Found problems: 85335
1990 Tournament Of Towns, (271) 5
The numerical sequence $\{x_n\}$ satisfies the condition $$x_{n+1}=|x_n|-x_{n-1}$$ for all $n > 1$. Prove that the sequence is periodic with period $9$, i.e. for any $n > 1$ we have $x_n = x_{n+9}$.
(M Kontsevich, Moscow)
2005 IMO Shortlist, 5
Let $x,y,z$ be three positive reals such that $xyz\geq 1$. Prove that
\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]
[i]Hojoo Lee, Korea[/i]
2014 Thailand Mathematical Olympiad, 5
Determine the maximal value of $k$ such that the inequality
$$\left(k +\frac{a}{b}\right) \left(k + \frac{b}{c}\right)\left(k + \frac{c}{a}\right)
\le \left( \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\right) \left( \frac{b}{a}+ \frac{c}{b}+ \frac{a}{c}\right)$$
holds for all positive reals $a, b, c$.
2020 Ecuador NMO (OMEC), 1
The country OMEC is divided in $5$ regions, each region is divided in $5$ districts, and, in each district, $1001$ people vote. Each person choose between $A$ or $B$. In a district, a candidate's letter wins if it's the letter with the most votes. In a region, a candidate's letter wins if it won in most districts. A candidate is the new president of OMEC if the candidate won in most regions. The candidate $A$ can rearrange the people of each district in each region (for example, A moves someone in District M to District N in region 1), but he can't change them to a different region.
Find the minimum number of votes that the candidate $A$ needs to become the new president.
2014 NIMO Problems, 2
In the Generic Math Tournament, $99$ people participate. One of the participants, Alfred, scores 16th in Algebra, 30th in Combinatorics, and 23rd in Geometry (and does not tie with anyone). The overall ranking is computed by adding the scores from all three tests. Given this information, let $B$ be the best ranking that Alfred could have achieved, and let $W$ be the worst ranking that he could have achieved. Compute $100B+W$.
[i]Proposed by Lewis Chen[/i]
2022 Mexican Girls' Contest, 7
Let $ABCD$ be a parallelogram (non-rectangle) and $\Gamma$ is the circumcircle of $\triangle ABD$. The points $E$ and $F$ are the intersections of the lines $BC$ and $DC$ with $\Gamma$ respectively. Define $P=ED\cap BA$, $Q=FB\cap DA$ and $R=PQ\cap CA$. Prove that
$$\frac{PR}{RQ}=(\frac{BC}{CD})^2$$
2004 AIME Problems, 12
Let $S$ be the set of ordered pairs $(x, y)$ such that $0<x\le 1$, $0<y\le 1$, and $\left[\log_2{\left(\frac 1x\right)}\right]$ and $\left[\log_5{\left(\frac 1y\right)}\right]$ are both even. Given that the area of the graph of $S$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. The notation $[z]$ denotes the greatest integer that is less than or equal to $z$.
1997 Swedish Mathematical Competition, 1
Let $AC$ be a diameter of a circle and $AB$ be tangent to the circle. The segment $BC$ intersects the circle again at $D$. Show that if $AC = 1$, $AB = a$, and $CD = b$, then $$\frac{1}{a^2+ \frac12 }< \frac{b}{a}< \frac{1}{a^2}$$
2020 Jozsef Wildt International Math Competition, W59
If $a_k>0~(k=1,2,\ldots,n)$ then prove that
$$\sum_{\text{cyc}}\left(\frac{(a_1+a_2+\ldots+a_{n-1})^2}{a_n}+\frac{a_n^2}{a_1}\right)\ge\frac{n^2}2\sum_{k=1}^na_k$$
[i]Proposed by Mihály Bencze[/i]
2014 NIMO Problems, 7
Ana and Banana play a game. First, Ana picks a real number $p$ with $0 \le p \le 1$. Then, Banana picks an integer $h$ greater than $1$ and creates a spaceship with $h$ hit points. Now every minute, Ana decreases the spaceship's hit points by $2$ with probability $1-p$, and by $3$ with probability $p$. Ana wins if and only if the number of hit points is reduced to exactly $0$ at some point (in particular, if the spaceship has a negative number of hit points at any time then Ana loses). Given that Ana and Banana select $p$ and $h$ optimally, compute the integer closest to $1000p$.
[i]Proposed by Lewis Chen[/i]
2019 IFYM, Sozopol, 8
Find all polynomials $f\in Z[X],$ such that for each odd prime $p$ $$f(p)|(p-3)!+\frac{p+1}{2}.$$
1992 Vietnam National Olympiad, 3
Label the squares of a $1991 \times 1992$ rectangle $(m, n)$ with $1 \leq m \leq 1991$ and $1 \leq n \leq 1992$. We wish to color all the squares red. The first move is to color red the squares $(m, n), (m+1, n+1), (m+2, n+1)$for some $m < 1990, n < 1992$. Subsequent moves are to color any three (uncolored) squares in the same row, or to color any three (uncolored) squares in the same column. Can we color all the squares in this way?
2022 AMC 8 -, 24
The figure below shows a polygon $ABCDEFGH$, consisting of rectangles and right triangles. When cut out and folded on the dotted lines, the polygon forms a triangular prism. Suppose that $AH = EF = 8$ and $GH = 14$. What is the volume of the prism?
[asy]
// djmathman diagram
unitsize(1cm);
defaultpen(linewidth(0.7)+fontsize(11));
real r = 2, s = 2.5, theta = 14;
pair G = (0,0), F = (r,0), C = (r,s), B = (0,s), M = (C+F)/2, I = M + s/2 * dir(-theta);
pair N = (B+G)/2, J = N + s/2 * dir(180+theta);
pair E = F + r * dir(- 45 - theta/2), D = I+E-F;
pair H = J + r * dir(135 + theta/2), A = B+H-J;
draw(A--B--C--I--D--E--F--G--J--H--cycle^^rightanglemark(F,I,C)^^rightanglemark(G,J,B));
draw(J--B--G^^C--F--I,linetype ("4 4"));
dot("$A$",A,N);
dot("$B$",B,1.2*N);
dot("$C$",C,N);
dot("$D$",D,dir(0));
dot("$E$",E,S);
dot("$F$",F,1.5*S);
dot("$G$",G,S);
dot("$H$",H,W);
dot("$I$",I,NE);
dot("$J$",J,1.5*S);
[/asy]
$\textbf{(A)} ~112\qquad\textbf{(B)} ~128\qquad\textbf{(C)} ~192\qquad\textbf{(D)} ~240\qquad\textbf{(E)} ~288\qquad$
2016 HMNT, 8
Alex has an $20 \times 16$ grid of lightbulbs, initially all off. He has $36$ switches, one for each row and column. Flipping the switch for the $i$th row will toggle the state of each lightbulb in the $i$th row (so that if it were on before, it would be off, and vice versa). Similarly, the switch for the $j$th column will toggle the state of each bulb in the $j$th column. Alex makes some (possibly empty) sequence of switch flips, resulting in some configuration of the lightbulbs and their states. How many distinct possible configurations of lightbulbs can Alex achieve with such a sequence? Two configurations are distinct if there exists a lightbulb that is on in one configuration and off in another.
2007 ITest, 50
A block $Z$ is formed by gluing one face of a solid cube with side length 6 onto one of the circular faces of a right circular cylinder with radius $10$ and height $3$ so that the centers of the square and circle coincide. If $V$ is the smallest convex region that contains Z, calculate $\lfloor\operatorname{vol}V\rfloor$ (the greatest integer less than or equal to the volume of $V$).
2006 Stanford Mathematics Tournament, 1
A finite sequence of positive integers $ m_i$ for $ i\equal{}1,2,...,2006$ are defined so that $ m_1\equal{}1$ and $ m_i\equal{}10m_{i\minus{}1} \plus{}1$ for $ i>1$. How many of these integers are divisible by $ 37$?
Kvant 2019, M2583
On the side $DE$ and on the diagonal $BE$ of the regular pentagon $ABCDE$ we consider the squares $DEFG$ and $BEHI$.
[list=a]
[*] Prove that $A,I,$ and $G$ are collinear.
[*] Prove that on this line lies also the centre $O$ of the square $BDJK$.
[/list]
2010 Morocco TST, 4
Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.
2022 VN Math Olympiad For High School Students, Problem 1
Given [i]Fibonacci[/i] sequence $(F_n)$
a) Prove that: for all $u,v\in \mathbb{N}, u\ge 1$, we have:$$F_{u+v}=F_{u-1}F_{v}+F_{u}F_{v+1}.$$
b) Prove that: for all $n\in \mathbb{N}, n\ge 1$, we have:$$F_{2n}=F_n(F_{n-1}+F_{n+1}),$$$$F_{2n+1}=F_n^2+F_{n+1}^2.$$
2000 France Team Selection Test, 1
Some squares of a $1999\times 1999$ board are occupied with pawns. Find the smallest number of pawns for which it is possible that for each empty square, the total number of pawns in the row or column of that square is at least $1999$.
2019 Danube Mathematical Competition, 3
Let be a sequence of $ 51 $ natural numbers whose sum is $ 100. $ Show that for any natural number $ 1\le k<100 $ there are some consecutive numbers from this sequence whose sum is $ k $ or $ 100-k. $
1989 Swedish Mathematical Competition, 1
Let $n$ be a positive integer. Prove that the numbers $n^2(n^2 + 2)^2$ and $n^4(n^2 + 2)^2$ are written in base $n^2 +1$ with the same digits but in opposite order.
Novosibirsk Oral Geo Oly VIII, 2019.1
Kikoriki live on the shores of a pond in the form of an equilateral triangle with a side of $600$ m, Krash and Wally live on the same shore, $300$ m from each other. In summer, Dokko to Krash walk $900$ m, and Wally to Rosa - also $900$ m. Prove that in winter, when the pond freezes and it will be possible to walk directly on the ice, Dokko will walk as many meters to Krash as Wally to Rosa.
[url=https://en.wikipedia.org/wiki/Kikoriki]about Kikoriki/GoGoRiki / Smeshariki [/url]
2005 Hungary-Israel Binational, 2
Let $F_{n}$ be the $n-$ th Fibonacci number (where $F_{1}= F_{2}= 1$). Consider the functions
$f_{n}(x)=\parallel . . . \parallel |x|-F_{n}|-F_{n-1}|-...-F_{2}|-F_{1}|, g_{n}(x)=| . . . \parallel x-1|-1|-...-1|$ ($F_{1}+...+F_{n}$ one’s).
Show that $f_{n}(x) = g_{n}(x)$ for every real number $x.$
1993 AIME Problems, 10
Euler's formula states that for a convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces, $V - E + F = 2$. A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces meet. What is the value of $100P + 10T + V$?