Found problems: 85335
2015 China Team Selection Test, 1
Let $x_1,x_2,\cdots,x_n$ $(n\geq2)$ be a non-decreasing monotonous sequence of positive numbers such that $x_1,\frac{x_2}{2},\cdots,\frac{x_n}{n}$ is a non-increasing monotonous sequence .Prove that
\[ \frac{\sum_{i=1}^{n} x_i }{n\left (\prod_{i=1}^{n}x_i \right )^{\frac{1}{n}}}\le \frac{n+1}{2\sqrt[n]{n!}}\]
2005 Tournament of Towns, 6
Two operations are allowed:
(i) to write two copies of number $1$;
(ii) to replace any two identical numbers $n$ by $(n + 1)$ and $(n - 1)$.
Find the minimal number of operations that required to produce the number $2005$ (at the beginning there are no numbers).
[i](8 points)[/i]
2014 Thailand TSTST, 2
Find the number of permutations $(a_1, a_2, . . . , a_{2013})$ of $(1, 2, \dots , 2013)$ such that there are exactly two indices $i \in \{1, 2, \dots , 2012\}$ where $a_i < a_{i+1}$.
2010 Postal Coaching, 3
Find all natural numbers $n$ such that the number $n(n+1)(n+2)(n+3)$ has exactly three different prime divisors.
2024 Sharygin Geometry Olympiad, 10.1
The diagonals of a cyclic quadrilateral $ABCD$ meet at point $P$. The bisector of angle $ABD$ meets $AC$ at point $E$, and the bisector of angle $ACD$ meets $BD$ at point $F$. Prove that the lines $AF$ and $DE$ meet on the median of triangle $APD$.
2024 Chile TST IMO, 1
Consider a set of \( n \geq 3 \) points in the plane where no three are collinear. Prove that the points can be labeled as \( P_1, P_2, \dots, P_n \) so that the angles \( \angle P_i P_{i+1} P_{i+2} \) are less than \( 90^\circ \) for all \( i \).
1989 IMO Longlists, 55
The set $ \{a_0, a_1, \ldots, a_n\}$ of real numbers satisfies the following conditions:
[b](i)[/b] $ a_0 \equal{} a_n \equal{} 0,$
[b](ii)[/b] for $ 1 \leq k \leq n \minus{} 1,$ \[ a_k \equal{} c \plus{} \sum^{n\minus{}1}_{i\equal{}k} a_{i\minus{}k} \cdot \left(a_i \plus{} a_{i\plus{}1} \right)\]
Prove that $ c \leq \frac{1}{4n}.$
2007 Princeton University Math Competition, 5
Let $f_n$ be the Fibonacci numbers, defined by $f_0 = 1$, $f_1 = 1$, and $f_n = f_{n-1}+f_{n-2}$. For each $i$, $1 \le i \le 200$, we calculate the greatest common divisor $g_i$ of $f_i$ and $f_{2007}$. What is the sum of the distinct values of $g_i$?
1990 All Soviet Union Mathematical Olympiad, 533
A game is played in three moves. The first player picks any real number, then the second player makes it the coefficient of a cubic, except that the coefficient of $x^3$ is already fixed at $1$. Can the first player make his choices so that the final cubic has three distinct integer roots?
2003 Purple Comet Problems, 10
How many gallons of a solution which is $15\%$ alcohol do we have to mix with a solution that is $35\%$ alcohol to make $250$ gallons of a solution that is $21\%$ alcohol?
1999 Putnam, 1
Right triangle $ABC$ has right angle at $C$ and $\angle BAC=\theta$; the point $D$ is chosen on $AB$ so that $|AC|=|AD|=1$; the point $E$ is chosen on $BC$ so that $\angle CDE=\theta$. The perpendicular to $BC$ at $E$ meets $AB$ at $F$. Evaluate $\lim_{\theta\to 0}|EF|$.
2022 Thailand TSTST, 1
Find all polynomials $f, g, h$ with real coefficients, such that $f(x)^2+(x+1)g(x)^2=(x^3+x)h(x)^2$
2014 Purple Comet Problems, 3
The cross below is made up of five congruent squares. The perimeter of the cross is $72$. Find its area.
[asy]
import graph;
size(3cm);
pair A = (0,0);
pair temp = (1,0);
pair B = rotate(45,A)*temp;
pair C = rotate(90,B)*A;
pair D = rotate(270,C)*B;
pair E = rotate(270,D)*C;
pair F = rotate(90,E)*D;
pair G = rotate(270,F)*E;
pair H = rotate(270,G)*F;
pair I = rotate(90,H)*G;
pair J = rotate(270,I)*H;
pair K = rotate(270,J)*I;
pair L = rotate(90,K)*J;
draw(A--B--C--D--E--F--G--H--I--J--K--L--cycle);
[/asy]
2019 Jozsef Wildt International Math Competition, W. 43
Consider the sequence of polynomials $P_0(x) = 2$, $P_1(x) = x$ and $P_n(x) = xP_{n-1}(x) - P_{n-2}(x)$ for $n \geq 2$. Let $x_n$ be the greatest zero of $P_n$ in the the interval $|x| \leq 2$. Show that $$\lim \limits_{n \to \infty}n^2\left(4-2\pi +n^2\int \limits_{x_n}^2P_n(x)dx\right)=2\pi - 4-\frac{\pi^3}{12}$$
2013 Math Prize For Girls Problems, 4
The MathMatters competition consists of 10 players $P_1$, $P_2$, $\dots$, $P_{10}$ competing in a ladder-style tournament. Player $P_{10}$ plays a game with $P_9$: the loser is ranked 10th, while the winner plays $P_8$. The loser of that game is ranked 9th, while the winner plays $P_7$. They keep repeating this process until someone plays $P_1$: the loser of that final game is ranked 2nd, while the winner is ranked 1st. How many different rankings of the players are possible?
1986 IMO, 1
Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.
2011 Albania Team Selection Test, 5
The sweeties shop called "Olympiad" sells boxes of $6,9$ or $20$ chocolates. Groups of students from a school that is near the shop collect money to buy a chocolate for each student; to make this they buy a box and than give to everybody a chocolate. Like this students can create groups of $15=6+9$ students, $38=2*9+20$ students, etc. The seller has promised to the students that he can satisfy any group of students, and if he will need to open a new box of chocolate for any group (like groups of $4,7$ or $10$ students) than he will give all the chocolates for free to this group. Can there be constructed the biggest group that profits free chocolates, and if so, how many students are there in this group?
2000 Austrian-Polish Competition, 1
Find all polynomials $P(x)$ with real coefficients having the following property: There exists a positive integer n such that the equality $$\sum_{k=1}^{2n+1}(-1)^k \left[\frac{k}{2}\right] P(x + k)=0$$
holds for infinitely many real numbers $x$.
2015 Postal Coaching, Problem 3
Let $a,b,c \in \mathbb{R^+}$ such that $abc=1$. Prove that
$$\sum_{a,b,c} \sqrt{\frac{a}{a+8}} \ge 1$$
1992 Romania Team Selection Test, 8
Let $m,n \ge 2$ be integers. The sides $A_{00}A_{0m}$ and $A_{nm}A_{n0}$ of a convex quadrilateral $A_{00}A_{0m}A_{nm}A_{n0}$ are divided into $m$ equal segments by points $A_{0j}$ and $A_{nj}$ respectively ($j = 1,...,m-1$). The other two sides are divided into $n$ equal segments by points $A_{i0}$ and $A_{im}$ ($i = 1,...,n -1$). Denote by $A_{ij}$ the intersection of lines $A_{0j}A{nj}$ and $A_{i0}A_{im}$, by $S_{ij}$ the area of quadrilateral $A_{ij}A_{i, j+1}A_{i+1, j+1}A_{i+1, j}$ and by $S$ the area of the big quadrilateral. Show that $S_{ij} +S_{n-1-i,m-1-j} =
\frac{2S}{mn}$
2014 AMC 8, 14
Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid, as shown. What is $DE$?
[asy]
size(250);
defaultpen(linewidth(0.8));
pair A=(0,5),B=origin,C=(6,0),D=(6,5),E=(18,0);
draw(A--B--E--D--cycle^^C--D);
draw(rightanglemark(D,C,E,30));
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,S);
label("$D$",D,N);
label("$E$",E,S);
label("$5$",A/2,W);
label("$6$",(A+D)/2,N);
[/asy]
$\textbf{(A) }12\qquad\textbf{(B) }13\qquad\textbf{(C) }14\qquad\textbf{(D) }15\qquad \textbf{(E) }16$
2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, A1
Find the smallest positive integer value of $N$ such that field $K=\mathbb{Q}(\sqrt{N},\ \sqrt{i+1})$, where $i=\sqrt{-1}$, is Galois extension on $\mathbb{Q}$, then find the Galois group $Gal(K/\mathbb{Q}).$
2022 Balkan MO Shortlist, A6
Determine all functions $f : \mathbb{R}^2 \to\mathbb {R}$ for which \[f(A)+f(B)+f(C)+f(D)=0,\]whenever $A,B,C,D$ are the vertices of a square with side-length one.
[i]Ilir Snopce[/i]
2023 Denmark MO - Mohr Contest, 1
A frog jumps around on the integers on the number line. If it lands on an even number $n$, it jumps to the number $\frac{n}{2}$ . If it lands on an odd number $n$, it jumps to the number $n + 5$. At some point it lands on the number $25$. At which numbers may it have been three jumps ago?
2024-IMOC, N6
Find all functions $f:\mathbb{Q}^+\to\mathbb{Q}^+$ such that
\[xy(f(x)-f(y))|x-f(f(y))\]
holds for all positive rationals $x$, $y$
(we define that $a|b$ if and only if exist $n \in \mathbb{Z}$ such that $b=an$)
[i]Proposed by supercarry & windleaf1A[/i]