Olympiad problems

1984 IMO Longlists, 60

Determine all pairs $(a, b)$ of positive real numbers with $a \neq 1$ such that \[\log_a b < \log_{a+1} (b + 1).\]

KoMaL A Problems 2019/2020, A. 759

Tags: combinatorics , permutation , expected value

We choose a random permutation of $1,2,\ldots,n$ with uniform distribution. Prove that the expected value of the length of the longest increasing subsequence in the permutation is at least $\sqrt{n}.$

2005 Miklós Schweitzer, 6

Tags: linear algebra , matrix , inequalities

$SU_2(\mathbb{C})=\left\{\begin{pmatrix} z & w \\ -\bar{w} & \bar{z} \end{pmatrix} : z,w\in\mathbb{C} , z\bar{z}+w\bar{w}=1\right\}$ A and B are 2 elements of the above matrix group and have eigenvalues $e^{i\theta_1}$ , $e^{-i\theta_1}$ and $e^{i\theta_2}$ , $e^{-i\theta_2}$respectively, where $0\leq\theta_i\leq\pi$ . Prove that if AB has eigenvalue $e^{i\theta_3}$ , then $\theta_3$ satisfies the inequality $|\theta_1-\theta_2|\leq\theta_3\leq \min\{\theta_1+\theta_2 , 2\pi-(\theta_1+\theta_2)\}$

2007 Iran Team Selection Test, 3

Tags: geometry , incenter , geometric transformation , homothety , trapezoid , analytic geometry , reflection

Let $\omega$ be incircle of $ABC$. $P$ and $Q$ are on $AB$ and $AC$, such that $PQ$ is parallel to $BC$ and is tangent to $\omega$. $AB,AC$ touch $\omega$ at $F,E$. Prove that if $M$ is midpoint of $PQ$, and $T$ is intersection point of $EF$ and $BC$, then $TM$ is tangent to $\omega$. [i]By Ali Khezeli[/i]

2008 Postal Coaching, 3

Tags: complex , inequalities , algebra

Let $a$ and $b$ be two complex numbers. Prove the inequality $$|1 + ab| + |a + b| \ge \sqrt{|a^2 - 1| \cdot |b^2 - 1|}$$

1950 AMC 12/AHSME, 1

Tags: ratio

If 64 is divided into three parts proportional to 2, 4, and 6, the smallest part is: $\textbf{(A)}\ 5\dfrac{1}{3} \qquad \textbf{(B)}\ 11 \qquad \textbf{(C)}\ 10\dfrac{2}{3} \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ \text{None of these answers}$

2020 Purple Comet Problems, 3

Tags: algebra

The mean number of days per month in $2020$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2006 China Second Round Olympiad, 9

Tags: ellipse , ratio , geometry , conic

Suppose points $F_1, F_2$ are the left and right foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{4}=1$ respectively, and point $P$ is on line $l:$, $x-\sqrt{3} y+8+2\sqrt{3}=0$. Find the value of ratio $\frac{|PF_1|}{|PF_2|}$ when $\angle F_1PF_2$ reaches its maximum value.

2018 Regional Olympiad of Mexico Southeast, 4

Tags: number theory , infinity

For every natural $n$ let $a_n=20\dots 018$ with $n$ ceros, for example, $a_1=2018, a_3=200018, a_7=2000000018$. Prove that there are infinity values of $n$ such that $2018$ divides $a_n$

2010 Tournament Of Towns, 5

Tags: invariant

$101$ numbers are written on a blackboard: $1^2, 2^2, 3^2, \cdots, 101^2$. Alex choses any two numbers and replaces them by their positive difference. He repeats this operation until one number is left on the blackboard. Determine the smallest possible value of this number.

2008 Mathcenter Contest, 8

Tags: geometry , combinatorics , combinatorial geometry

Prove that there are different points $A_0 \,\, ,A_1 \,\, , \cdots A_{2550}$ on the $XY$ plane corresponding to the following properties simultaneously. (i) Any three points are not on the same line. (ii) If $ d(A_i,A_j)$ represents the distance between $A_i\,\, , A_j $ then $$ \sum_{0 \leq i < j \leq 2550}\{d(A_i,A_j)\} < 10^{-2008}$$ Note : $ \{x \}$ represents the decimal part of x e.g. $ \{ 3.16\} = 0.16$. [i] (passer-by)[/i]

2018 Hong Kong TST, 1

Tags: algebra , number theory

Find all positive integer(s) $n$ such that $n^2+32n+8$ is a perfect square.

2008 SDMO (Middle School), 2

Tags:

Find the sum of the first $55$ terms of the sequence $$\binom{0}{0},\quad\binom{1}{0},\quad\binom{1}{1},\quad\binom{2}{0},\quad\binom{2}{1},\quad\binom{2}{2},\quad\binom{3}{0},\quad\ldots.$$ Note: For nonnegative integers $n$ and $k$ where $0\leq k\leq n$, $$\binom{n}{k}=\frac{n!}{k!\left(n-k\right)!}.$$

1966 IMO Shortlist, 12

Tags: number theory , decimal representation , equation

Find digits $x, y, z$ such that the equality \[\sqrt{\underbrace{\overline{xx\cdots x}}_{2n \text{ times}}-\underbrace{\overline{yy\cdots y}}_{n \text{ times}}}=\underbrace{\overline{zz\cdots z}}_{n \text{ times}}\] holds for at least two values of $n \in \mathbb N$, and in that case find all $n$ for which this equality is true.

2011 AMC 10, 12

Tags: algebra , system of equations

The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws did they make? $\textbf{(A)}\,13 \qquad\textbf{(B)}\,14 \qquad\textbf{(C)}\,15 \qquad\textbf{(D)}\,16 \qquad\textbf{(E)}\,17$

2017 AMC 10, 7

Tags: percent

Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip? $\textbf{(A)}\ 30 \%\qquad\textbf{(B)}\ 40 \%\qquad\textbf{(C)}\ 50 \%\qquad\textbf{(D)}\ 60 \%\qquad\textbf{(E)}\ 70 \%$

2012 Iran Team Selection Test, 2

Tags: geometry , circumcircle , incenter , trigonometry , angle bisector , power of a point , config geo

Consider $\omega$ is circumcircle of an acute triangle $ABC$. $D$ is midpoint of arc $BAC$ and $I$ is incenter of triangle $ABC$. Let $DI$ intersect $BC$ in $E$ and $\omega$ for second time in $F$. Let $P$ be a point on line $AF$ such that $PE$ is parallel to $AI$. Prove that $PE$ is bisector of angle $BPC$. [i]Proposed by Mr.Etesami[/i]

2020 Dutch IMO TST, 2

Tags: combinatorics , game , winning strategy , game strategy

Ward and Gabrielle are playing a game on a large sheet of paper. At the start of the game, there are $999$ ones on the sheet of paper. Ward and Gabrielle each take turns alternatingly, and Ward has the first turn. During their turn, a player must pick two numbers a and b on the sheet such that $gcd(a, b) = 1$, erase these numbers from the sheet, and write the number $a + b$ on the sheet. The first player who is not able to do so, loses. Determine which player can always win this game.

2012 India PRMO, 15

Tags: algebra , floor function , function

How many non-negative integral values of $x$ satisfy the equation $ \lfloor \frac{x}{5}\rfloor = \lfloor \frac{x}{7}\rfloor $

2024 HMNT, 9

Tags: guts

Compute the remainder when $$1002003004005006007008009$$ is divided by $13.$

1980 USAMO, 5

Tags: inequalities , algebra , convexity

Prove that for numbers $a,b,c$ in the interval $[0,1]$, \[\frac{a}{b+c+1}+\frac{b}{c+a+1}+\frac{c}{a+b+1}+(1-a)(1-b)(1-c) \le 1.\]

2017 CMIMC Team, 8

Tags: team

Alice and Bob have a fair coin with sides labeled $C$ and $M$, and they flip the coin repeatedly while recording the outcomes; for example, if they flip two $C$'s then an $M$, they have $CCM$ recorded. They play the following game: Alice chooses a four-character string $\mathcal A$, then Bob chooses two distinct three-character strings $\mathcal B_1$ and $\mathcal B_2$ such that neither is a substring of $\mathcal A$. Bob wins if $\mathcal A$ shows up in the running record before either $\mathcal B_1$ or $\mathcal B_2$ do, and otherwise Alice wins. Given that Alice chooses $\mathcal A = CMMC$ and Bob plays optimally, compute the probability that Bob wins.

1999 National Olympiad First Round, 3

Tags:

Four boxes with ball capacity $3, 5, 7,$ and $8$ are given. In how many ways can $19$ same balls be put into these boxes? $\textbf{(A)}\ 34 \qquad\textbf{(B)}\ 35 \qquad\textbf{(C)}\ 36 \qquad\textbf{(D)}\ 40 \qquad\textbf{(E)}\ \text{None}$

PEN K Problems, 24

Tags: functional equation , function , induction

A function $f$ is defined on the positive integers by \[\left\{\begin{array}{rcl}f(1) &=& 1, \\ f(3) &=& 3, \\ f(2n) &=& f(n), \\ f(4n+1) &=& 2f(2n+1)-f(n), \\ f(4n+3) &=& 3f(2n+1)-2f(n), \end{array}\right.\] for all positive integers $n$. Determine the number of positive integers $n$, less than or equal to 1988, for which $f(n) = n$.

2001 239 Open Mathematical Olympiad, 2

Tags: geometry , circumcircle , diagonal , equal angles

In a convex quadrangle $ ABCD $, the rays $ DA $ and $ CB $ intersect at point $ Q $, and the rays $ BA $ and $ CD $ at the point $ P $. It turned out that $ \angle AQB = \angle APD $. The bisectors of the angles $ \angle AQB $ and $ \angle APD $ intersect the sides quadrangle at points $ X $, $ Y $ and $ Z $, $ T $ respectively. Circumscribed circles of triangles $ ZQT $ and $ XPY $ intersect at $ K $ inside quadrangle. Prove that $ K $ lies on the diagonal $ AC $.