Olympiad problems

2020 AMC 10, 2

Tags: geometry , 3d geometry

Carl has $5$ cubes each having side length $1$, and Kate has $5$ cubes each having side length $2$. What is the total volume of the $10$ cubes? $\textbf{(A) }24 \qquad \textbf{(B) }25 \qquad \textbf{(C) } 28\qquad \textbf{(D) } 40\qquad \textbf{(E) } 45$

2018 Romanian Masters in Mathematics, 3

Tags: combinatorics

Ann and Bob play a game on the edges of an infinite square grid, playing in turns. Ann plays the first move. A move consists of orienting any edge that has not yet been given an orientation. Bob wins if at any point a cycle has been created. Does Bob have a winning strategy?

2003 Tournament Of Towns, 5

Tags: geometry , rectangle

What is the largest number of squares on $9 \times 9$ square board that can be cut along their both diagonals so that the board does not fall apart into several pieces?

2005 Junior Tuymaada Olympiad, 7

Tags: midpoint , incircle , geometry

The point $ I $ is the center of the inscribed circle of the triangle $ ABC $. The points $ B_1 $ and $ C_1 $ are the midpoints of the sides $ AC $ and $ AB $, respectively. It is known that $ \angle BIC_1 + \angle CIB_1 = 180^\circ $. Prove the equality $ AB + AC = 3BC $

2001 CentroAmerican, 3

Tags: modular arithmetic

In a circumference of a circle, $ 10000$ points are marked, and they are numbered from $ 1$ to $ 10000$ in a clockwise manner. $ 5000$ segments are drawn in such a way so that the following conditions are met: 1. Each segment joins two marked points. 2. Each marked point belongs to one and only one segment. 3. Each segment intersects exactly one of the remaining segments. 4. A number is assigned to each segment that is the product of the number assigned to each end point of the segment. Let $ S$ be the sum of the products assigned to all the segments. Show that $ S$ is a multiple of $ 4$.

2013 Sharygin Geometry Olympiad, 2

Tags: geometry , circumcircle , circles

Two circles with centers $O_1$ and $O_2$ meet at points $A$ and $B$. The bisector of angle $O_1AO_2$ meets the circles for the second time at points $C $and $D$. Prove that the distances from the circumcenter of triangle $CBD$ to $O_1$ and to $O_2$ are equal.

2012 Today's Calculation Of Integral, 827

Tags: calculus , integration , limit , trigonometry , calculus computations

Find $\lim_{n\to\infty}\sum_{k=0}^{\infty} \int_{2k\pi}^{(2k+1)\pi} xe^{-x}\sin x\ dx.$

2008 AMC 8, 10

Tags:

The average age of the $6$ people in Room A is $40$. The average age of the $4$ people in Room B is $25$. If the two groups are combined, what is the average age of all the people? $\textbf{(A)}\ 32.5 \qquad \textbf{(B)}\ 33 \qquad \textbf{(C)}\ 33.5 \qquad \textbf{(D)}\ 34\qquad \textbf{(E)}\ 35$

1965 AMC 12/AHSME, 20

Tags: arithmetic sequence

For every $ n$ the sum of $ n$ terms of an arithmetic progression is $ 2n \plus{} 3n^2$. The $ r$th term is: $ \textbf{(A)}\ 3r^2 \qquad \textbf{(B)}\ 3r^2 \plus{} 2r \qquad \textbf{(C)}\ 6r \minus{} 1 \qquad \textbf{(D)}\ 5r \plus{} 5 \qquad \textbf{(E)}\ 6r \plus{} 2 \qquad$

1996 IMO Shortlist, 5

Tags: algebra , polynomial , inequalities , function , maximization

Let $ P(x)$ be the real polynomial function, $ P(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d.$ Prove that if $ |P(x)| \leq 1$ for all $ x$ such that $ |x| \leq 1,$ then \[ |a| \plus{} |b| \plus{} |c| \plus{} |d| \leq 7.\]

Geometry Mathley 2011-12, 16.2

Tags: geometry , orthocenter , collinear

Let $ABCD$ be a quadrilateral and $P$ a point in the plane of the quadrilateral. Let $M,N$ be on the sides $AC,BD$ respectively such that $PM \parallel BC, PN \parallel AD$. $AC$ meets $BD$ at $E$. Prove that the orthocenter of triangles $EBC, EAD, EMN$ are collinear if and only if $P$ is on the line $AB$. Đỗ Thanh Sơn PS. Instead of the word [b]collinear[/b], it was written [b]concurrent[/b], probably a typo.

2003 Tournament Of Towns, 1

Tags: induction , combinatorics

Two players in turns color the sides of an $n$-gon. The first player colors any side that has $0$ or $2$ common vertices with already colored sides. The second player colors any side that has exactly $1$ common vertex with already colored sides. The player who cannot move, loses. For which $n$ the second player has a winning strategy?

2015 Chile TST Ibero, 4

Tags: inequalities , algebra

Let $x, y \in \mathbb{R}^+$. Prove that: \[ \left( 1 + \frac{1}{x} \right) \left( 1 + \frac{1}{y} \right) \geq \left( 1 + \frac{2}{x + y} \right)^2. \]

2008 Argentina Iberoamerican TST, 2

Tags: number theory , prime number , relatively prime

Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.

2021 Iran RMM TST, 2

Tags: function , algebra

Let $f : \mathbb{R}^+\to\mathbb{R}$ satisfying $f(x)=f(x+2)+2f(x^2+2x)$. Prove that if for all $x>1400^{2021}$, $xf(x) \le 2021$, then $xf(x) \le 2021$ for all $x \in \mathbb {R}^+$ Proposed by [i]Navid Safaei[/i]

2015 SDMO (High School), 1

Tags:

Let $\mathcal{C}$ be a circle and let $P$ and $Q$ be points inside $\mathcal{C}$. Prove that there are infinitely many circle through $P$ and $Q$ that are completely contained inside of $\mathcal{C}$.

2010 Peru MO (ONEM), 1

Tags: combinatorics

In each of the $9$ small circles of the following figure we write positive integers less than $10$, without repetitions. In addition, it is true that the sum of the $5$ numbers located around each one of the $3$ circles is always equal to $S$. Find the largest possible value of $S$. [img]https://cdn.artofproblemsolving.com/attachments/6/6/2db2c1ac7f45022606fb0099f24e6287977d10.png[/img]

1999 Baltic Way, 10

Tags: combinatorics

May the points of a disc of radius $1$ (including its circumference) be partitioned into three subsets in such a way that no subset contains two points separated by a distance $1$?

2005 Austrian-Polish Competition, 1

Tags: polygon , enumeration , combinatorics

For a convex $n$-gon $P_n$, we say that a convex quadrangle $Q$ is a [i]diagonal-quadrangle[/i] of $P_n$, if its vertices are vertices of $P_n$ and its sides are diagonals of $P_n$. Let $d_n$ be the number of diagonal-quadrangles of a convex $n$-gon. Determine $d_n$ for all $n\geq 8$.

1962 AMC 12/AHSME, 2

Tags:

The expression $ \sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}}$ is equal to: $ \textbf{(A)}\ \frac{\sqrt{3}}{6} \qquad \textbf{(B)}\ \frac{-\sqrt{3}}{6} \qquad \textbf{(C)}\ \frac{\sqrt{-3}}{6} \qquad \textbf{(D)}\ \frac{5 \sqrt{3}}{6} \qquad \textbf{(E)}\ 1$

2018 HMNT, 4

Tags:

Let $a$ and $b$ be real numbers greater than 1 such that $ab=100$. The maximum possible value of $a^{(\log_{10}b)^2}$ can be written in the form $10^x$ for some real number $x$. Find $x$.

2012 ITAMO, 3

Tags: induction , combinatorics

Let $n$ be an integer greater than or equal to $2$. There are $n$ people in one line, each of which is either a [i]scoundrel[/i] (who always lie) or a [i]knight[/i] (who always tells the truth). Every person, except the first, indicates a person in front of him/her and says "This person is a scoundrel" or "This person is a knight." Knowing that there are strictly more scoundrel than knights, seeing the statements show that it is possible to determine each person whether he/she is a scoundrel or a knight.

2021 AMC 10 Fall, 15

Tags:

Isosceles triangle $ABC$ has $AB = AC = 3\sqrt6$, and a circle with radius $5\sqrt2$ is tangent to line $AB$ at $B$ and to line $AC$ at $C$. What is the area of the circle that passes through vertices $A$, $B$, and $C?$ $\textbf{(A) }24\pi\qquad\textbf{(B) }25\pi\qquad\textbf{(C) }26\pi\qquad\textbf{(D) }27\pi\qquad\textbf{(E) }28\pi$

1999 Italy TST, 2

Tags: inequalities , geometry

Let $D$ and $E$ be points on sides $AB$ and $AC$ respectively of a triangle $ABC$ such that $DE$ is parallel to $BC$ and tangent to the incircle of $ABC$. Prove that \[DE\le\frac{1}{8}(AB+BC+CA) \]

1967 IMO Shortlist, 2

Tags: trigonometry , inequality , trigonometric inequality , minimum

Let $n$ and $k$ be positive integers such that $1 \leq n \leq N+1$, $1 \leq k \leq N+1$. Show that: \[ \min_{n \neq k} |\sin n - \sin k| < \frac{2}{N}. \]