Found problems: 85335
The smallest value of the function $f(x) =|x| +\left|\frac{1 - 2013x}{2013 - x}\right|$ where $x \in [-1, 1] $ is:
(A): $\frac{1}{2012}$, (B): $\frac{1}{2013}$, (C): $\frac{1}{2014}$, (D): $\frac{1}{2015}$, (E): None of the above.
Let $A=(a_{ij})$ and $B=(b_{ij})$ be two real $10\times10$ matrices such that $a_{ij}=b_{ij}+1$ for all $i,j$ and $A^3=0$. Prove that $\det B=0$.
Let $x,y,z$ be positive numbers with $x+y+z=1$. Show that
$$yz+zx+xy\ge4\left(y^2z^2+z^2x^2+x^2y^2\right)+5xyz.$$When does equality hold?
Let $ x_1, x_2, ..., x_5$ be a non-negative real numbers such that $ x_1 \plus{} x_2 \plus{} \cdots \plus{} x_5 \equal{} 100.$ Let $ M$ be a maximum of the numbers $ x_1 \plus{} x_2, x_2 \plus{} x_3, x_3 \plus{} x_4,$ and $ x_4 \plus{} x_5$. The least possible value of $ M$ lies in the interval
A. [0,32)
B. [32, 34)
C. [34, 36)
D. [36, 38)
E. [38, 40]
Let $N$ be the number of integral solutions of the equation
\[x^2 - y^2 = z^3 - t^3\]
satisfying the condition $0 \leq x, y, z, t \leq 10^6$, and let $M$ be the number of integral solutions of the equation
\[x^2 - y^2 = z^3 - t^3 + 1\]
satisfying the condition $0 \leq x, y, z, t \leq 10^6$. Prove that $N >M.$
Let $a,b,c$ be positive real numbers. Prove that
$\sqrt{\frac{a}{b+c}+\frac{b}{c+a}}+\sqrt{\frac{b}{c+a}+\frac{c}{a+b}}+\sqrt{\frac{c}{a+b}+\frac{a}{b+c}}\ge 3$
For what real numbers $p$ has the system of equations
$$\begin{cases} x_1^4+\dfrac{1}{x_1^2}=px_2 \\ \\ x_2^4+\dfrac{1}{x_2^2}=px_3 \\ ... \\ x_{2004}^4+\dfrac{1}{x_{2004}^2}=px_{2005} \\ \\ x_{2005}^4+\dfrac{1}{x_{2005}^2}=px_{1}\end{cases}$$
just one solution $(x_1,x_2,...,x_{2005})$, where $x_1,x_2,...,x_{2005}$ are real numbers?
A machine accepts coins of $k{}$ values $1 = a_1 <\cdots < a_k$ and sells $k{}$ different drinks with prices $0<b_1 < \cdots < b_k$. It is known that if we start inserting coins into the machine in an arbitrary way, sooner or later the total value of the coins will be equal to the price of a drink. For which sets of numbers $(a_1,\ldots,a_k;b_1,\ldots,b_k)$ does this property hold?
Consider an acute angled triangle $\vartriangle ABC$ with side lengths $7$, $8$, and $9$. Let $H$ be the orthocenter of $ABC$. Let $\Gamma_A$, $\Gamma_B$, and $\Gamma_C$ be the circumcircles of $\vartriangle BCH$, $\vartriangle CAH$, and $\vartriangle ABH$ respectively. Find the area of the region $\Gamma_A \cup \Gamma_B \cup \Gamma_C$ (the set of all points contained in at least one of $\Gamma_A$, $\Gamma_B$, and $\Gamma_C$).
The $n$ contestant of EGMO are named $C_1, C_2, \cdots C_n$. After the competition, they queue in front of the restaurant according to the following rules.
[list]
[*]The Jury chooses the initial order of the contestants in the queue.
[*]Every minute, the Jury chooses an integer $i$ with $1 \leq i \leq n$.
[list]
[*]If contestant $C_i$ has at least $i$ other contestants in front of her, she pays one euro to the Jury and moves forward in the queue by exactly $i$ positions.
[*]If contestant $C_i$ has fewer than $i$ other contestants in front of her, the restaurant opens and process ends.
[/list]
[/list]
[list=a]
[*]Prove that the process cannot continue indefinitely, regardless of the Jury’s choices.
[*]Determine for every $n$ the maximum number of euros that the Jury can collect by cunningly choosing the initial order and the sequence of moves.
[/list]
[asy]size(100);
draw((0,0)--(5,0),MidArrow);
draw((5,0)--(10,0),MidArrow);
draw((5,5sqrt(3))--(2.5,2.5sqrt(3)),MidArrow);
draw((2.5,2.5sqrt(3))--(0,0),MidArrow);
draw((5,5sqrt(3))--(7.5,2.5sqrt(3)),MidArrow);
draw((7.5,2.5sqrt(3))--(10,0),MidArrow);
draw((7.5,2.5sqrt(3))--(2.5,2.5sqrt(3)),MidArrow);
draw((7.5,2.5sqrt(3))--(5,0),MidArrow);
draw((2.5,2.5sqrt(3))--(5,0),MidArrow);
label("D",(0,0),SW);
label("C",(5,0),S);
label("N",(10,0),SE);
label("A",(2.5,2.5sqrt(3)),W);
label("B",(7.5,2.5sqrt(3)),E);
label("M",(5,5sqrt(3)),N);[/asy]
Using only the paths and the directions shown, how many different routes are there from $ M$ to $ N$?
\[ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 5 \qquad
\textbf{(E)}\ 6
\]
The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller?
$\textbf{(A) }\dfrac54\qquad\textbf{(B) }\dfrac32\qquad\textbf{(C) }\dfrac95\qquad\textbf{(D) }2\qquad\textbf{(E) }\dfrac52$
The figure shows an arbitrary (green) triangle in the center. White squares were built on its sides to the outside. Some of their vertices were connected by segments, white squares were built on them again to the outside, and so on. In the spaces between the squares, triangles and quadrilaterals were formed, which were painted in different colors. Prove that
[list=a]
[*]all colored quadrilaterals are trapezoids;
[*]the areas of all polygons of the same color are equal;
[*]the ratios of the bases of one-color trapezoids are equal;
[*]if $S_0=1$ is the area of the original triangle, and $S_i$ is the area of the colored polygons at the $i^{\text{th}}$ step, then $S_1=1$, $S_2=5$ and for $n\geqslant 3$ the equality $S_n=5S_{n-1}-S_{n-2}$ is satisfied.
[/list]
[i]Proposed by F. Nilov[/i]
[center][img width="40"]https://i.ibb.co/n8gt0pV/Screenshot-2023-03-09-174624.png[/img][/center]
For each integer $ n\geq 2 $, let $ S_n $ be the sum of all products $ jk $, where $ j $ and $ k $ are integers and $ 1\leq j<k\leq n $. What is the sum of the 10 least values of $ n $ such that $ S_n $ is divisible by $ 3 $?
$\textbf{(A) }196\qquad\textbf{(B) }197\qquad\textbf{(C) }198\qquad\textbf{(D) }199\qquad\textbf{(E) }200$
Let $\omega_1$ and $\omega_2$ be two circles intersecting at distinct points $A$ and $B.$ Point $X$ varies along $\omega_1,$ and point $Y$ is chosen on $\omega_2$ such that $AB$ bisects angle $\angle{XAY}.$ Prove that as $X$ varies along $\omega_1,$ the circumcenter of $\triangle{AXY}$ (if it exists) varies along a fixed line.
Find the last three digits of the sum of all the real values of $m$ such that the ellipse $x^2+xy+y^2=m$ intersects the hyperbola $xy=n$ only at its two vertices, as $n$ ranges over all non-zero integers $-81\le n \le 81$.
[i]Proposed by [b]AOPS12142015[/b][/i]
Let $K$ be the midpoint of the median $AM$ of a triangle $ABC$. Points $X, Y$ lie on $AB, AC$, respectively, such that $\angle KXM =\angle ACB$, $AX>BX$ and similarly $\angle KYM =\angle ABC$, $AY>CY$. Prove that $B, C, X, Y$ are concyclic.
[i]Proposed by Mykhailo Shtandenko[/i]
Let us define a function $f:\mathbb N\to\mathbb N_0$ by $f(1)=0$ and, for all $n\in\mathbb N$,
$$f(2n)=2f(n)+1,\qquad f(2n+1)=2f(n).$$Given a positive integer $p$, define a sequence $(u_n)$ by $u_0=p$ and $u_{k+1}=f(u_k)$ whenever $u_k\ne0$.
(a) Prove that, for each $p\in\mathbb N$, there is a unique integer $v(p)$ such that $u_{v(p)}=0$.
(b) Compute $v(1994)$. What is the smallest integer $p>0$ for which $v(p)=v(1994)$.
(c) Given an integer $N$, determine the smallest integer $p$ such that $v(p)=N$.
Let $f$, $g$, $h : [a, b] \to \mathbb{R}$, three integrable functions such that:$$\int \limits_a^b fgdx=\int \limits_a^bghdx=\int \limits_a^bhfdx=\int \limits_a^bg^2dx\int \limits_a^bh^2dx=1$$Then$$\int \limits_a^bg^2dx=\int \limits_a^bh^2dx=1$$
Finite number of dwarfs excavates ore in the mine with infinite number of levels. Each day at the same time one dwarf from each level, inhabited with exactly $n = 1, 2, 3, ...$ dwarfs, move $n$ levels down. Prove that after some moment there will be no more then one dwarf on each level.
Let $ABCD$ be a cyclic quadrilateral. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ such that $KLMN$ is a rhombus with $KL \parallel AC$ and $LM \parallel BD$. Let $\omega_A, \omega_B, \omega_C, \omega_D$ be the incircles of $\triangle ANK, \triangle BKL, \triangle CLM, \triangle DMN$.
Prove that the common internal tangents to $\omega_A$, and $\omega_C$ and the common internal tangents to $\omega_B$ and $\omega_D$ are concurrent.
The Euclidean Algorithm on inputs $a$ and $b$ is a way to find the greatest common divisor $\gcd(a,b)$. Suppose WLOG that $a>b$. On each step of the Euclidan Algorithm, we solve the equation $a=bq+r$ for integers $q,r$ such that $0\leq r<b$, and repeat on $b$ and $r$. Thus $\gcd(a,b)=\gcd(b,r)$, and we repeat. If $r=0$, we are done. For example, $\gcd(100,15)=\gcd(15,10)=\gcd(10,5)=5$, because $100=15\cdot6+10$, $15=10\cdot1+5$, and $10=5\cdot2+0$. Thus, the Euclidean Algorithm here takes $3$ steps. What is the largest number of steps that the Euclidean Algorithm can take on some integer inputs $a,b$ where $0<a,b<10^{2016}$?
Let $C$ be the actual answer and $A$ be the answer you submit. If $\tfrac{|A-C|}{C}>\tfrac{1}{2}$, then your score will be $0$. Otherwise, your score will be given by $\max\{0,\lceil25-2(\tfrac{|A-C|}{20})^{1/2.2}\rceil\}$.
Solve the following system of equations:
$$xy^2 = 108$$
$$\frac{x^3}{y}= 10^{10}$$
For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A [i]tromino[/i] is an $L$-shape formed by three connected unit squares.
$(a)$ For which values of $n$ is it possible to cover all the black squares with non-overlapping trominoes lying entirely on the chessboard?
$(b)$ When it is possible, find the minimum number of trominoes needed.
In a lottery there are $14$ balls, numbered from $1$ to $14$. Four of these balls are drawn at random. D'Angelo wins the lottery if he can split the four balls into two disjoint pairs, where the two balls in each pair have difference at least $5$. The probability that D'Angelo wins the lottery can be expressed as $\frac{m}{n}$, with $m,n$ relatively prime. Find $m+n$.
[i]Proposed by Richard Chen[/i]