Found problems: 85335
2004 Regional Competition For Advanced Students, 2
Solve the following equation for real numbers:
$ \sqrt{4\minus{}x\sqrt{4\minus{}(x\minus{}2)\sqrt{1\plus{}(x\minus{}5)(x\minus{}7)}}}\equal{}\frac{5x\minus{}6\minus{}x^2}{2}$
(all square roots are non negative)
2015 Purple Comet Problems, 11
The Purple Plant Garden Store sells grass seed in ten-pound bags and fifteen-pound bags. Yesterday half
of the grass seed they had was in ten-pound bags. This morning the store received a shipment of 27 more
ten-pound bags, and now they have twice as many ten-pound bags as fifteen-pound bags. Find the total
weight in pounds of grass seed the store now has.
2019 IFYM, Sozopol, 8
We are given a $\Delta ABC$. Point $D$ on the circumscribed circle k is such that $CD$ is a symmedian in $\Delta ABC$. Let $X$ and $Y$ be on the rays $\overrightarrow{CB}$ and $\overrightarrow{CA}$, so that $CX=2CA$ and $CY=2CB$. Prove that the circle, tangent externally to $k$ and to the lines $CA$ and $CB$, is tangent to the circumscribed circle of $\Delta XDY$.
2022 China Team Selection Test, 3
Given a positive integer $n \ge 2$. Find all $n$-tuples of positive integers $(a_1,a_2,\ldots,a_n)$, such that $1<a_1 \le a_2 \le a_3 \le \cdots \le a_n$, $a_1$ is odd, and
(1) $M=\frac{1}{2^n}(a_1-1)a_2 a_3 \cdots a_n$ is a positive integer;
(2) One can pick $n$-tuples of integers $(k_{i,1},k_{i,2},\ldots,k_{i,n})$ for $i=1,2,\ldots,M$ such that for any $1 \le i_1 <i_2 \le M$, there exists $j \in \{1,2,\ldots,n\}$ such that $k_{i_1,j}-k_{i_2,j} \not\equiv 0, \pm 1 \pmod{a_j}$.
1993 Hungary-Israel Binational, 5
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.
Let $H \leq G, |H | = 3.$ What can be said about $|N_{G}(H ) : C_{G}(H )|$?
LMT Speed Rounds, 2010.14
On the team round, an LMT team of six students wishes to divide itself into two distinct groups of three, one group to work on part $1,$ and one group to work on part $2.$ In addition, a captain of each group is designated. In how many ways can this be done?
2003 IMC, 4
Determine the set of all pairs (a,b) of positive integers for which the set of positive integers can be decomposed into 2 sets A and B so that $a\cdot A=b\cdot B$.
2010 Math Prize For Girls Problems, 16
Let $P$ be the quadratic function such that $P(0) = 7$, $P(1) = 10$, and $P(2) = 25$. If $a$, $b$, and $c$ are integers such that every positive number $x$ less than 1 satisfies
\[
\sum_{n = 0}^\infty P(n) x^n = \frac{ax^2 + bx + c}{{(1 - x)}^3},
\]
compute the ordered triple $(a, b, c)$.
2005 Thailand Mathematical Olympiad, 1
A point $A$ is chosen outside a circle with diameter $BC$ so that $\vartriangle ABC$ is acute. Segments $AB$ and $AC$ intersect the circle at $D$ and $E$, respectively, and $CD$ intersects $BE$ at $F$. Line $AF$ intersects the circle again at $G$ and intersects $BC$ at $H$. Prove that $AH \cdot F H = GH^2$.
.
1985 IMO Shortlist, 17
The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by
\[f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)\]
Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$
2023 Sharygin Geometry Olympiad, 8.5
The median $CM$ and the altitude $AH$ of an acute-angled triangle $ABC$ meet at point $O$. A point $D$ lies outside the triangle in such a way that $AOCD$ is a parallelogram. Find the length of $BD$, if $MO= a$, $OC = b$.
1988 IMO Longlists, 62
Let $x = p, y = q, z = r, w = s$ be the unique solution of the system of linear equations \[ x + a_i \cdot y + a^2_i \cdot z + a^3_i \cdot w = a^4_i, i = 1,2,3,4. \] Express the solutions of the following system in terms of $p,q,r$ and $s:$ \[ x + a^2_i \cdot y + a^4_i \cdot z + a^6_i \cdot w = a^8_i, i = 1,2,3,4. \] Assume the uniquness of the solution.
2021 China Second Round A1, 4
There are 100 points on a circle that are about to be colored in two colors: red or blue. Find the largest number $k$ such that no matter how I select and color $k$ points, you can always color the remaining $100-k$ points such that you can connect 50 pairs of points of the same color with lines in a way such that no two lines intersect.
1999 Mongolian Mathematical Olympiad, Problem 3
At each vertex of a $4\times5$ rectangle there is a house. Find the path of the minimum length connecting all these houses.
1995 Czech And Slovak Olympiad IIIA, 3
Five distinct points and five distinct lines are given in the plane. Prove that one can select two of the points and two of the lines so that none of the selected lines contains any of the selected points.
2021 Estonia Team Selection Test, 1
Juku has the first $100$ volumes of the Harrie Totter book series at his home. For every$ i$ and $j$, where $1 \le i < j \le 100$, call the pair $(i, j)$ reversed if volume No. $j$ is before volume No, $i$ on Juku’s shelf. Juku wants to arrange all volumes of the series to one row on his shelf in such a way that there does not exist numbers $i, j, k$, where $1 \le i < j < k \le 100$, such that pairs $(i, j)$ and $(j, k)$ are both reversed. Find the largest number of reversed pairs that can occur under this condition
1989 Bundeswettbewerb Mathematik, 2
Find all pairs $(a,b)$ of real numbers such that
$$|\sqrt{1-x^2 }-ax-b| \leq \frac{\sqrt{2} -1}{2}$$
holds for all $x\in [0,1]$.
2000 Romania National Olympiad, 3
A function $ f:\mathbb{R}^2\longrightarrow\mathbb{R} $ is [i]olympic[/i] if, any finite number of pairwise distinct elements of $ \mathbb{R}^2 $ at which the function takes the same value represent in the plane the vertices of a convex polygon.
Prove that if $ p $ if a complex polynom of degree at least $ 1, $ then the function $ \mathbb{R}^2\ni (x,y)\mapsto |p(x+iy)| $ is olympic if and only if the roots of $ p $ are all equal.
2014 European Mathematical Cup, 1
Prove that there exist infinitely many positive integers which cannot be written in form $a^{d(a)}+b^{d(b)}$ for some positive integers $a$ and $b$
For positive integer $d(a)$ denotes number of positive divisors of $a$
[i]Proposed by Borna Vukorepa[/i]
2017 CCA Math Bonanza, L2.1
Adam and Mada are playing a game of one-on-one basketball, in which participants may take $2$-point shots (worth $2$ points) or $3$-point shots (worth $3$ points). Adam makes $10$ shots of either value while Mada makes $11$ shots of either value. Furthermore, Adam made the same number of $2$-point shots as Mada made $3$-point shots. At the end of the game, the two basketball players realize that they have the exact same number of points! How many total points were scored in the game?
[i]2017 CCA Math Bonanza Lightning Round #2.1[/i]
1969 Putnam, A5
Let $u(t)$ be a continuous function in the system of differential equations
$$\frac{dx}{dt} =-2y +u(t),\;\;\; \frac{dy}{dt}=-2x+ u(t).$$
Show that, regardless of the choice of $u(t)$, the solution of the system which satisfies $x=x_0 , y=y_0$
at $t=0$ will never pass through $(0, 0)$ unless $x_0 =y_0.$ When $x_0 =y_0 $, show that, for any positive value
$t_0$ of $t$, it is possible to choose $u(t)$ so the solution is equal to $(0,0)$ when $t=t_0 .$
2023 Belarusian National Olympiad, 9.8
On the faces of a cube several positive integer numbers are written. On every edge the sum of the numbers of it's two faces is written, and in every vertex the sum of numbers on the three faces that have this vertex. It turned out that all the written numbers are different.
Find the smallest possible amount of the sum of all written numbers.
2021 Polish Junior MO First Round, 1
Is there a six-digit number where every two consecutive digits make up a certain number two-digit number that is the square of an integer? Justify your answer.
2014 Saudi Arabia GMO TST, 3
Let $ABC$ be a triangle, $I$ its incenter, and $\omega$ a circle of center $I$. Points $A',B', C'$ are on $\omega$ such that rays $IA', IB', IC',$ starting from $I$ intersect perpendicularly sides $BC, CA, AB$, respectively. Prove that lines $AA', BB', CC'$ are concurrent.
2017 Junior Balkan Team Selection Tests - Romania, 2
Given $x_1,x_2,...,x_n$ real numbers, prove that there exists a real number $y$, such that,
$$\{y-x_1\}+\{y-x_2\}+...+\{y-x_n\} \leq \frac{n-1}{2}$$