Olympiad problems

2019 Regional Olympiad of Mexico Southeast, 3

Tags: probability , combinatorics

Eight teams are competing in a tournament all against all (every pair of team play exactly one time among them). There are not ties and both results of every game are equally probable. What is the probability that in the tournament every team had lose at least one game and won at least one game?

2012 Hitotsubashi University Entrance Examination, 5

Tags: probability , expected value , probability and stats

At first a fair dice is placed in such way the spot $1$ is shown on the top face. After that, select a face from the four sides at random, the process that the side would be the top face is repeated $n$ times. Note the sum of the spots of opposite face is 7. (1) Find the probability such that the spot $1$ would appear on the top face after the $n$-repetition. (2) Find the expected value of the number of the spot on the top face after the $n$-repetition.

IV Soros Olympiad 1997 - 98 (Russia), 11.3

Tags: trigonometry , analytic geometry

Draw on the coordinate plane the set of points whose coordinates satisfy the equation $$\sin x \cos^2 y +\sin y \cos^2 x =0$$

2004 Bulgaria Team Selection Test, 3

Tags: induction , linear algebra , matrix , combinatorics

In any cell of an $n \times n$ table a number is written such that all the rows are distinct. Prove that we can remove a column such that the rows in the new table are still distinct.

2011 Hanoi Open Mathematics Competitions, 4

Tags: algebra , inequalities

Among the five statements on real numbers below, how many of them are correct? "If $a < b < 0$ then $a < b^2$" , "If $0 < a < b$ then $a < b^2$", "If $a^3 < b^3$ then $a < b$", "If $a^2 < b^2$ then $a < b$", "If $|a| < |b|$ then $a < b$", (A) $0$, (B) $1$, (C) $2$, (D) $3$, (E) $4$

2010 IMC, 1

Tags: integration , real analysis

Let $0 < a < b$. Prove that $\int_a^b (x^2+1)e^{-x^2} dx \geq e^{-a^2} - e^{-b^2}$.

2009 German National Olympiad, 2

Tags: number theory

Find all positive interger $ n$ so that $ n^3\minus{}5n^2\plus{}9n\minus{}6$ is perfect square number.

2015 AMC 10, 16

Tags: vieta , algebra , polynomial , symmetry , rational root theorem , system of equations

If $y+4 = (x-2)^2, x+4 = (y-2)^2$, and $x \neq y$, what is the value of $x^2+y^2$? $ \textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }\text{30} $

2013 NIMO Problems, 3

Tags: college , search

At Stanford in 1988, human calculator Shakuntala Devi was asked to compute $m = \sqrt[3]{61{,}629{,}875}$ and $n = \sqrt[7]{170{,}859{,}375}$. Given that $m$ and $n$ are both integers, compute $100m+n$. [i]Proposed by Evan Chen[/i]

2000 National Olympiad First Round, 25

Tags: geometry

The area of a convex quadrilateral $ABCD$ is $18$. If $|AB|+|BD|+|DC|=12$, then what is $|AC|$? $ \textbf{(A)}\ 9 \qquad\textbf{(B)}\ 6\sqrt 3 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 6\sqrt 2 $

2019 Ukraine Team Selection Test, 3

Tags: geometry , circumcircle , orthocenter , altitude , tangent circle

Given an acute triangle $ABC$ . It's altitudes $AA_1 , BB_1$ and $CC_1$ intersect at a point $H$ , the orthocenter of $\vartriangle ABC$. Let the lines $B_1C_1$ and $AA_1$ intersect at a point $K$, point $M$ be the midpoint of the segment $AH$. Prove that the circumscribed circle of $\vartriangle MKB_1$ touches the circumscribed circle of $\vartriangle ABC$ if and only if $BA1 = 3A1C$. (Bondarenko Mykhailo)

2021 Science ON all problems, 4

Tags: equality case , set , combinatorics

Take $k\in \mathbb{Z}_{\ge 1}$ and the sets $A_1,A_2,\dots, A_k$ consisting of $x_1,x_2,\dots ,x_k$ positive integers, respectively. For any two sets $A$ and $B$, define $A+B=\{a+b~|~a\in A,~b\in B\}$. Find the least and greatest number of elements the set $A_1+A_2+\dots +A_k$ may have. [i] (Andrei Bâra)[/i]

OIFMAT II 2012, 2

Tags: number theory , functional , functional equation in n

Find all functions $ f: N \rightarrow N $ such that: $\bullet$ $ f (m) = 1 \iff m = 1 $; $\bullet$ If $ d = \gcd (m, n) $, then $ f (mn) = \frac {f (m) f (n)} {f (d)} $; and $\bullet$ $ \forall m \in N $, we have $ f ^ {2012} (m) = m $. Clarification: $f^n (a) = f (f^{n-1} (a))$

2011 ELMO Shortlist, 4

Tags: modular arithmetic , ellipse , geometric transformation , reflection , geometry , conic

Prove that for any convex pentagon $A_1A_2A_3A_4A_5$, there exists a unique pair of points $\{P,Q\}$ (possibly with $P=Q$) such that $\measuredangle{PA_i A_{i-1}} = \measuredangle{A_{i+1}A_iQ}$ for $1\le i\le 5$, where indices are taken $\pmod5$ and angles are directed $\pmod\pi$. [i]Calvin Deng.[/i]

2016 China Northern MO, 7

Tags: number theory

Define sequence $(a_n):a_n=2^n+3^n+6^n+1(n\in\mathbb{Z}_+)$. Are there intenger $k\geq2$, satisfying that $\gcd(k,a_i)=1$ for all $k\in\mathbb{Z}_+$? If yes, find the smallest $k$. If not, prove this.

2005 Morocco National Olympiad, 2

Tags: quadratic , induction , number theory

Find all the positive integers $x,y,z$ satisfiing : $x^{2}+y^{2}+z^{2}=2xyz$

2020 Thailand Mathematical Olympiad, 4

Tags: geometry

Let $\triangle ABC$ be a triangle with altitudes $AD,BE,CF$. Let the lines $AD$ and $EF$ meet at $P$, let the tangent to the circumcircle of $\triangle ADC$ at $D$ meet the line $AB$ at $X$, and let the tangent to the circumcircle of $\triangle ADB$ at $D$ meet the line $AC$ at $Y$. Prove that the line $XY$ passes through the midpoint of $DP$.

2014 Stars Of Mathematics, 1

Tags: number theory , diophantine equation

Prove that for any integer $n>1$ there exist infinitely many pairs $(x,y)$ of integers $1<x<y$, such that $x^n+y \mid x+y^n$. ([i]Dan Schwarz[/i])

2019 Ramnicean Hope, 2

Tags: complex number , algebra

Let be three complex numbers $ a,b,c $ such that $ |a|=|b|=|c|=1=a^2+b^2+c^2. $ Calculate $ \left| a^{2019} +b^{2019} +c^{2019} \right| . $ [i]Costică Ambrinoc[/i]

2016 India Regional Mathematical Olympiad, 2

Tags: maximum , algebra , combinatorics

On a stormy night ten guests came to dinner party and left their shoes outside the room in order to keep the carpet clean. After the dinner there was a blackout, and the gusts leaving one by one, put on at random, any pair of shoes big enough for their feet. (Each pair of shoes stays together). Any guest who could not find a pair big enough spent the night there. What is the largest number of guests who might have had to spend the night there?

2020 Kazakhstan National Olympiad, 2

Tags: algebra , functional equation

Find all functions $ f: \mathbb {R} ^ + \to \mathbb {R} ^ + $ such that for any $ x, y \in \mathbb {R} ^ + $ the following equality holds: \[f (x) f (y) = f \left (\frac {xy} {x f (x) + y} \right). \] $ \mathbb {R} ^ + $ denotes the set of positive real numbers.

2012-2013 SDML (Middle School), 15

Tags: geometry

Pentagon $ABCDE$ is inscribed in a circle such that $ACDE$ is a square with area $12$. What is the largest possible area of pentagon $ABCDE$? $\text{(A) }9+3\sqrt{2}\qquad\text{(B) }13\qquad\text{(C) }12+\sqrt{2}\qquad\text{(D) }14\qquad\text{(E) }12+\sqrt{6}-\sqrt{3}$

2024 USAMO, 3

Tags: geometry

Let $m$ be a positive integer. A triangulation of a polygon is [i]$m$-balanced[/i] if its triangles can be colored with $m$ colors in such a way that the sum of the areas of all triangles of the same color is the same for each of the $m$ colors. Find all positive integers $n$ for which there exists an $m$-balanced triangulation of a regular $n$-gon. [i]Note[/i]: A triangulation of a convex polygon $\mathcal{P}$ with $n \ge 3$ sides is any partitioning of $\mathcal{P}$ into $n-2$ triangles by $n-3$ diagonals of $\mathcal{P}$ that do not intersect in the polygon's interior. [i]Proposed by Krit Boonsiriseth[/i]

2010 Contests, 2

Tags: trigonometry , limit , real analysis

Calculate the sum of the series $\sum_{-\infty}^{\infty}\frac{\sin^33^k}{3^k}$.

2007 Estonia Team Selection Test, 5

Tags: functional , continuous , functional equation , algebra

Find all continuous functions $f: R \to R$ such that for all reals $x$ and $y$, $f(x+f(y)) = y+f(x+1)$.