
label("$B$",B,W);
label("$C$",C,E);
label("$D$",D,W);
label("$E$",E,N);
label("Figure not drawn to scale",(2,-2.5),S);
[/asy]
1979 Putnam, A5
Denote by $\lceil x \rceil$ the greatest integer less than or equal to $x$ and by $S(x)$ the sequence $\lceil x \rceil, \lceil 2x \rceil, \lceil 3x \rceil, \dots.$ Prove that there are distinct real solutions $\alpha$ and $\beta$ of the equation $$x^3-10x^2+29x-25=0$$ such that infinitely many positive integers appear both in $S(\alpha)$ and in $S(\beta).$
2010 AMC 10, 5
A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$
1996 AMC 8, 16
$1-2-3+4+5-6-7+8+9-10-11+\cdots + 1992+1993-1994-1995+1996=$
$\text{(A)}\ -998 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 998$
1994 China Team Selection Test, 3
Find the smallest $n \in \mathbb{N}$ such that if any 5 vertices of a regular $n$-gon are colored red, there exists a line of symmetry $l$ of the $n$-gon such that every red point is reflected across $l$ to a non-red point.
2001 Tournament Of Towns, 5
In a chess tournament, every participant played with each other exactly once, receiving $1$ point for a win, $1/2$ for a draw and $0$ for a loss.
[list][b](a)[/b] Is it possible that for every player $P$, the sum of points of the players who were beaten by P is greater than the sum of points of the players who beat $P$?
[b](b)[/b] Is it possible that for every player $P$, the first sum is less than the second one?[/list]
2001 Junior Balkan MO, 3
Let $ABC$ be an equilateral triangle and $D$, $E$ points on the sides $[AB]$ and $[AC]$ respectively. If $DF$, $EF$ (with $F\in AE$, $G\in AD$) are the interior angle bisectors of the angles of the triangle $ADE$, prove that the sum of the areas of the triangles $DEF$ and $DEG$ is at most equal with the area of the triangle $ABC$. When does the equality hold?
[i]Greece[/i]