Olympiad problems

2003 JHMMC 8, 15

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Evaluate $\frac{100-99+98-97\cdots +4-3+2-1}{1-2+3-4\cdots +97-98+99-100}$.

1998 South africa National Olympiad, 6

Tags: geometry , combinatorics

You are given $n$ squares, not necessarily all of the same size, which have total area 1. Is it always possible to place them without overlapping in a square of area 2?

Ukrainian TYM Qualifying - geometry, 2011.2

Tags: right triangle , equal circles , geometry

Eight circles of radius $r$ located in a right triangle $ABC$ (angle $C$ is right) as shown in figure (each of the circles touches the respactive sides of the triangle and the other circles). Find the radius of the inscribed circle of triangle $ABC$. [img]https://cdn.artofproblemsolving.com/attachments/4/7/1b1cd7d6bc7f5004b8e94468d723ed16e9a039.png[/img]

2011 F = Ma, 5

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A crude approximation is that the Earth travels in a circular orbit about the Sun at constant speed, at a distance of $\text{150,000,000 km}$ from the Sun. Which of the following is the closest for the acceleration of the Earth in this orbit? (A) $\text{exactly 0 m/s}^2$ (B) $\text{0.006 m/s}^2$ (C) $\text{0.6 m/s}^2$ (D) $\text{6 m/s}^2$ (E) $\text{10 m/s}^2$

2014 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: algebra , inequality , inequalities

Let $a$, $b$ and $c$ be positive real numbers such that $ab+bc+ca=1$. Prove the inequality: $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq 3(a+b+c)$$

2018 Dutch IMO TST, 3

Tags: geometry , right angle , equal segments

Let $ABC$ be an acute triangle, and let $D$ be the foot of the altitude through $A$. On $AD$, there are distinct points $E$ and $F$ such that $|AE| = |BE|$ and $|AF| =|CF|$. A point$ T \ne D$ satises $\angle BTE = \angle CTF = 90^o$. Show that $|TA|^2 =|TB| \cdot |TC|$.

2024 MMATHS, 1

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On a planet, far, far away, the Yaliens have defined: $x$ "equals" $y$ if and only if $|x-y| \le 3.$ Let $S$ be a set of positive integers. What is the smallest possible number of elements in $S$ such that, for any positive integer $r,$ where $1 \le r \le 2024,$ $r$ "equals" some element in $S$?

Russian TST 2019, P1

Tags: geometry

Point $M{}$ is the middle of the side side $AB$ of the isosceles triangle $ABC$. On the extension of the base $AC$, point $D{}$ is marked such that $C{}$ is between $A{}$ and $D{}$, and point $E{}$ is marked on the segment $BM$. The circumcircle of the triangle $CDE$ intersects the segment $ME$ a second time at point $F$. Prove that it is possible to make a triangle from the segments $AD, DE$ and $AF$.

2010 National Olympiad First Round, 20

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Starting from $0$, at each step we take $1$ more or $2$ times of the previous number. Which one below can be get in a less number of steps? $ \textbf{(A)}\ 2011 \qquad\textbf{(B)}\ 2010 \qquad\textbf{(C)}\ 2009 \qquad\textbf{(D)}\ 2008 \qquad\textbf{(E)}\ 2007 $

2007 Estonia Team Selection Test, 3

Tags: number theory , power of prime , prime

Let $n$ be a natural number, $n > 2$. Prove that if $\frac{b^n-1}{b-1}$ is a prime power for some positive integer $b$ then $n$ is prime.

1995 Irish Math Olympiad, 1

Tags: combinatorics

There are $ n^2$ students in a class. Each week all the students participate in a table quiz. Their teacher arranges them into $ n$ teams of $ n$ players each. For as many weeks as possible, this arrangement is done in such a way that any pair of students who were members of the same team one week are not in the same team in subsequent weeks. Prove that after at most $ n\plus{}2$ weeks, it is necessary for some pair of students to have been members of the same team in at least two different weeks.

2019 Estonia Team Selection Test, 6

Tags: lattice , combinatorics , geometry , rhombus

It is allowed to perform the following transformations in the plane with any integers $a$: (1) Transform every point $(x, y)$ to the corresponding point $(x + ay, y)$, (2) Transform every point $(x, y)$ to the corresponding point $(x, y + ax)$. Does there exist a non-square rhombus whose all vertices have integer coordinates and which can be transformed to: a) Vertices of a square, b) Vertices of a rectangle with unequal side lengths?

1999 China Team Selection Test, 2

Tags: prime number , number theory

Find all prime numbers $p$ which satisfy the following condition: For any prime $q < p$, if $p = kq + r, 0 \leq r < q$, there does not exist an integer $q > 1$ such that $a^{2} \mid r$.

2001 Belarusian National Olympiad, 1

Tags: analytic geometry , geometry

On the Cartesian coordinate plane, the graph of the parabola $y = x^2$ is drawn. Three distinct points $A$, $B$, and $C$ are marked on the graph with $A$ lying between $B$ and $C$. Point $N$ is marked on $BC$ so that $AN$ is parallel to the y-axis. Let $K_1$ and $K_2$ are the areas of triangles $ABN$ and $ACN$, respectively. Express $AN$ in terms of $K_1$ and $K_2$.

1978 Czech and Slovak Olympiad III A, 2

Tags: inequalities , algebra , parameterization

Determine (at least one) pair of real numbers $k,q$ such that the inequality \[2\left|\sqrt{1-x^2}-kx-q\right|\le\sqrt2-1\] holds for all $x\in[0,1].$

1956 Putnam, A5

Tags: binomial coefficient , combinatorics

Call a subset of $\{1,2,\ldots, n\}$ [i]unfriendly[/i] if no two of its elements are consecutive. Show that the number of unfriendly subsets with $k$ elements is $\binom{n-k+1}{k}.$

2013 AMC 10, 18

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The number $2013$ has the property that its units digit is the sum of its other digits, that is $2+0+1=3$. How many integers less than $2013$ but greater than $1000$ share this property? ${ \textbf{(A)}\ 33 \qquad\textbf{(B)}\ 34 \qquad\textbf{(C)}\ 45 \qquad\textbf{(D)}}\ 46\qquad\textbf{(E)}\ 58 $

2012 India IMO Training Camp, 2

Tags: number theory

Find the least positive integer that cannot be represented as $\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, d$.

2023 LMT Spring, 8

Tags: algebra

Let $x, y$, and $z$ be positive reals that satisfy the system $$\begin{cases} x^2 + x y + y^2 = 10 \\ x^2 + xz + z^2 = 20 \\ y^2 + yz + z^2 = 30\end{cases}$$ Find $x y + yz + xz$.

2017 Harvard-MIT Mathematics Tournament, 9

Tags: geometry

Let $ABC$ be a triangle, and let $BCDE$, $CAFG$, $ABHI$ be squares that do not overlap the triangle with centers $X$, $Y$, $Z$ respectively. Given that $AX=6$, $BY=7$, and $CA=8$, find the area of triangle $XYZ$.

2022 Stanford Mathematics Tournament, 7

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Let $n_0$ be the product of the first $25$ primes. Now, choose a random divisor $n_1$ of $n_0$, where a choice $n_1$ is taken with probability proportional to $\phi(n_1)$. ($\phi(m)$ is the number of integers less than $m$ which are relatively prime to $m$.) Given this $n_1$, we let $n_2$ be a random divisor of $n_1$, again chosen with probability proportional to $\phi(n_2)$. Compute the probability that $n_2\equiv0\pmod{2310}$.

2007 Tuymaada Olympiad, 1

Tags: number theory

Positive integers $ a<b$ are given. Prove that among every $ b$ consecutive positive integers there are two numbers whose product is divisible by $ ab$.

1974 Putnam, A1

Tags: relatively prime , integer

Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. What is the largest number of elements in any "conspiratorial" subset of the integers $1$ to $16$?

1997 Irish Math Olympiad, 5

Tags: inequalities , number theory

Let $ S$ be the set of odd integers greater than $ 1$. For each $ x \in S$, denote by $ \delta (x)$ the unique integer satisfying the inequality $ 2^{\delta (x)}<x<2^{\delta (x) \plus{}1}$. For $ a,b \in S$, define: $ a \ast b\equal{}2^{\delta (a)\minus{}1} (b\minus{}3)\plus{}a.$ Prove that if $ a,b,c \in S$, then: $ (a)$ $ a \ast b \in S$ and $ (b)$ $ (a \ast b)\ast c\equal{}a \ast (b \ast c)$.

1977 Swedish Mathematical Competition, 4

Tags: algebra , trigonometry

Show that if \[ \frac{\cos x}{\cos y}+\frac{\sin x}{\sin y}=-1 \] then \[ \frac{\cos^3 y}{\cos x}+\frac{\sin^3 y}{\sin x}=1 \]