Olympiad problems

2000 India Regional Mathematical Olympiad, 4

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All the $7$ digit numbers containing each of the digits $1,2,3,4,5,6,7$ exactly once , and not divisible by $5$ are arranged in increasing order. Find the $200th$ number in the list.

2018 Danube Mathematical Competition, 2

Prove that there are infinitely many pairs of positive integers $(m, n)$ such that simultaneously $m$ divides $n^2 + 1$ and $n$ divides $m^2 + 1$.

2018 AMC 12/AHSME, 5

Tags: algebra , polynomial

What is the sum of all possible values of $k$ for which the polynomials $x^2 - 3x + 2$ and $x^2 - 5x + k$ have a root in common? $ \textbf{(A) }3 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5 \qquad \textbf{(D) }6 \qquad \textbf{(E) }10 \qquad $

2012 IFYM, Sozopol, 6

Tags: algebra , binomial sum , sum , calculate

Calculate the sum $1+\frac{\binom{2}{1}}{8}+\frac{\binom{4}{2}}{8^2}+\frac{\binom{6}{3}}{8^3}+...+\frac{\binom{2n}{n}}{8^n}+...$

2020 Canada National Olympiad, 3

Tags: combinatorics

There are finite many coins in David’s purse. The values of these coins are pair wisely distinct positive integers. Is that possible to make such a purse, such that David has exactly $2020$ different ways to select the coins in his purse and the sum of these selected coins is $2020$?

2023 239 Open Mathematical Olympiad, 4

Tags: combinatorics , game strategy

There are a million numbered chairs at a large round table. The Sultan has seated a million wise men on them. Each of them sees the thousand people following him in clockwise order. Each of them was given a cap of black or white color, and they must simultaneously write down on their own piece of paper a guess about the color of their cap. Those who do not guess will be executed. The wise men had the opportunity to agree on a strategy before the test. What is the largest number of survivors that they can guarantee?

May Olympiad L1 - geometry, 1997.2

Tags: geometry , rectangle , area of a triangle

In the rectangle $ABCD, M, N, P$ and $Q$ are the midpoints of the sides. If the area of the shaded triangle is $1$, calculate the area of the rectangle $ABCD$. [img]https://2.bp.blogspot.com/-9iyKT7WP5fc/XNYuXirLXSI/AAAAAAAAKK4/10nQuSAYypoFBWGS0cZ5j4vn_hkYr8rcwCK4BGAYYCw/s400/may3.gif[/img]

1980 Poland - Second Round, 2

Tags: algebra , inequalities

Prove that for any real numbers $ x_1, x_2, x_3, \ldots, x_n $ the inequality is true $$ x_1x_2x_3\ldots x_n \leq \frac{x_1^2}{2} + \frac{x_2^4}{4} + \frac{x_3^8}{8} + \ldots + \frac{x_n^{2^ n}}{2^n} + \frac{1}{2^n}$$

2004 AMC 8, 2

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How many different four-digit numbers can be formed by rearranging the four digits in $2004$? $\textbf{(A)}\ 4\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 24\qquad \textbf{(E)}\ 81$

2016 AMC 8, 14

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Karl's car uses a gallon of gas every $35$ miles, and his gas tank holds $14$ gallons when it is full. One day, Karl started with a full tank of gas, drove $350$ miles, bought $8$ gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. How many miles did Karl drive that day? $\textbf{(A)}\mbox{ }525\qquad\textbf{(B)}\mbox{ }560\qquad\textbf{(C)}\mbox{ }595\qquad\textbf{(D)}\mbox{ }665\qquad\textbf{(E)}\mbox{ }735$

2016 Saudi Arabia GMO TST, 1

Tags: tangent , circumcircle , geometry

Let $ABC$ be an acute, non-isosceles triangle which is inscribed in a circle $(O)$. A point $I$ belongs to the segment $BC$. Denote by $H$ and $K$ the projections of $I$ on $AB$ and $AC$, respectively. Suppose that the line $HK$ intersects $(O)$ at $M, N$ ($H$ is between $M, K$ and $K$ is between $H, N$). Prove the following assertions: a) If $A$ is the center of the circle $(IMN)$, then $BC$ is tangent to $(IMN)$. b) If $I$ is the midpoint of $BC$, then $BC$ is equal to $4$ times of the distance between the centers of two circles $(ABK)$ and $(ACH)$.

2014 Putnam, 3

Tags: limit , induction , inequalities , algebra , difference of squares

Let $a_0=5/2$ and $a_k=a_{k-1}^2-2$ for $k\ge 1.$ Compute \[\prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right)\] in closed form.

2019 Saudi Arabia Pre-TST + Training Tests, 1.2

Tags: number theory , divide , arithmetic sequence

Determine all arithmetic sequences $a_1, a_2,...$ for which there exists integer $N > 1$ such that for any positive integer $k$ the following divisibility holds $a_1a_2 ...a_k | a_{N+1}a_{N+2}...a_{N+k}$ .

2015 Princeton University Math Competition, A5/B7

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Alice has an orange $\text{3-by-3-by-3}$ cube, which is comprised of $27$ distinguishable, $\text{1-by-1-by-1}$ cubes. Each small cube was initially orange, but Alice painted $10$ of the small cubes completely black. In how many ways could she have chosen $10$ of these smaller cubes to paint black such that every one of the $27$ $\text{3-by-1-by-1}$ sub-blocks of the $\text{3-by-3-by-3}$ cube contains at least one small black cube?

2012 Turkey MO (2nd round), 3

Tags: function , functional equation , algebra

Find all non-decreasing functions from real numbers to itself such that for all real numbers $x,y$ $f(f(x^2)+y+f(y))=x^2+2f(y)$ holds.

LMT Accuracy Rounds, 2021 F7

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Find the number of ways to tile a $12 \times 3$ board with $1 \times 4$ and $2 \times 2$ tiles with no overlap or uncovered space.

2009 Belarus Team Selection Test, 2

Tags: number theory , modular arithmetic , sequence , divisibility

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

2008 JBMO Shortlist, 6

Tags: inequalities , algebra

If the real numbers $a, b, c, d$ are such that $0 < a,b,c,d < 1$, show that $1 + ab + bc + cd + da + ac + bd > a + b + c + d$.

1996 Taiwan National Olympiad, 5

Tags: number theory , integer sequence

Dertemine integers $a_{1},a_{2},...,a_{99}=a_{0}$ satisfying $|a_{k}-a_{k-1}|\geq 1996$ for all $k=1,2,...,99$, such that $m=\max_{1\leq k\leq 99} |a_{k}-a_{k-1}|$ is minimum possible, and find the minimum value $m^{*}$ of $m$.

2010 Contests, 3

Tags: geometry

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be externally tangent at a point $A$. A line tangent to $\mathcal{C}_1$ at $B$ intersects $\mathcal{C}_2$ at $C$ and $D$; then the segment $AB$ is extended to intersect $\mathcal{C}_2$ at a point $E$. Let $F$ be the midpoint of $\overarc{CD}$ that does not contain $E$, and let $H$ be the intersection of $BF$ with $\mathcal{C}_2$. Show that $CD$, $AF$, and $EH$ are concurrent.

ICMC 4, 2

Tags: number theory , mod p

Let $p > 3$ be a prime number. A sequence of $p-1$ integers $a_1,a_2, \dots, a_{p-1}$ is called [i]wonky[/i] if they are distinct modulo $p$ and $a_ia_{i+2} \not\equiv a_{i+1}^2 \pmod p$ for all $i \in \{1, 2, \dots, p-1\}$, where $a_p = a_1$ and $a_{p+1} = a_2$. Does there always exist a wonky sequence such that $$a_1a_2, \qquad a_1a_2+a_2a_3, \qquad \dots, \qquad a_1a_2+\cdots +a_{p-1}a_1,$$ are all distinct modulo $p$? [i]Proposed by Harun Khan[/i]

2014 Contests, 3

Tags: geometry , rectangle

Let $ABCD$ be a rectangle and $P$ a point outside of it such that $\angle{BPC} = 90^{\circ}$ and the area of the pentagon $ABPCD$ is equal to $AB^{2}$. Show that $ABPCD$ can be divided in 3 pieces with straight cuts in such a way that a square can be built using those 3 pieces, without leaving any holes or placing pieces on top of each other. Note: the pieces can be rotated and flipped over.

2016 Balkan MO Shortlist, C3

Tags: combinatorics , coloring

The plane is divided into squares by two sets of parallel lines, forming an infinite grid. Each unit square is coloured with one of $1201$ colours so that no rectangle with perimeter $100$ contains two squares of the same colour. Show that no rectangle of size $1\times1201$ or $1201\times1$ contains two squares of the same colour. [i]Note: Any rectangle is assumed here to have sides contained in the lines of the grid.[/i] [i](Bulgaria - Nikolay Beluhov)[/i]

1990 Tournament Of Towns, (256) 4

Tags: combinatorics

A set of $103$ coins that look alike is given. Two coins (whose weights are equal) are counterfeit. The other $101$ (genuine) coins also have the same weight, but a different weight from that of the counterfeit coins. However it is not known whether it is the genuine coins or the counterfeit coins which are heavier. How can this question be resolved by three weighings on the one balance? (It is not required to separate the counterfeit coins from the genuine ones.) (D. Fomin, Leningrad)

2011 Irish Math Olympiad, 1

Tags: algebra

Suppose $abc\neq 0$. Express in terms of $a,b,$ and $c$ the solutions $x,y,z,u,v,w$ of the equations $$x+y=a,\quad z+u=b,\quad v+w=c,\quad ay=bz,\quad ub=cv,\quad wc=ax.\quad$$