Olympiad problems

2017 Romania EGMO TST, P4

Tags: combinatorics

In $p{}$ of the vertices of the regular polygon $A_0A_1\ldots A_{2016}$ we write the number $1{}$ and in the remaining ones we write the number $-1.{}$ Let $x_i{}$ be the number written on the vertex $A_i{}.$ A vertex is [i]good[/i] if \[x_i+x_{i+1}+\cdots+x_j>0\quad\text{and}\quad x_i+x_{i-1}+\cdots+x_k>0,\]for any integers $j{}$ and $k{}$ such that $k\leqslant i\leqslant j.$ Note that the indices are taken modulo $2017.$ Determine the greatest possible value of $p{}$ such that, regardless of numbering, there always exists a good vertex.

1954 Putnam, A1

Tags: linear algebra , matrix , permutation

Let $n$ be an odd integer greater than $1.$ Let $A$ be an $n\times n$ symmetric matrix such that each row and column consists of some permutation of the integers $1,2, \ldots, n.$ Show that each of the integers $1,2, \ldots, n$ must appear in the main diagonal of $A$.

2015 Junior Regional Olympiad - FBH, 3

Tags: geometry

Find the area of quadrilateral $ABCD$ if: two opposite angles are right;two sides which form right angle are of equal length and sum of lengths of other two sides is $10$

1964 Spain Mathematical Olympiad, 7

Tags: combinatorics

A table with 1000 cards on a line, numbered from 1 to 1000, is considered. The cards are ordered in the usual way. Now, we proceed in the following way. The first card (which is 1) is put just before the last card (between 999 and 1000) and, after, the new first card (which is 2) is put after the last card (which was 1000). Show that after 1000 movements, the cards are ordered again in the usual way. Show that the analogous result ($n$ movements for $n$ cards) does not hold when $n$ is odd.

2011 Kyrgyzstan National Olympiad, 8

Tags: algebra

Given a sequence $x_1,x_2,...,x_n$ of real numbers with ${x_{n + 1}}^3 = {x_n}^3 - 3{x_n}^2 + 3{x_n}$, where $(n=1,2,3,...)$. What must be value of $x_1$, so that $x_{100}$ and $x_{1000}$ becomes equal?

1991 National High School Mathematics League, 3

Tags: number theory

Let $a_n$ be the number of such numbers $N$: sum of all digits of $N$ is $n$, and each digit can only be $1,3,4$. Prove that $a_{2n}$ is a perfect square for all $n\in\mathbb{Z}_+$.

1995 All-Russian Olympiad, 4

Tags: combinatorics

Can the numbers from 1 to 81 be written in a 9×9 board, so that the sum of numbers in each 3×3 square is the same? [i]S. Tokarev[/i]

PEN H Problems, 55

Tags: diophantine equation , floor function

Given that \[34! = 95232799cd96041408476186096435ab000000_{(10)},\] determine the digits $a, b, c$, and $d$.

2002 Moldova National Olympiad, 4

Tags:

All the internal phone numbers in a certain company have four digits. The director wants the phone numbers of the administration offices to consist of digits $ 1$, $ 2$, $ 3$ only, and that any of these phone numbers coincide in at most one position. What is the maximum number of distinct phone numbers that these offices can have ?

Indonesia MO Shortlist - geometry, g8

Tags: geometry , geometric inequality , angle bisector

Suppose the points $D, E, F$ lie on sides $BC, CA, AB$, respectively, so that $AD, BE, CF$ are angle bisectors. Define $P_1$, $P_2$, $P_3$ respectively as the intersection point of $AD$ with $EF$, $BE$ with $DF$, $CF$ with $DE$ respectively. Prove that $$\frac{AD}{AP_1}+\frac{BE}{BP_2}+\frac{CF}{CP_3} \ge 6$$

2011 Purple Comet Problems, 4

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Five non-overlapping equilateral triangles meet at a common vertex so that the angles between adjacent triangles are all congruent. What is the degree measure of the angle between two adjacent triangles? [asy] size(100); defaultpen(linewidth(0.7)); path equi=dir(300)--dir(240)--origin--cycle; for(int i=0;i<=4;i=i+1) draw(rotate(72*i,origin)*equi); [/asy]

1985 Putnam, B5

Tags:

Evaluate $\textstyle\int_{0}^{\infty} t^{-1 / 2} e^{-1985\left(t+t^{-1}\right)} d t.$ You may assume that $\textstyle\int_{-\infty}^{\infty} e^{-x^{2}} d x=\sqrt{\pi}.$

2019 AMC 10, 17

Tags: probability

A red ball and a green ball are randomly and independently tossed into bins numbered with positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k=1,2,3,\ldots.$ What is the probability that the red ball is tossed into a higher-numbered bin than the green ball? $\textbf{(A) } \frac{1}{4} \qquad\textbf{(B) } \frac{2}{7} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{3}{8} \qquad\textbf{(E) } \frac{3}{7}$

2018 MIG, 4

Tags:

In regular hexagon $ABCDEF$, lines $AC$ and $BE$ are drawn, and their intersection is labeled $G$. What fraction of the area of $ABCDEF$ is contained in triangle $ABG$? [center][img]https://cdn.artofproblemsolving.com/attachments/1/a/47d6226b18cb2b7c941287b1628b56423909b8.png[/img][/center]

2015 India IMO Training Camp, 3

Tags: floor function , sequence , parity , number theory

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

JOM 2015 Shortlist, A7

Tags: inequalities

Given positive reals $ a, b, c $ that satisfy $ a + b + c = 1 $, show that $$ \displaystyle \sum^{}_{cyc}\frac{a^3+bc}{a^2+bc}\ge 2 $$

1993 Dutch Mathematical Olympiad, 3

Tags: limit , algebra , polynomial

A sequence of numbers is defined by $ u_1\equal{}a, u_2\equal{}b$ and $ u_{n\plus{}1}\equal{}\frac{u_n\plus{}u_{n\minus{}1}}{2}$ for $ n \ge 2$. Prove that $ \displaystyle\lim_{n\to\infty}u_n$ exists and express its value in terms of $ a$ and $ b$.

2003 AMC 12-AHSME, 21

Tags: probability , geometry , trigonometry

An object moves $ 8$ cm in a straight line from $ A$ to $ B$, turns at an angle $ \alpha$, measured in radians and chosen at random from the interval $ (0,\pi)$, and moves $ 5$ cm in a straight line to $ C$. What is the probability that $ AC<7$? $ \textbf{(A)}\ \frac{1}{6} \qquad \textbf{(B)}\ \frac{1}{5} \qquad \textbf{(C)}\ \frac{1}{4} \qquad \textbf{(D)}\ \frac{1}{3} \qquad \textbf{(E)}\ \frac{1}{2}$

2018 IOM, 1

Tags: algebra

Solve the system of equations in real numbers: \[ \begin{cases*} (x - 1)(y - 1)(z - 1) = xyz - 1,\\ (x - 2)(y - 2)(z - 2) = xyz - 2.\\ \end{cases*} \] [i]Vladimir Bragin[/i]

2018 AMC 12/AHSME, 2

Tags:

While exploring a cave, Carl comes across a collection of $5$-pound rocks worth $\$14$ each, $4$-pound rocks worth $\$11$ each, and $1$-pound rocks worth $\$2$ each. There are at least $20$ of each size. He can carry at most $18$ pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave? $\textbf{(A) } 48 \qquad \textbf{(B) } 49 \qquad \textbf{(C) } 50 \qquad \textbf{(D) } 51 \qquad \textbf{(E) } 52 $

1998 AMC 8, 10

Tags:

Each of the letters $W$, $X$, $Y$,and $Z$ represents a different integer in the set $ \{ 1,2,3,4\} $, but not necessarily in that order. If , $ \frac{\text{W}}{\text{X}}-\frac{\text{Y}}{\text{Z}}=1 $ then the sum of $W$ and $Y$ is $ \text{(A)}\ 3\qquad\text{(B)}\ 4\qquad\text{(C)}\ 5\qquad\text{(D)}\ 6\qquad\text{(E)}\ 7 $

Oliforum Contest V 2017, 1

Tags: number theory , digit , multiple

We know that there exists a positive integer with $7$ distinct digits which is multiple of each of them. What are its digits? (Paolo Leonetti)

2002 Moldova National Olympiad, 1

Tags:

We are given three nuggets of weights $ 1$ kg, $ 2$ kg and $ 3$ kg, containing different percentages of gold, and need to cut each nugget into two parts so that the obtained parts can be alloyed into two ingots of weights $ 1$ kg ande $ 5$ kg containing the same proportion of gold. How we can do that?

2007 Miklós Schweitzer, 5

Tags: real analysis , function

Let $D=\{ (x,y) \mid x>0, y\neq 0\}$ and let $u\in C^1(\overline {D})$ be a bounded function that is harmonic on $D$ and for which $u=0$ on the $y$-axis. Prove that $u$ is identically zero. (translated by Miklós Maróti)

1992 IMO Shortlist, 4

Tags: combinatorics , ramsey theory , graph theory , coloring , extremal combinatorics

Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $\,n\,$ such that whenever exactly $\,n\,$ edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.