Olympiad problems

2023 Romania Team Selection Test, P1

Tags: geometry

Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$. Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$

2022 Chile TST IMO, 3

Tags: order , number theory , primitive root

Let $n$ be a natural number with more than $2021$ digits, none of which are $8$ or $9$. Suppose that $n$ has no common factors with $2021$. Prove that it is possible to increase one of the digits of $n$ by at most $2$ so that the resulting number is a multiple of 2021.

2007 Romania Team Selection Test, 2

Tags: inequalities , function , algebra

Let $f: \mathbb{Q}\rightarrow \mathbb{R}$ be a function such that \[|f(x)-f(y)|\leq (x-y)^{2}\] for all $x,y \in\mathbb{Q}$. Prove that $f$ is constant.

2004 Gheorghe Vranceanu, 1

Tags: limit , real analysis , sequence

Let be the sequence $ \left( x_n \right)_{n\ge 1} $ defined as $$ x_n= \frac{4009}{4018020} x_{n-1} -\frac{1}{4018020} x_{n-2} + \left( 1+\frac{1}{n} \right)^n. $$ Prove that $ \left( x_n \right)_{n\ge 1} $ is convergent and determine its limit.

1999 AMC 8, 7

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The third exit on a highway is located at milepost 40 and the tenth exit is at milepost 160. There is a service center on the highway located three-fourths of the way from the third exit to the tenth exit. At what milepost would you expect to find this service center? $ \text{(A)}\ 90\qquad\text{(B)}\ 100\qquad\text{(C)}\ 110\qquad\text{(D)}\ 120\qquad\text{(E)}\ 130 $

2023 Indonesia TST, 1

Tags: combinatorics , integer sequence

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

Kvant 2020, M2602

Tags: geometry , combinatorial geometry

For a given natural number $k{}$, a convex polygon is called $k{}$[i]-triangular[/i] if it is the intersection of some $k{}$ triangles. [list=a] [*]What is the largest $n{}$ for which there exist a $k{}$-triangular $n{}$-gon? [*]What is the largest $n{}$ for which any convex $n{}$-gon is $k{}$-triangular? [/list] [i]Proposed by P. Kozhevnikov[/i]

1993 AMC 8, 7

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$3^3+3^3+3^3 = $ $\text{(A)}\ 3^4 \qquad \text{(B)}\ 9^3 \qquad \text{(C)}\ 3^9 \qquad \text{(D)}\ 27^3 \qquad \text{(E)}\ 3^{27}$

2002 National High School Mathematics League, 13

Tags: parabola , conic

$A(0,2)$, and two points $B,C$ on parabola $y^2=x+4$ satisfy that $AB\perp BC$. Find the range value of $y_C$.

2008 Finnish National High School Mathematics Competition, 5

Tags: combinatorics

The closed line segment $I$ is covered by finitely many closed line segments. Show that one can choose a subfamily $S$ of the family of line segments having the properties: (1) the chosen line segments are disjoint, (2) the sum of the lengths of the line segments of S is more than half of the length of $I.$ Show that the claim does not hold any more if the line segment $I$ is replaced by a circle and other occurences of the compound word ''line segment" by the word ''circular arc".

2011 Dutch IMO TST, 2

Tags: functional equation , algebra

Find all functions $f : R\to R$ satisfying $xf(x + xy) = xf(x) + f(x^2)f(y)$ for all $x, y \in R$.

2019 Israel Olympic Revenge, G

Tags: geometry , excircle

Let $\omega$ be the $A$-excircle of triangle $ABC$ and $M$ the midpoint of side $BC$. $G$ is the pole of $AM$ w.r.t $\omega$ and $H$ is the midpoint of segment $AG$. Prove that $MH$ is tangent to $\omega$.

2010 Saudi Arabia IMO TST, 3

Tags: geometry

Consider a circle of center $O$ and a chord $AB$ of it (not a diameter). Take a point $T$ on the ray $OB$. The perpendicular at $T$ onto $OB$ meets the chord $AB$ at $C$ and the circle at $D$ and $E$. Denote by $S$ the orthogonal projection of $T$ onto the chord $AB$. Prove that $AS \cdot BC = T E \cdot TD$.

2012 Iran MO (3rd Round), 1

Tags: polynomial , induction , function , modular arithmetic , algebra

Suppose $0<m_1<...<m_n$ and $m_i \equiv i (\mod 2)$. Prove that the following polynomial has at most $n$ real roots. ($\forall 1\le i \le n: a_i \in \mathbb R$). \[a_0+a_1x^{m_1}+a_2x^{m_2}+...+a_nx^{m_n}.\]

2005 Manhattan Mathematical Olympiad, 1

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In how many regions can four straight lines divide the plane? List all possible cases.

2016 NIMO Problems, 2

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Find the greatest positive integer $n$ such that $2^n$ divides \[\text{lcm}\left(1^1,2^2,3^3,\ldots,2016^{2016}\right).\] [i]Proposed by Michael Tang[/i]

2004 Manhattan Mathematical Olympiad, 4

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Twenty points are marked on the circumference of a circle. Two players play the following game. On each turn, one connects two of the $20$ points with a segment, according to the following rules: [list] [*] a segment can only appear once during the game; [*] no two segments can intersect, except at the endpoints; [*] the player who has no choice left loses the game.[/list] Assuming both players use their best strategy, which one (first or second) is certain to win the game?

2016 AMC 8, 1

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The longest professional tennis match ever played lasted a total of $11$ hours and $5$ minutes. How many minutes was this? $\textbf{(A) }605\qquad\textbf{(B) }655\qquad\textbf{(C) }665\qquad\textbf{(D) }1005\qquad \textbf{(E) }1105$

2008 Singapore MO Open, 1

Tags: number theory

Find all pairs of positive integers $ (n,k)$ so that $ (n\plus{}1)^k\minus{}1\equal{}n!$.

2019 MIG, 16

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For some constant $b$, the graph of $y = x^2 + b^2 + 2bx - b + 2$ has only one $x$ intercept. What is the value of $b$? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }10$

1996 AMC 12/AHSME, 13

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Sunny runs at a steady rate, and Moonbeam runs $m$ times as fast, where $m$ is a number greater than 1. If Moonbeam gives Sunny a head start of $h$ meters, how many meters must Moonbeam run to overtake Sunny? $\text{(A)}\ hm \qquad \text{(B)}\ \frac{h}{h+m} \qquad \text{(C)}\ \frac{h}{m-1} \qquad \text{(D)}\ \frac{hm}{m-1} \qquad \text{(E)}\ \frac{h+m}{m-1}$

2015 ASDAN Math Tournament, 35

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Let $S$ be the set of positive integers less than $10^6$ that can be written as the sum of two perfect squares. Compute the number of elements in $S$. Your score will be given by $\max\{\lfloor75(\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}-\tfrac{2}{3})\rfloor,0\}$, where $A$ is your answer and $C$ is the actual answer.

2023 Canada National Olympiad, 5

Tags: combinatorics

A country with $n$ cities has some two-way roads connecting certain pairs of cities. Someone notices that if the country is split into two parts in any way, then there would be at most $kn$ roads between the two parts (where $k$ is a fixed positive integer). What is the largest integer $m$ (in terms of $n$ and $k$) such that there is guaranteed to be a set of $m$ cities, no two of which are directly connected by a road?

2024 ELMO Shortlist, N5

Tags: number theory

Let $T$ be a finite set of squarefree integers. (a) Show that there exists an integer polynomial $P(x)$ such that the set of squarefree numbers in the range of $P(n)$ across all $n \in \mathbb{Z}$ is exactly $T$. (b) Suppose that $T$ is allowed to be infinite. Is it still true that for all choices of $T$, such an integer polynomial $P(x)$ exists? [i]Allen Wang[/i]

1992 IMTS, 2

Tags: search , algebra

Prove that if $a,b,c$ are positive integers such that $c^2 = a^2+b^2$, then both $c^2+ab$ and $c^2-ab$ are also expressible as the sums of squares of two positive integers.