Olympiad problems

2016 Purple Comet Problems, 4

Tags:

One side of a rectangle has length 18. The area plus the perimeter of the rectangle is 2016. Find the perimeter of the rectangle.

2022 Junior Balkan Team Selection Tests - Romania, P1

Tags: number theory

Let $p$ be an odd prime number. Prove that there exist nonnegative integers $x,y,z,t$ not all of which are $0$ such that $t<p$ and \[x^2+y^2+z^2=tp.\]

2016 Saint Petersburg Mathematical Olympiad, 2

Tags: algebra , inequalities , positive real

Given the positive numbers $x_1, x_2,..., x_n$, such that $x_i \le 2x_j$ with $1 \le i < j \le n$. Prove that there are positive numbers $y_1\le y_2\le...\le y_n$, such that $x_k \le y_k \le 2x_k$ for all $k=1,2,..., n$

2010 Finnish National High School Mathematics Competition, 3

Tags: polynomial , algebra

Let $P(x)$ be a polynomial with integer coefficients and roots $1997$ and $2010$. Suppose further that $|P(2005)|<10$. Determine what integer values $P(2005)$ can get.

2008 JBMO Shortlist, 5

Tags: geometry

Is it possible to cover a given square with a few congruent right-angled triangles with acute angle equal to ${{30}^{o}}$? (The triangles may not overlap and may not exceed the margins of the square.)

1998 India Regional Mathematical Olympiad, 3

Tags: inequalities

Prove that for every natural number $n > 1$ \[ \frac{1}{n+1} \left( 1 + \frac{1}{3} +\frac{1}{5} + \ldots + \frac{1}{2n-1} \right) > \frac{1}{n} \left( \frac{1}{2} + \frac{1}{4} + \ldots + \frac{1}{2n} \right) . \]

2007 Nicolae Păun, 3

Tags: limit , permutation , group theory , group of permutations , centralizer , real analysis

In the following exercise, $ C_G (e) $ denotes the centralizer of the element $ e $ in the group $ G. $ [b]a)[/b] Prove that $ \max_{\sigma\in S_n\setminus\{1\}} \left| C_{S_n} (\sigma ) \right| <\frac{n!}{2} , $ for any natural number $ n\ge 4. $ [b]b)[/b] Show that $ \lim_{n\to\infty} \left(\frac{1}{n!}\cdot\max_{\sigma\in S_n\setminus\{1\}} \left| C_{S_n} (\sigma ) \right|\right) =0. $ [i]Alexandru Cioba[/i]

2024 ELMO Shortlist, G7

Tags: geometry , rectangle

Let $ABC$ be a triangle. Construct rectangles $BA_1A_2C$, $CB_1B_2A$, and $AC_1C_2B$ outside $ABC$ such that $\angle BCA_1=\angle CAB_1=\angle ABC_1$. Let $A_1B_2$ and $A_2C_1$ intersect at $A'$ and define $B',C'$ similarly. Prove that line $AA'$ bisects $B'C'$. [i]Linus Tang[/i]

2021 AMC 12/AHSME Spring, 19

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How many solutions does the equation $\sin \left( \frac{\pi}2 \cos x\right)=\cos \left( \frac{\pi}2 \sin x\right)$ have in the closed interval $[0,\pi]$? $\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3\qquad \textbf{(E) }4$

2022 Argentina National Olympiad, 6

Tags: algebra , polynomial

For every positive integer $n$, we consider the polynomial of real coefficients, of $2n+1$ terms, $$P(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x+a_0$$ where all coefficients are real numbers satisfying $100 \le a_i \le 101$ for $0 \le i \le 2n$. Find the smallest possible value of $n$ such that the polynomial can have at least one real root.

2013 Purple Comet Problems, 17

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For positive integers $m$ and $n$, the decimal representation for the fraction $\tfrac{m}{n}$ begins $0.711$ followed by other digits. Find the least possible value for $n$.

1966 IMO Shortlist, 1

Tags: combinatorial geometry , circles , geometry

Given $n>3$ points in the plane such that no three of the points are collinear. Does there exist a circle passing through (at least) $3$ of the given points and not containing any other of the $n$ points in its interior ?

1975 AMC 12/AHSME, 13

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The equation $ x^6 \minus{} 3x^5 \minus{} 6x^3 \minus{} x \plus{} 8$ has $ \textbf{(A)}\ \text{no real roots} \qquad$ $ \textbf{(B)}\ \text{exactly two distinct negative roots} \qquad$ $ \textbf{(C)}\ \text{exactly one negative root} \qquad$ $ \textbf{(D)}\ \text{no negative roots, but at least one positive root} \qquad$ $ \textbf{(E)}\ \text{none of these}$

2000 National Olympiad First Round, 20

Tags: algebra , polynomial

For every real $x$, the polynomial $p(x)$ whose roots are all real satisfies $p(x^2-1)=p(x)p(-x)$. What can the degree of $p(x)$ be at most? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ \text{There is no upper bound for the degree of } p(x) \qquad\textbf{(E)}\ \text{None} $

Kyiv City MO Juniors 2003+ geometry, 2006.8.3

Tags: geometry , right triangle , equal segments , perpendicular , llllllllllllllllllllllllllllll

On the legs $AC, BC$ of a right triangle $\vartriangle ABC$ select points $M$ and $N$, respectively, so that $\angle MBC = \angle NAC$. The perpendiculars from points $M$ and $C$ on the line $AN$ intersect $AB$ at points $K$ and $L$, respectively. Prove that $KL=LB$. (O. Clurman)

2016 Postal Coaching, 1

Tags: number theory

Let $n$ be an odd positive integer such that $\varphi (n)$ and $\varphi (n+1)$ are both powers of $2$ (here $\varphi(n)$ denotes Euler’s totient function). Prove that $n+1$ is a power of $2$ or $n = 5$.

2022 Canadian Mathematical Olympiad Qualification, 7

Tags: geometry

Let $ABC$ be a triangle with $|AB| < |AC|$, where $| · |$ denotes length. Suppose $D, E, F$ are points on side $BC$ such that $D$ is the foot of the perpendicular on $BC$ from $A$, $AE$ is the angle bisector of $\angle BAC$, and $F$ is the midpoint of $BC$. Further suppose that $\angle BAD = \angle DAE = \angle EAF = \angle FAC$. Determine all possible values of $\angle ABC$.

2021 EGMO, 5

Tags: triangle , geometry , combinatorial geometry

A plane has a special point $O$ called the origin. Let $P$ be a set of 2021 points in the plane such that [list] [*] no three points in $P$ lie on a line and [*] no two points in $P$ lie on a line through the origin. [/list] A triangle with vertices in $P$ is [i]fat[/i] if $O$ is strictly inside the triangle. Find the maximum number of fat triangles.

2023 CMIMC Team, 15

Tags: team

Equilateral triangle $T_0$ with side length $3$ is on a plane. Given triangle $T_n$ on the plane, triangle $T_{n+1}$ is constructed on the plane by translating $T_n$ by $1$ unit, in one of six directions parallel to one of the sides of $T_n$. The direction is chosen uniformly at random. Let $a$ be the least integer such that at most one point on the plane is in or on all of $T_0, T_1, T_2, \ldots, T_a$. It can be shown that $a$ exists with probability $1$. Find the probability that $a$ is even. [i]Proposed by Justin Hseih[/i]

2009 Stanford Mathematics Tournament, 10

Tags: algebra

Evaluate $\sum_{n=2009}^{\infty} \frac{ {n \choose 2009}}{2^n}$

2021 Indonesia TST, G

Tags: geometry , geometric transformation

Let $P$ be a point in the plane of $\triangle ABC$, and $\gamma$ a line passing through $P$. Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.

2016 CCA Math Bonanza, T9

Tags: geometry , angle bisector

Let ABC be a triangle with $AB = 8$, $BC = 9$, and $CA = 10$. The line tangent to the circumcircle of $ABC$ at $A$ intersects the line $BC$ at $T$, and the circle centered at $T$ passing through $A$ intersects the line $AC$ for a second time at $S$. If the angle bisector of $\angle SBA$ intersects $SA$ at $P$, compute the length of segment $SP$. [i]2016 CCA Math Bonanza Team #9[/i]

Mathley 2014-15, 3

Tags: projection , circumcircle , orthocenter , game strategy

A point $P$ is interior to the triangle $ABC$ such that $AP \perp BC$. Let $E, F$ be the projections of $CA, AB$. Suppose that the tangents at $E, F$ of the circumcircle of triangle $AEF$ meets at a point on $BC$. Prove that $P$ is the orthocenter of triangle $ABC$. Do Thanh Son, High School of Natural Sciences, National University, Hanoi

VMEO II 2005, 9

Tags: chessboard , combinatorics , queen

On a board with $64$ ($8 \times 8$) squares, find a way to arrange $9$ queens and $ 1$ king so that every queen cannot capture another queen.

2016 AMC 10, 25

Tags: floor function

Let $f(x)=\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor)$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to $r$. How many distinct values does $f(x)$ assume for $x \ge 0$? $\textbf{(A)}\ 32\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 45\qquad\textbf{(D)}\ 46\qquad\textbf{(E)}\ \text{infinitely many}$