Found problems: 85335
2016 Singapore Senior Math Olympiad, 5
For each integer $n > 1$, find a set of $n$ integers $\{a_1, a_2,..., a_n\}$ such that the set of numbers $\{a_1+a_j | 1 \le i \le j \le n\}$ leave distinct remainders when divided by $n(n + 1)/2$. If such a set of integers does not exist, give a proof.
2024 Chile Junior Math Olympiad, 6
In a regular polygon with 100 vertices, 10 vertices are painted blue, and 10 other vertices are painted red.
1. Prove that there exist two distinct blue vertices \( A_1 \) and \( A_2 \), and two distinct red vertices \( R_1 \) and \( R_2 \), such that the distance between \( A_1 \) and \( R_1 \) is equal to the distance between \( A_2 \) and \( R_2 \).
2. Prove that there exist two distinct blue vertices \( A_1 \) and \( A_2 \), and two distinct red vertices \( R_1 \) and \( R_2 \), such that the distance between \( A_1 \) and \( A_2 \) is equal to the distance between \( R_1 \) and \( R_2 \).
2024 Kyiv City MO Round 1, Problem 4
For a positive integer $n$, does there exist a permutation of all its positive integer divisors $(d_1 , d_2 , \ldots, d_k)$ such that the equation $d_kx^{k-1} + \ldots + d_2x + d_1 = 0$ has a rational root, if:
a) $n = 2024$;
b) $n = 2025$?
[i]Proposed by Mykyta Kharin[/i]
VII Soros Olympiad 2000 - 01, 10.7
The President of the Bank "Glavny Central" Gerasim Shchenkov announced that from January $2$, $2001$ until January $31$ of the same year, the dollar exchange rate would not go beyond the boundaries of the corridor of $27$ rubles $50$ kopecks. and $28$ rubles $30$ kopecks for the dollar. On January $2$, the rate will be a multiple of $5$ kopecks, and starting from January 3, each day will differ from the rate of the previous day by exactly $5$ kopecks. Mr. Shchenkov suggested that citizens try to guess what the dollar exchange rate will be during the specified period. Anyone who can give an accurate forecast for at least one day, he promised to give a cash prize. One interesting person lives in our house, a tireless arguer. For his passion for arguments and constant winnings, he was even nicknamed Zhora Sporos. Zhora claims that he can give such a forecast of the dollar exchange rate for every day from January 424 to January 4314, which he will surely guess at least once, if, of course, the banker strictly acts in accordance with the announced rules. Is Zhora right?
Note: 1 ruble =100 kopecks
[hide=original wording]10-I-7. Президент банка "Главный централ" Герасим Щенков объявил, что со 2-го января 2001 года и до 31-го января этого же года курс доллара не будет выходить за границы коридора 27 руб. 50 коп. и 28 руб. 30 коп. за доллар. 2-го января курс будет кратен 5 копейкам, а, начиная с 3-го января, каждый день будет отличаться от курса предыдущего дня ровно на 5 копеек. Господин Щенков предложил гражданам попробовать угадать, каким будет курс доллара в течение указанного периода. Тому, кто сумеет дать точный прогноз хотя бы на один день, он обещал выдать денежный приз. В нашем доме живет один интересный человек, неутомимый спорщик. За страсть к спорам и постоянные выигрыши его даже прозвали Жора Спорос. Жора утверждает, что может дать такой прогноз курса доллара на каждый день со 2-го по 31-е января, что обязательно хотя бы один раз угадает, если, конечно, банкир будет строго действовать в соответствии с объявленными правилами. Прав ли Жора?
[/hide]
Maryland University HSMC part II, 2018
[b]p1.[/b] I have $6$ envelopes full of money. The amounts (in dollars) in the $6$ envelopes are six consecutive integers. I give you one of the envelopes. The total amount in the remaining $5$ envelopes is $\$2018$. How much money did I give you?
[b]p2. [/b]Two tangents $AB$ and $AC$ are drawn to a circle from an exterior point $A$. Let $D$ and $E$ be the midpoints of the line segments $AB$ and $AC$. Prove that the line DE does not intersect the circle.
[b]p3.[/b] Let $n \ge 2$ be an integer. A subset $S$ of {0, 1, . . . , n − 2} is said to be closed whenever it satisfies all of the following properties:
• $0 \in S$
• If $x \in S$ then $n - 2 - x \in S$
• If $x \in S$, $y \ge 0$, and $y + 1$ divides $x + 1$ then $y \in S$.
Prove that $\{0, 1, . . . , n - 2\}$ is the only closed subset if and only if $n$ is prime.
(Note: “$\in$” means “belongs to”.)
[b]p4.[/b] Consider the $3 \times 3$ grid shown below
$\begin{tabular}{|l|l|l|l|}
\hline
A & B & C \\ \hline
D & E & F \\ \hline
G & H & I \\ \hline
\end{tabular}$
A knight move is a pair of elements $(s, t)$ from $\{A, B, C, D, E, F, G, H, I\}$ such that $s$ can be reached from $t$ by moving either two spaces horizontally and one space vertically, or by moving one space horizontally and two spaces vertically. (For example, $(B, I)$ is a knight move, but $(G, E)$ is not.) A knight path of length $n$ is a sequence $s_0$, $s_1$, $s_2$, $. . . $, $s_n$ drawn from the set $\{A, B, C, D, E, F, G, H, I\}$ (with repetitions allowed) such that each pair $(s_i , s_{i+1})$ is a knight move.
Let $N$ be the total number of knight paths of length $2018$ that begin at $A$ and end at $A$. Let $M$ be the total number of knight paths of length $2018$ that begin at $A$ and end at $I$. Compute the value $(N- M)$, with proof. (Your answer must be in simplified form and may not involve any summations.)
[b]p5.[/b] A strip is defined to be the region of the plane lying on or between two parallel lines. The width of the strip is the distance between the two lines. Consider a finite number of strips whose widths sum to a number $d < 1$, and let $D$ be a circular closed disk of diameter $1$. Prove or disprove: no matter how the strips are placed in the plane, they cannot entirely cover the disk $D$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2004 South East Mathematical Olympiad, 2
In $\triangle$ABC, points D, M lie on side BC and AB respectively, point P lies on segment AD. Line DM intersects segments BP, AC (extended part), PC (extended part) at E, F and N respectively. Show that if DE=DF, then DM=DN.
2009 AMC 8, 7
The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land ACD?
[asy]
size(250);
defaultpen(linewidth(0.55));
pair A=(-6,0), B=origin, C=(0,6), D=(0,12);
pair ac=C+2.828*dir(45),
ca=A+2.828*dir(225),
ad=D+2.828*dir(A--D),
da=A+2.828*dir(D--A),
ab=(2.828,0),
ba=(-6-2.828, 0);
fill(A--C--D--cycle, gray);
draw(ba--ab);
draw(ac--ca);
draw(ad--da);
draw((0,-1)--(0,15));
draw((1/3, -1)--(1/3, 15));
int i;
for(i=1; i<15; i=i+1) {
draw((-1/10, i)--(13/30, i));
}
label("$A$", A, SE);
label("$B$", B, SE);
label("$C$", C, SE);
label("$D$", D, SE);
label("$3$", (1/3,3), E);
label("$3$", (1/3,9), E);
label("$3$", (-3,0), S);
label("Main", (-3,0), N);
label(rotate(45)*"Aspen", A--C, SE);
label(rotate(63.43494882)*"Brown", A--D, NW);
[/asy]
$\textbf{(A)}\ 2\qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 4.5 \qquad
\textbf{(D)}\ 6 \qquad
\textbf{(E)}\ 9$
1986 All Soviet Union Mathematical Olympiad, 433
Find the relation of the black part length and the white part length for the main diagonal of the
a) $100\times 99$ chess-board;
b) $101\times 99$ chess-board.
2014 Contests, 3
Find all $(m,n)$ in $\mathbb{N}^2$ such that $m\mid n^2+1$ and $n\mid m^2+1$.
2015 China Western Mathematical Olympiad, 8
Let $k$ be a positive integer, and $n=\left(2^k\right)!$ .Prove that $\sigma(n)$ has at least a prime divisor larger than $2^k$, where $\sigma(n)$ is the sum of all positive divisors of $n$.
2021 Saint Petersburg Mathematical Olympiad, 1
Solve the following system of equations $$\sin^2{x} + \cos^2{y} = y^4. $$ $$\sin^2{y} + \cos^2{x} = x^2. $$
[i]A. Khrabov[/i]
2004 Thailand Mathematical Olympiad, 20
Two pillars of height $a$ and $b$ are erected perpendicular to the ground. On each pillar, a straight cable is placed connecting the top of the pillar to the base of the other pillar; the two lines of cable intersect at a point above ground. What is the height of this point?
2004 Junior Tuymaada Olympiad, 1
A positive rational number is written on the blackboard. Every minute Vasya replaces the number $ r $ written on the board with $ \sqrt {r + 1} $. Prove that someday he will get an irrational number.
2013 Saint Petersburg Mathematical Olympiad, 4
There are $100$ numbers from $(0,1)$ on the board. On every move we replace two numbers $a,b$ with roots of $x^2-ax+b=0$(if it has two roots). Prove that process is not endless.
1965 Dutch Mathematical Olympiad, 3
Given are the points $A$ and $B$ in the plane. If $x$ is a straight line is in that plane, and $x$ does not coincide with the perpendicular bisectror of $AB$, then denote the number of points $C$ located at $x$ such that $\vartriangle ABC$ is isosceles, as the "weight of the line $x$”.
Prove that the weight of any line $x$ is at most $5$ and determine the set of points $P$ which has a line with weight $1$, but none with weight $0$.
2011 District Olympiad, 1
Find the real numbers $x$ and $y$ such that
$$(x^2 -x +1)(3y^2-2y + 3) -2=0.$$
Russian TST 2019, P3
Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.
2015 Peru IMO TST, 5
We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ .
[i]Proposed by Abbas Mehrabian, Iran[/i]
1985 Dutch Mathematical Olympiad, 4
A convex hexagon $ ABCDEF$ is such that each of the diagonals $ AD,BE,CF$ divides the hexagon into two parts of equal area. Prove that these three diagonals are concurrent.
2023 Irish Math Olympiad, P5
The positive integers $a, b, c, d$ satisfy
(i) $a + b + c + d = 2023$
(ii) $2023 \text{ } | \text{ } ab - cd$
(iii) $2023 \text{ } | \text{ } a^2 + b^2 + c^2 + d^2.$
Assuming that each of the numbers $a, b, c, d$ is divisible by $7$, prove that each of the numbers $a, b, c, d$ is divisible by $17$.
2012 Today's Calculation Of Integral, 803
Answer the following questions:
(1) Evaluate $\int_{-1}^1 (1-x^2)e^{-2x}dx.$
(2) Find $\lim_{n\to\infty} \left\{\frac{(2n)!}{n!n^n}\right\}^{\frac{1}{n}}.$
MMPC Part II 1958 - 95, 1981
[b]p1.[/b] A canoeist is paddling upstream in a river when she passes a log floating downstream,, She continues upstream for awhile, paddling at a constant rate. She then turns around and goes downstream and paddles twice as fast. She catches up to the same log two hours after she passed it. How long did she paddle upstream?
[b]p2.[/b] Let $g(x) =1-\frac{1}{x}$ and define $g_1(x) = g(x)$ and $g_{n+1}(x) = g(g_n(x))$ for $n = 1,2,3, ...$. Evaluate $g_3(3)$ and $g_{1982}(l982)$.
[b]p3.[/b] Let $Q$ denote quadrilateral $ABCD$ where diagonals $AC$ and $BD$ intersect. If each diagonal bisects the area of $Q$ prove that $Q$ must be a parallelogram.
[b]p4.[/b] Given that: $a_1, a_2, ..., a_7$ and $b_1, b_2, ..., b_7$ are two arrangements of the same seven integers, prove that the product $(a_1-b_1)(a_2-b_2)...(a_7-b_7)$ is always even.
[b]p5.[/b] In analyzing the pecking order in a finite flock of chickens we observe that for any two chickens exactly one pecks the other. We decide to call chicken $K$ a king provided that for any other chicken $X, K$ necks $X$ or $K$ pecks a third chicken $Y$ who in turn pecks $X$. Prove that every such flock of chickens has at least one king. Must the king be unique?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2011 Bogdan Stan, 2
Solve the system
$$ \left\{\begin{matrix} ax=b\\bx=a \end{matrix}\right. $$
independently of the fixed elements $ a,b $ of a group of odd order.
[i]Marian Andronache[/i]
1990 Turkey Team Selection Test, 6
Let $k\geq 2$ and $n_1, \dots, n_k \in \mathbf{Z}^+$. If $n_2 | (2^{n_1} -1)$, $n_3 | (2^{n_2} -1)$, $\dots$, $n_k | (2^{n_{k-1}} -1)$, $n_1 | (2^{n_k} -1)$, show that $n_1 = \dots = n_k =1$.
2023 Regional Olympiad of Mexico West, 2
We have $n$ guinea pigs placed on the vertices of a regular polygon with $n$ sides inscribed in a circumference, one guinea pig in each vertex. Each guinea pig has a direction assigned, such direction is either "clockwise" or "anti-clockwise", and a velocity between $1 km/h$, $2km/h$,..., and $n km/h$, each one with a distinct velocity, and each guinea pig has a counter starting from $0$. They start moving along the circumference with the assigned direction and velocity, everyone at the same time, when 2 or more guinea pigs meet a point, all of the guinea pigs at that point follow the same direction of the fastest guinea pig and they keep moving (with the same velocity as before); each time 2 guinea pigs meet for the first time in the same point, the fastest guinea pig adds 1 to its counter. Prove that, at some moment, for each $1\leq i\leq n$ we have that the $i-$th guinea pig has $i-1$ in its counter.