Olympiad problems

2019 LIMIT Category A, Problem 10

Tags: algebra , equation

Number of solutions of the equation $3^x+4^x=8^x$ in reals is $\textbf{(A)}~0$ $\textbf{(B)}~1$ $\textbf{(C)}~2$ $\textbf{(D)}~\infty$

2020 SIME, 5

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Let $ABCD$ be a rectangle with side lengths $\overline{AB} = \overline{CD} = 6$ and $\overline{BC} = \overline{AD} = 3$. A circle $\omega$ with center $O$ and radius $1$ is drawn inside rectangle $ABCD$ such that $\omega$ is tangent to $\overline{AB}$ and $\overline{AD}$. Suppose $X$ and $Y$ are points on $\omega$ that are not on the perimeter of $ABCD$ such that $BX$ and $DY$ are tangent to $\omega$. If the value of $XY^2$ can be expressed as a common fraction in the form $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m + n$.

2011 Germany Team Selection Test, 2

Tags: geometry , circumcircle , reflection , parallelogram

Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$ [i]Proposed by Nazar Serdyuk, Ukraine[/i]

1997 Italy TST, 1

Tags: algebra

Let $x,y,z,t$ be real numbers with $x,y,z$ not all equal such that \[x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}=t.\] Find all possible values of $ t$ such that $xyz+t=0$.

1996 Tournament Of Towns, (499) 1

Tags: 3d geometry , cube , geometry

Does there exist a cube in space such that the perpendiculars dropped from its eight vertices to a given plane are of length $0, 1, 2, 3, 4, 5, 6$ and $7$? (V Proizvolov)

2003 AMC 10, 5

Tags: vieta , quadratic , algebra , polynomial

Let $ d$ and $ e$ denote the solutions of $ 2x^2\plus{}3x\minus{}5\equal{}0$. What is the value of $ (d\minus{}1)(e\minus{}1)$? $ \textbf{(A)}\ \minus{}\frac{5}{2} \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

MOAA Team Rounds, 2023.10

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Let $S$ be the set of lattice points $(a,b)$ in the coordinate plane such that $1\le a\le 30$ and $1\le b\le 30$. What is the maximum number of lattice points in $S$ such that no four points form a square of side length 2? [i]Proposed by Harry Kim[/i]

2020 MBMT, 4

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Ken has a six sided die. He rolls the die, and if the result is not even, he rolls the die one more time. Find the probability that he ends up with an even number. [i]Proposed by Gabriel Wu[/i]

2016 NIMO Problems, 1

Tags: arithmetic sequence

Find the value of $\lfloor 1 \rfloor + \lfloor 1.7 \rfloor +\lfloor 2.4 \rfloor +\lfloor 3.1 \rfloor +\cdots+\lfloor 99 \rfloor$. [i]Proposed by Jack Cornish[/i]

2018 CMIMC Team, 3-1/3-2

Tags: team

Let $\Omega$ be a semicircle with endpoints $A$ and $B$ and diameter 3. Points $X$ and $Y$ are located on the boundary of $\Omega$ such that the distance from $X$ to $AB$ is $\frac{5}{4}$ and the distance from $Y$ to $AB$ is $\frac{1}{4}$. Compute \[(AX+BX)^2 - (AY+BY)^2.\] Let $T = TNYWR$. $T$ people each put a distinct marble into a bag; its contents are mixed randomly and one marble is distributed back to each person. Given that at least one person got their own marble back, what is the probability that everyone else also received their own marble?

2013 Moldova Team Selection Test, 4

Tags: number theory

$p$ is a 4k+3 prime. Prove that there are infinite $p$ which satisfies $p|2^ny+1$. $y$ is an random integer.

2007 China Team Selection Test, 3

Tags: function , induction , combinatorics

Let $ n$ be positive integer, $ A,B\subseteq[0,n]$ are sets of integers satisfying $ \mid A\mid \plus{} \mid B\mid\ge n \plus{} 2.$ Prove that there exist $ a\in A, b\in B$ such that $ a \plus{} b$ is a power of $ 2.$

1949-56 Chisinau City MO, 35

Tags: arithmetic sequence , algebra

The numbers $a^2, b^2, c^2$ form an arithmetic progression. Show that the numbers $\frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b}$ also form arithmetic progression.

2002 Kazakhstan National Olympiad, 1

Tags: geometry , incenter , equal segments , angle

Let $ O $ be the center of the inscribed circle of the triangle $ ABC $, tangent to the side of $ BC $. Let $ M $ be the midpoint of $ AC $, and $ P $ be the intersection point of $ MO $ and $ BC $. Prove that $ AB = BP $ if $ \angle BAC = 2 \angle ACB $.

2015 ASDAN Math Tournament, 8

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Lynnelle and Moor love toy cars, and together, they have $27$ red cars, $27$ purple cars, and $27$ green cars. The number of red cars Lynnelle has individually is the same as the number of green cars Moor has individually. In addition, Lynnelle has $17$ more cars of any color than Moor has of any color. How many purple cars does Lynnelle have?

2025 Belarusian National Olympiad, 8.3

Tags: number theory

A positive integer with three digits is written on the board. Each second the number $n$ on the board gets replaced by $n+\frac{n}{p}$, where $p$ is the largest prime divisor of $n$. Prove that either after 999 seconds or 1000 second the number on the board will be a power of two. [i]A. Voidelevich[/i]

2021 Nigerian Senior MO Round 3, 3

Tags: number theory

Find all pairs of natural numbers $(p,n)$ with $p$ prime such that $p^6+p^5+n^3+n=n^5+n^2$

2015 Online Math Open Problems, 2

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At a national math contest, students are being housed in single rooms and double rooms; it is known that $75\%$ of the students are housed in double rooms. What percentage of the rooms occupied are double rooms? [i]Proposed by Evan Chen[/i]

2005 Olympic Revenge, 6

Tags: combinatorics

Zé Roberto and Humberto are playing the Millenium Game! There are 30 empty boxes in a queue, and each box have a capacity of one blue stome. Each player takes a blue stone and places it in a box (and it is a [i]move[/i]). The winner is who, in its move, obtain three full consecutive boxes. If Zé Roberto is the first player, who has the winner strategy?

2016 Polish MO Finals, 3

Tags: combinatorics , counting

Let $a, \ b \in \mathbb{Z_{+}}$. Denote $f(a, b)$ the number sequences $s_1, \ s_2, \ ..., \ s_a$, $s_i \in \mathbb{Z}$ such that $|s_1|+|s_2|+...+|s_a| \le b$. Show that $f(a, b)=f(b, a)$.

2014 Harvard-MIT Mathematics Tournament, 4

Tags: probability , expected value

[4] Let $D$ be the set of divisors of $100$. Let $Z$ be the set of integers between $1$ and $100$, inclusive. Mark chooses an element $d$ of $D$ and an element $z$ of $Z$ uniformly at random. What is the probability that $d$ divides $z$?

1998 Dutch Mathematical Olympiad, 5

Tags: function

Find all real solutions of the following equation: \[ (x + 1995)(x + 1997)(x + 1999)(x + 2001) + 16 = 0. \]

2010 Singapore Senior Math Olympiad, 1

Tags: geometry

In the $\triangle ABC$ with $AC>BC$ and $\angle B<90^{\circ}$, $D$ is the foot of the perpendicular from $A$ onto $BC$ and $E$ is the foot of perpendicular from $D$ onto $AC$. Let $F$ be the point on the segment $DE$ such that $EF \cdot DC=BD \cdot DE$. Prove that $AF$ is perpendicular to $BF$.

1995 Tournament Of Towns, (454) 3

Tags: perpendicular , geometry

Triangle $ABC$ is inscribed in a circle with center $O$. Let $q$ be the circle passing through $A$, $O$ and $B$. The lines $CA$ and $CB$ intersect $q$ at the points $D$ and $E$ (different from $A$ and $B$). Prove that the lines $CO$ and $DE$ are perpendicular to each other. (S Markelov)

1972 All Soviet Union Mathematical Olympiad, 166

Tags: combinatorial geometry , geometry , square

Each of the $9$ straight lines divides the given square onto two quadrangles with the areas ratio as $2:3$. Prove that there exist three of them intersecting in one point