Olympiad problems

1987 AMC 8, 3

Tags:

$2(81+83+85+87+89+91+93+95+97+99)=$ $\text{(A)}\ 1600 \qquad \text{(B)}\ 1650 \qquad \text{(C)}\ 1700 \qquad \text{(D)}\ 1750 \qquad \text{(E)}\ 1800$

2022 USA TSTST, 4

Let $\mathbb N$ denote the set of positive integers. A function $f\colon\mathbb N\to\mathbb N$ has the property that for all positive integers $m$ and $n$, exactly one of the $f(n)$ numbers \[f(m+1),f(m+2),\ldots,f(m+f(n))\] is divisible by $n$. Prove that $f(n)=n$ for infinitely many positive integers $n$.

2019 Estonia Team Selection Test, 1

Tags: number theory , power of prime

Some positive integer $n$ is written on the board. Andrey and Petya is playing the following game. Andrey finds all the different representations of the number n as a product of powers of prime numbers (values degrees greater than 1), in which each factor is greater than all previous or equal to the previous one. Petya finds all different representations of the number $n$ as a product of integers greater than $1$, in which each factor is divisible by all the previous factors. The one who finds more performances wins, if everyone finds the same number of representations, the game ends in a draw. Find all positive integers $n$ for which the game will end in a draw. Note. The representation of the number $n$ as a product is also considered a representation consisting of a single factor $n$.

2003 National High School Mathematics League, 12

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$M_n=\{\overline{0.a_1a_2\cdots a_n}|a_i\in{0,1},i=1,2,\cdots,n,a_n=1\}$. $T_n=|M_n|,S_n=\sum_{x\in M_n}x$, then $\lim_{n\to\infty}\frac{S_n}{T_n}=$________.

2010 Contests, 1

Tags: geometry , rhombus , trapezoid , midpoint

Let $ABCD$ be a trapezoid with $AB // CD$, $2|AB| = |CD|$ and $BD \perp BC$. Let $M$ be the midpoint of $CD$ and let $E$ be the intersection $BC$ and $AD$. Let $O$ be the intersection of $AM$ and $BD$. Let $N$ be the intersection of $OE$ and $AB$. (a) Prove that $ABMD$ is a rhombus. (b) Prove that the line $DN$ passes through the midpoint of the line segment $BE$.

2019 SAFEST Olympiad, 3

Tags: algebra , polynomial , IMO Shortlist

Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.

1952 Polish MO Finals, 6

Tags: geometry , 3D geometry

In a circular tower with an internal diameter of $ 2$ m, there is a spiral staircase with a height of $ 6$ m. The height of each stair step is $ 0.15$ m. In the horizontal projection, the steps form adjacent circular sections with an angle of $ 18^\circ $. The narrower ends of the steps are mounted in a round pillar with a diameter of $ 0.64$ m, the axis of which coincides with the axis of the tower. Calculate the greatest length of a straight rod that can be moved up these stairs from the bottom to the top (do not take into account the thickness of the rod or the thickness of the boards from which the stairs are made).

2024 Simon Marais Mathematical Competition, A1

Tags: algebra

Let $a,b,c$ be real number greater than 1 satisfying $$\lfloor a\rfloor b = \lfloor b \rfloor c = \lfloor c\rfloor a.$$Prove that $a=b=c$ (Here, $\lfloor x \rfloor$ denotes the laregst integer that is less than or equal to $x$.)

2004 Romania National Olympiad, 3

Tags: geometry , trapezoid

Let $ABCD$ be an orthodiagonal trapezoid such that $\measuredangle A = 90^{\circ}$ and $AB$ is the larger base. The diagonals intersect at $O$, $\left( OE \right.$ is the bisector of $\measuredangle AOD$, $E \in \left( AD \right)$ and $EF \| AB$, $F \in \left( BC \right)$. Let $P,Q$ the intersections of the segment $EF$ with $AC,BD$. Prove that: (a) $EP=QF$; (b) $EF=AD$. [i]Claudiu-Stefan Popa[/i]

2022 Nigerian Senior MO Round 2, Problem 6

Tags: number theory , Diophantine equation

Let $k, l, m, n$ be positive integers. Given that $k+l+m+n=km=ln$, find all possible values of $k+l+m+n$.

2006 Romania Team Selection Test, 2

Tags: quadratics , real analysis , complex numbers , algebra , polynomial , sum of roots , number theory proposed

Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]

2011 Indonesia TST, 3

Tags: combinatorics , Coloring

Given a board consists of $n \times n$ unit squares ($n \ge 3$). Each unit square is colored black and white, resembling a chessboard. In each step, TOMI can choose any $2 \times 2$ square and change the color of every unit square chosen with the other color (white becomes black and black becomes white). Find every $n$ such that after a finite number of moves, every unit square on the board has a same color.

2018 Brazil Undergrad MO, 22

Tags: Brazilian Undergrad MO , college contest , calculus , integration

What is the value of the improper integral $ \int_0 ^ {\pi} \log (\sin (x)) dx$?

2011 NIMO Summer Contest, 12

Tags: geometry

In triangle $ABC$, $AB = 100$, $BC = 120$, and $CA = 140$. Points $D$ and $F$ lie on $\overline{BC}$ and $\overline{AB}$, respectively, such that $BD = 90$ and $AF = 60$. Point $E$ is an arbitrary point on $\overline{AC}$. Denote the intersection of $\overline{BE}$ and $\overline{CF}$ as $K$, the intersection of $\overline{AD}$ and $\overline{CF}$ as $L$, and the intersection of $\overline{AD}$ and $\overline{BE}$ as $M$. If $[KLM] = [AME] + [BKF] + [CLD]$, where $[X]$ denotes the area of region $X$, compute $CE$. [i]Proposed by Lewis Chen [/i]

2011 IMAR Test, 1

Tags: geometry , Locus , concurrency , concurrent , Locus problems

Let $A_0A_1A_2$ be a triangle and let $P$ be a point in the plane, not situated on the circle $A_0A_1A_2$. The line $PA_k$ meets again the circle $A_0A_1A_2$ at point $B_k, k = 0, 1, 2$. A line $\ell$ through the point $P$ meets the line $A_{k+1}A_{k+2}$ at point $C_k, k = 0, 1, 2$. Show that the lines $B_kC_k, k = 0, 1, 2$, are concurrent and determine the locus of their concurrency point as the line $\ell$ turns about the point $P$.

Kyiv City MO Juniors Round2 2010+ geometry, 2020.8.2

Tags: geometry , equal segments , equal angles , right angle

Given a convex quadrilateral $ABCD$, in which $\angle CBD = 90^o$, $\angle BCD =\angle CAD$ and $AD= 2BC$. Prove that $CA =CD$. (Anton Trygub)

2017-2018 SDML (Middle School), 11

Tags: SDML Middle School , 2017-18 SDML Round 1

How many three-digit numbers leave remainder $2$ when divided by $5$ and leave remainder $7$ when divided by $9$? $\mathrm{(A) \ } 20 \qquad \mathrm{(B) \ } 21 \qquad \mathrm {(C) \ } 22 \qquad \mathrm{(D) \ } 23 \qquad \mathrm{(E) \ } 24$

2010 Greece National Olympiad, 4

Tags: combinatorics proposed , combinatorics

On the plane are given $ k\plus{}n$ distinct lines , where $ k>1$ is integer and $ n$ is integer as well.Any three of these lines do not pass through the same point . Among these lines exactly $ k$ are parallel and all the other $ n$ lines intersect each other.All $ k\plus{}n$ lines define on the plane a partition of triangular , polygonic or not bounded regions. Two regions are colled different, if the have not common points or if they have common points only on their boundary.A regions is called ''good'' if it contained in a zone between two parallel lines . If in a such given configuration the minimum number of ''good'' regionrs is $ 176$ and the maximum number of these regions is $ 221$, find $ k$ and $ n$. Babis

2021 Princeton University Math Competition, A5 / B7

Tags: algebra

Consider the sum $$S =\sum^{2021}_{j=1} \left|\sin \frac{2\pi j}{2021}\right|.$$ The value of $S$ can be written as $\tan \left( \frac{c\pi}{d} \right)$ for some relatively prime positive integers $c, d$, satisfying $2c < d$. Find the value of $c + d$.

2019 Sharygin Geometry Olympiad, 7

Tags: geometry , incenter , Locus , incircle

Let $P$ be an arbitrary point on side $BC$ of triangle $ABC$. Let $K$ be the incenter of triangle $PAB$. Let the incircle of triangle $PAC$ touch $BC$ at $F$. Point $G$ on $CK$ is such that $FG // PK$. Find the locus of $G$.

2020 Princeton University Math Competition, B1

Tags: number theory

The number $2021$ leaves a remainder of $11$ when divided by a positive integer. Find the smallest such integer.

2010 Malaysia National Olympiad, 5

Tags: ratio , areas , geometry

A circle and a square overlap such that the overlapping area is $50\%$ of the area of the circle, and is $25\%$ of the area of the square, as shown in the figure. Find the ratio of the area of the square outside the circle to the area of the whole figure. [img]https://cdn.artofproblemsolving.com/attachments/e/2/c209a95f457dbf3c46f66f82c0a45cc4b5c1c8.png[/img]

2004 USAMTS Problems, 4

Tags: USAMTS , LaTeX , algebra , system of equations

Find, with proof, all integers $n$ such that there is a solution in nonnegative real numbers $(x,y,z)$ to the system of equations \[2x^2+3y^2+6z^2=n\text{ and }3x+4y+5z=23.\]

2010 All-Russian Olympiad Regional Round, 9.8

Tags: quadratics , number theory , prime numbers , number theory proposed

For every positive integer $n$, let $S_n$ be the sum of the first $n$ prime numbers: $S_1 = 2, S_2 = 2 + 3 = 5, S_3 = 2 + 3 + 5 = 10$, etc. Can both $S_n$ and $S_{n+1}$ be perfect squares?

1996 Singapore Senior Math Olympiad, 2

Tags: max , semicircle , triangle area , Sum , geometry , trigonometry , angles

Let $180^o < \theta_1 < \theta_2 <...< \theta_n = 360^o$. For $i = 1,2,..., n$, $P_i = (\cos \theta_i^o, \sin \theta_i^o)$ is a point on the circle $C$ with centre $(0,0)$ and radius $1$. Let $P$ be any point on the upper half of $C$. Find the coordinates of $P$ such that the sum of areas $[PP_1P_2] + [PP_2P_3] + ...+ [PP_{n-1}P_n]$ attains its maximum.