Olympiad problems

2009 Olympic Revenge, 3

Let $ABC$ to be a triangle with incenter $I$. $\omega_{A}$, $\omega_{B}$ and $\omega_{C}$ are the incircles of the triangles $BIC$, $CIA$ and $AIB$, repectively. After all, $T$ is the tangent point between $\omega_{A}$ and $BC$. Prove that the other internal common tangent to $\omega_{B}$ and $\omega_{C}$ passes through the point $T$.

2021 MIG, 19

Tags: 2021 Individual

Aprameya graphs the equation $2x = y + 4$ on the coordinate plane. It turns out that there is a unique point with a positive integer coordinate and a negative integer coordinate lying on Aprameya's graph. What is the sum of the coordinates of this point? $\textbf{(A) }{-}3\qquad\textbf{(B) }{-}1\qquad\textbf{(C) }0\qquad\textbf{(D) }1\qquad\textbf{(E) }2$

1988 AMC 8, 7

Tags:

$ 2.46\times 8.163\times (5.17+4.829) $ is closest to: $ \text{(A)}\ 100\qquad\text{(B)}\ 200\qquad\text{(C)}\ 300\qquad\text{(D)}\ 400\qquad\text{(E)}\ 500 $

2000 Tournament Of Towns, 2

Tags: geometry , isosceles , inradius

In triangle $ABC, AB = AC$. A line is drawn through $A$ parallel to $BC$. Outside triangle $ABC$, a circle is drawn tangent to this line, to the line $BC$, to $AB$ and to the incircle of $ABC$. If the radius of this circle is $1$ , determine the inradius of $ABC$. (RK Gordin)

1993 Romania Team Selection Test, 2

Tags: geometry proposed , geometry

Let $ABC$ be a triangle inscribed in the circle $\mathcal{C}(O,R)$ and circumscribed to the circle $\mathcal{C}(L,r)$. Denote $d=\dfrac{Rr}{R+r}$. Show that there exists a triangle $DEF$ such that for any interior point $M$ in $ABC$ there exists a point $X$ on the sides of $DEF$ such that $MX\le d$. [i]Dan Brânzei[/i]

1972 AMC 12/AHSME, 3

Tags: AMC

If $x=\dfrac{1-i\sqrt{3}}{2}$ where $i=\sqrt{-1}$, then $\dfrac{1}{x^2-x}$ is equal to $\textbf{(A) }-2\qquad\textbf{(B) }-1\qquad\textbf{(C) }1+i\sqrt{3}\qquad\textbf{(D) }1\qquad \textbf{(E) }2$

2021 Mexico National Olympiad, 2

Tags: geometry

Let $ABC$ be a triangle with $\angle ACB > 90^{\circ}$, and let $D$ be a point on $BC$ such that $AD$ is perpendicular to $BC$. Consider the circumference $\Gamma$ with with diameter $BC$. A line $\ell$ passes through $D$ and is tangent to $\Gamma$ at $P$, cuts $AC$ at $M$ (such that $M$ is in between $A$ and $C$), and cuts the side $AB$ at $N$. Prove that $M$ is the midpoint of $DP$ if and only if $N$ is the midpoint of $AB$.

2015 VJIMC, 4

Tags: polynomial , college contests , number theory

[b]Problem 4 [/b] Let $m$ be a positive integer and let $p$ be a prime divisor of $m$. Suppose that the complex polynomial $a_0 + a_1x + \ldots + a_nx^n$ with $n < \frac{p}{p-1}\varphi(m)$ and $a_n \neq 0$ is divisible by the cyclotomic polynomial $\phi_m(x)$. Prove that there are at least $p$ nonzero coefficients $a_i\ .$ The cyclotomic polynomial $\phi_m(x)$ is the monic polynomial whose roots are the $m$-th primitive complex roots of unity. Euler’s totient function $\varphi(m)$ denotes the number of positive integers less than or equal to $m$ which are coprime to $m$.

2006 CHKMO, 4

Tags: search , number theory solved , number theory

Show that there exist infinitely many square-free positive integers $n$ that divide $2005^n-1$.

1953 Poland - Second Round, 4

Tags: algebra , system of equations

Solve the system of equations $$ \qquad \begin{array}{c} x_1x_2 = 1\\ x_2x_3 = 2\\ x_3x_4 = 3\\ \ldots\\ x_nx_1 = n \end{array}$$

2021 AMC 12/AHSME Spring, 15

Tags: AUKAAT

A choir director must select a group of singers from among his $6$ tenors and $8$ basses. The only requirements are that the difference between the number of tenors and basses must be a multiple of $4$, and the group must have at least one singer. Let $N$ be the number of groups that can be selected. What is the remainder when $N$ is divided by $100$? $\textbf{(A)}\ 47 \qquad\textbf{(B)}\ 48 \qquad\textbf{(C)}\ 83 \qquad\textbf{(D)}\ 95 \qquad\textbf{(E)}\ 96$

2022 Harvard-MIT Mathematics Tournament, 7

Tags: algebra , functional

Find, with proof, all functions $f : R - \{0\} \to R$ such that $$f(x)^2 - f(y)f(z) = x(x + y + z)(f(x) + f(y) + f(z))$$ for all real $x, y, z$ such that $xyz = 1$.

2012 Today's Calculation Of Integral, 831

Tags: calculus , integration , limit , induction , calculus computations

Let $n$ be a positive integer. Answer the following questions. (1) Find the maximum value of $f_n(x)=x^{n}e^{-x}$ for $x\geq 0$. (2) Show that $\lim_{x\to\infty} f_n(x)=0$. (3) Let $I_n=\int_0^x f_n(t)\ dt$. Find $\lim_{x\to\infty} I_n(x)$.

2023-24 IOQM India, 21

Tags: function

For $n \in \mathbb{N}$, consider non-negative valued functions $f$ on $\{1,2, \cdots , n\}$ satisfying $f(i) \geqslant f(j)$ for $i>j$ and $\sum_{i=1}^{n} (i+ f(i))=2023.$ Choose $n$ such that $\sum_{i=1}^{n} f(i)$ is at least. How many such functions exist in that case?

2006 MOP Homework, 4

Tags: geometry , geometric inequality , excenter , right triangle

Let $ABC$ be a right triangle with$ \angle A = 90^o$. Point $D$ lies on side $BC$ such that $\angle BAD = \angle CAD$. Point $I_a$ is the excenter of the triangle opposite $A$. Prove that $\frac{AD}{DI_a } \le \sqrt{2} -1$

2013 JBMO TST - Macedonia, 5

Tags: number theory , TST

Let $ p, r $ be prime numbers, and $ q $ natural. Solve the equation $ (p+q+r)^2=2p^2+2q^2+r^2 $.

2002 AMC 10, 11

Tags:

Jamal wants to store $ 30$ computer files on floppy disks, each of which has a capacity of $ 1.44$ megabytes (MB). Three of his files require $ 0.8$ MB of memory each, $ 12$ more require $ 0.7$ MB each, and the remaining $ 15$ require $ 0.4$ MB each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 14 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 16$

2011 LMT, 19

Tags: number theory

A positive six-digit integer begins and ends in $8$, and is also the product of three consecutive even numbers. What is the sum of the three even numbers?

MMPC Part II 1958 - 95, 1977

Tags: MMPC , algebra , geometry , number theory , combinatorics

[b]p1.[/b] A teenager coining home after midnight heard the hall clock striking the hour. At some moment between $15$ and $20$ minutes later, the minute hand hid the hour hand. To the nearest second, what time was it then? [b]p2.[/b] The ratio of two positive integers $a$ and $b$ is $2/7$, and their sum is a four digit number which is a perfect cube. Find all such integer pairs. [b]p3.[/b] Given the integers $1, 2 , ..., n$ , how many distinct numbers are of the form $\sum_{k=1}^n( \pm k) $ , where the sign ($\pm$) may be chosen as desired? Express answer as a function of $n$. For example, if $n = 5$ , then we may form numbers: $ 1 + 2 + 3- 4 + 5 = 7$ $-1 + 2 - 3- 4 + 5 = -1$ $1 + 2 + 3 + 4 + 5 = 15$ , etc. [b]p4.[/b] $\overline{DE}$ is a common external tangent to two intersecting circles with centers at $O$ and $O'$. Prove that the lines $AD$ and $BE$ are perpendicular. [img]https://cdn.artofproblemsolving.com/attachments/1/f/40ffc1bdf63638cd9947319734b9600ebad961.png[/img] [b]p5.[/b] Find all polynomials $f(x)$ such that $(x-2) f(x+1) - (x+1) f(x) = 0$ for all $x$ . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

2014 Gulf Math Olympiad, 4

Tags: combinatorics , Sum , Chessboard

The numbers from $1$ to $64$ must be written on the small squares of a chessboard, with a different number in each small square. Consider the $112$ numbers you can make by adding the numbers in two small squares which have a common edge. Is it possible to write the numbers in the squares so that these $112$ sums are all different?

2023 ELMO Shortlist, C6

Tags: Elmo , combinatorics

For a set $S$ of positive integers and a positive integer $n$, consider the game of [i]$(n,S)$-nim[/i], which is as follows. A pile starts with $n$ watermelons. Two players, Deric and Erek, alternate turns eating watermelons from the pile, with Deric going first. On any turn, the number of watermelons eaten must be an element of $S$. The last player to move wins. Let $f(S)$ denote the set of positive integers $n$ for which Deric has a winning strategy in $(n,S)$-nim. Let $T$ be a set of positive integers. Must the sequence \[T, \; f(T), \; f(f(T)), \;\ldots\] be eventually constant? [i]Proposed by Brandon Wang and Edward Wan[/i]

1983 IMO Longlists, 41

Tags: geometry , combinatorics , Coloring , counting , 3D geometry , IMO Shortlist

Let $E$ be the set of $1983^3$ points of the space $\mathbb R^3$ all three of whose coordinates are integers between $0$ and $1982$ (including $0$ and $1982$). A coloring of $E$ is a map from $E$ to the set {red, blue}. How many colorings of $E$ are there satisfying the following property: The number of red vertices among the $8$ vertices of any right-angled parallelepiped is a multiple of $4$ ?

2010 JBMO Shortlist, 1

Tags: geometry , circumcircle , reflection

$\textbf{Problem G1}$ Consider a triangle $ABC$ with $\angle ACB=90^{\circ}$. Let $F$ be the foot of the altitude from $C$. Circle $\omega$ touches the line segment $FB$ at point $P$, the altitude $CF$ at point $Q$ and the circumcircle of $ABC$ at point $R$. Prove that points $A, Q, R$ are collinear and $AP = AC$.

2011 Princeton University Math Competition, A3 / B5

Tags: number theory

What is the sum of all primes $p$ such that $7^p - 6^p + 2$ is divisible by 43?

1987 IMO Shortlist, 12

Tags: geometry , circumcircle , incenter , collinearity , IMO Shortlist

Given a nonequilateral triangle $ABC$, the vertices listed counterclockwise, find the locus of the centroids of the equilateral triangles $A'B'C'$ (the vertices listed counterclockwise) for which the triples of points $A,B', C'; A',B, C';$ and $A',B', C$ are collinear. [i]Proposed by Poland.[/i]