Found problems: 85335
2024 Malaysian IMO Training Camp, 2
The sequence $1, 2, \dots, 2023, 2024$ is written on a whiteboard. Every second, Megavan chooses two integers $a$ and $b$, and four consecutive numbers on the whiteboard. Then counting from the left, he adds $a$ to the 1st and 3rd of those numbers, and adds $b$ to the 2nd and 4th of those numbers. Can he achieve the sequence $2024, 2023, \dots, 2, 1$ in a finite number of moves?
[i](Proposed by Avan Lim Zenn Ee)[/i]
2020 SJMO, 6
We say a positive integer $n$ is [i]$k$-tasty[/i] for some positive integer $k$ if there exists a permutation $(a_0, a_1, a_2, \ldots , a_n)$ of $(0,1,2, \ldots, n)$ such that $|a_{i+1} - a_i| \in \{k, k+1\}$ for all $0 \le i \le n-1$. Prove that for all positive integers $k$, there exists a constant $N$ such that all integers $n \geq N$ are $k$-tasty.
[i]Proposed by Anthony Wang[/i]
2023 NMTC Junior, P6
The sum of squares of four reals $x,y,z,u$ is $1$. Find the minimum value of the expression $E=(x-y)(y-z)(z-u)(u-x)$.
Find also the minimum values of $x$, $y$, $z$ and $u$ when this minimum occurs.
2016 CCA Math Bonanza, L1.2
What is the largest prime factor of $729-64$?
[i]2016 CCA Math Bonanza Lightning #1.2[/i]
2016 PUMaC Geometry A, 7
Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$ and let $AC$ and $BD$ intersect at $X$. Let the line through $A$ parallel to $BD$ intersect line $CD$ at $E$ and $\omega$ at $Y \ne A$. If $AB = 10, AD = 24, XA = 17$, and $XB = 21$, then the area of $\vartriangle DEY$ can be written in simplest form as $\frac{m}{n}$ . Find $m + n$.
2009 All-Russian Olympiad, 6
Can be colored the positive integers with 2009 colors if we know that each color paints infinitive integers and that we can not find three numbers colored by three different colors for which the product of two numbers equal to the third one?
2018 Turkey Team Selection Test, 3
A Retired Linguist (R.L.) writes in the first move a word consisting of $n$ letters, which are all different. In each move, he determines the maximum $i$, such that the word obtained by reversing the first $i$ letters of the last word hasn't been written before, and writes this new word. Prove that R.L. can make $n!$ moves.
2021 Dutch Mathematical Olympiad, 4
In triangle $ABC$ we have $\angle ACB = 90^o$. The point $M$ is the midpoint of $AB$. The line through $M$ parallel to $BC$ intersects $AC$ in $D$. The midpoint of line segment $CD$ is $E$. The lines $BD$ and $CM$ are perpendicular.
(a) Prove that triangles $CME$ and $ABD$ are similar.
(b) Prove that $EM$ and $AB$ are perpendicular.
[asy]
unitsize(1 cm);
pair A, B, C, D, E, M;
A = (0,0);
B = (4,0);
C = (2.6,2);
M = (A + B)/2;
D = (A + C)/2;
E = (C + D)/2;
draw(A--B--C--cycle);
draw(C--M--D--B);
dot("$A$", A, SW);
dot("$B$", B, SE);
dot("$C$", C, N);
dot("$D$", D, NW);
dot("$E$", E, NW);
dot("$M$", M, S);
[/asy]
[i]Be aware: the figure is not drawn to scale.[/i]
Russian TST 2019, P1
A convex pentagon $APBCQ$ is given such that $AB < AC$. The circle $\omega$ centered at point $A{}$ passes through $P{}$ and $Q{}$ and touches the segment $BC$ at point $R{}$. Let the circle $\Gamma$ centered at the point $O{}$ be the circumcircle of the triangle $ABC$. It is known that $AO \perp P Q$ and $\angle BQR = \angle CP R$. Prove that the tangents at points $P{}$ and $Q{}$ to the circle $\omega$ intersect on $\Gamma$.
2009 F = Ma, 14
A wooden block (mass $M$) is hung from a peg by a massless rope. A speeding bullet (with mass $m$ and initial speed $v_\text{0}$) collides with the block at time $t = \text{0}$ and embeds in it. Let $S$ be the system consisting of the block and bullet. Which quantities are conserved between $t = -\text{10 s}$ and $ t = \text{+10 s}$?
[asy]
// Code by riben
draw(circle((0,0),0.3),linewidth(2));
filldraw(circle((0,0),0.3),gray);
draw((0,-0.8)--(0,-15.5),linewidth(2));
draw((5,-15.5)--(-5,-15.5)--(-5,-20.5)--(5,-20.5)--cycle,linewidth(2));
filldraw((5,-15.5)--(-5,-15.5)--(-5,-20.5)--(5,-20.5)--cycle,gray);
draw((-15,-18)--(-16,-17)--(-18,-17)--(-18,-19)--(-16,-19)--cycle,linewidth(2));
filldraw((-15,-18)--(-16,-17)--(-18,-17)--(-18,-19)--(-16,-19)--cycle,gray);
[/asy]
(A) The total linear momentum of $S$.
(B) The horizontal component of the linear momentum of $S$.
(C) The mechanical energy of $S$.
(D) The angular momentum of $S$ as measured about a perpendicular axis through the peg.
(E) None of the above are conserved.
2012 Kosovo National Mathematical Olympiad, 4
The right triangle $ABC$ with a right angle at $C$. From all the rectangles $CA_1MB_1$, where $A_1\in BC, M\in AB$ and $B_1\in AC$ which one has the biggest area?
2019 SIMO, Q1
[i]George the grasshopper[/i] lives of the real line, starting at $0$ . He is given the following sequence of numbers: $2, 3, 4, 8, 9, ... ,$ which are all the numbers of the form $2^k$ or $3^l$, $k, l \in \mathbb{N}$, arranged in increasing order. Starting from $2$, for each number $x$ in the sequence in order, he (currently at $a$) must choose to jump to either $a+x$ or $a-x$. Show that [i]George the grasshopper[/i] can jump in a way that he reaches every integer on the real line.
2020 Greece Junior Math Olympiad, 4
We are having 99 equal circles in a row and in the interior, we write inside them all the numbers from 1 up to 99 (one number in each circle).We color each of the circles with one of the two colors available: red and green.
A coloring is good if it has the ability:
Red circles lying in the interval of the numbers from 1 up to 50 are more than the red circles lying in the interval of the numbers from 51 up to 99 .
a) Find how many different colorings can be constructed.
b) Find how many different good colorings can be constructed.
(Note: Two colorings are different, if they have different color in at least one of their circles.)
2004 Bulgaria Team Selection Test, 1
Let $n$ be a positive integer. Find all positive integers $m$ for which there exists a polynomial $f(x) = a_{0} + \cdots + a_{n}x^{n} \in \mathbb{Z}[X]$ ($a_{n} \not= 0$) such that $\gcd(a_{0},a_{1},\cdots,a_{n},m)=1$ and $m|f(k)$ for each $k \in \mathbb{Z}$.
2014 Contests, 1
The diagram below shows a circle with center $F$. The angles are related with $\angle BFC = 2\angle AFB$, $\angle CFD = 3\angle AFB$, $\angle DFE = 4\angle AFB$, and $\angle EFA = 5\angle AFB$. Find the degree measure of $\angle BFC$.
[asy]
size(4cm);
pen dps = fontsize(10);
defaultpen(dps);
dotfactor=4;
draw(unitcircle);
pair A,B,C,D,E,F;
A=dir(90);
B=dir(66);
C=dir(18);
D=dir(282);
E=dir(210);
F=origin;
dot("$F$",F,NW);
dot("$A$",A,dir(90));
dot("$B$",B,dir(66));
dot("$C$",C,dir(18));
dot("$D$",D,dir(306));
dot("$E$",E,dir(210));
draw(F--E^^F--D^^F--C^^F--B^^F--A);
[/asy]
2023-24 IOQM India, 6
Let $X$ be the set of all even positive integers $n$ such that the measure of the angle of some regular polygon is $n$ degrees. Find the number of elements in $X$.
2020 Tuymaada Olympiad, 3
Each edge of a complete graph with $101$ vertices is marked with $1$ or $-1$. It is known that absolute value of the sum of numbers on all the edges is less than $150$. Prove that the graph contains a path visiting each vertex exactly once such that the sum of numbers on all edges of this path is zero.
[i](Y. Caro, A. Hansberg, J. Lauri, C. Zarb)[/i]
1978 All Soviet Union Mathematical Olympiad, 268
Consider a sequence $$x_n=(1+\sqrt2+\sqrt3)^n$$ Each member can be represented as $$x_n=q_n+r_n\sqrt2+s_n\sqrt3+t_n\sqrt6$$ where $q_n, r_n, s_n, t_n$ are integers. Find the limits of the fractions $r_n/q_n, s_n/q_n, t_n/q_n$.
2015 Peru IMO TST, 14
Let $ n$ be a positive integer and let $ a_1,a_2,\ldots,a_n$ be positive real numbers such that:
\[ \sum^n_{i \equal{} 1} a_i \equal{} \sum^n_{i \equal{} 1} \frac {1}{a_i^2}.
\]
Prove that for every $ i \equal{} 1,2,\ldots,n$ we can find $ i$ numbers with sum at least $ i$.
2019 Singapore Junior Math Olympiad, 5
Let $n$ be a positive integer and consider an arrangement of $2n$ blocks in a straight line, where $n$ of them are red and the rest blue. A swap refers to choosing two consecutive blocks and then swapping their positions. Let $A$ be the minimum number of swaps needed to make the first $n$ blocks all red and $B$ be the minimum number of swaps needed to make the first $n$ blocks all blue. Show that $A+B$ is independent of the starting arrangement and determine its value.
2008 Irish Math Olympiad, 4
How many sequences $ a_1,a_2,...,a{}_2{}_0{}_0{}_8$ are there such that each of the numbers $ 1,2,...,2008$ occurs once in the sequence, and $ i \in (a_1,a_2,...,a_i)$ for each $ i$ such that $ 2\le i \le2008$?
2000 Stanford Mathematics Tournament, 14
The author of this question was born on April 24, 1977. What day of the week was that?
2016 NIMO Problems, 8
For a complex number $z \neq 3$,$4$, let $F(z)$ denote the real part of $\tfrac{1}{(3-z)(4-z)}$. If \[
\int_0^1 F \left( \frac{\cos 2 \pi t + i \sin 2 \pi t}{5} \right) \; dt = \frac mn
\] for relatively prime positive integers $m$ and $n$, find $100m+n$.
[i]Proposed by Evan Chen[/i]
2008 AMC 10, 3
Assume that $ x$ is a positive real number. Which is equivalent to $ \sqrt[3]{x\sqrt{x}}$?
$ \textbf{(A)}\ x^{1/6} \qquad
\textbf{(B)}\ x^{1/4} \qquad
\textbf{(C)}\ x^{3/8} \qquad
\textbf{(D)}\ x^{1/2} \qquad
\textbf{(E)}\ x$
2020 LMT Spring, 3
Let $LMT$ represent a 3-digit positive integer where $L$ and $M$ are nonzero digits. Suppose that the 2-digit number $MT$ divides $LMT$. Compute the difference between the maximum and minimum possible values of $LMT$.