Olympiad problems

2015 HMNT, 5

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Consider a $5 \times 5$ grid of squares. Vladimir colors some of these squares red, such that the centers of any four red squares do $\textbf{not}$ form an axis-parallel rectangle (i.e. a rectangle whose sides are parallel to those of the squares). What is the maximum number of squares he could have colored red?

MBMT Guts Rounds, 2015.25

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Three real numbers $a$, $b$, and $c$ between $0$ and $1$ are chosen independently and at random. What is the probability that $a + 2b + 3c > 5$?

2001 Tuymaada Olympiad, 1

$16$ chess players held a tournament among themselves: every two chess players played exactly one game. For victory in the party was given $1$ point, for a draw $0.5$ points, for defeat $0$ points. It turned out that exactly 15 chess players shared the first place. How many points could the sixteenth chess player score?

2003 All-Russian Olympiad Regional Round, 8.3

Tags: number theory , combinatorics

Two people take turns writing natural numbers from $1$ to $1000$. On the first move, the first player writes the number $1$ on the board. Then with your next move you can write either the number $2a$ or the number $a+1$ on the board if number $a$ is already written on the board. In this case, it is forbidden to write down numbers that are already written on the board. The one who writes out wins the number $1000$ on the board. Who wins if played correctly?

2023 LMT Fall, 2

Tags: speed , alg

Eddie has a study block that lasts $1$ hour. It takes Eddie $25$ minutes to do his homework and $5$ minutes to play a game of Clash Royale. He can’t do both at the same time. How many games can he play in this study block while still completing his homework? [i]Proposed by Edwin Zhao[/i] [hide=Solution] [i]Solution.[/i] $\boxed{7}$ Study block lasts 60 minutes, thus he has 35 minutes to play Clash Royale, during which he can play $\frac{35}{5}=\boxed{7}$ games. [/hide]

2008 German National Olympiad, 2

Tags: rectangle , geometry

The triangle $ \triangle SFA$ has a right angle at $ F$. The points $ P$ and $ Q$ lie on the line $ SF$ such that the point $ P$ lies between $ S$ and $ F$ and the point $ F$ is the midpoint of the segment $ [PQ]$. The circle $ k_1$ is th incircle of the triangle $ \triangle SPA$. The circle $ k_2$ lies outside the triangle $ \triangle SQA$ and touches the segment $ [QA]$ and the lines $ SQ$ and $ SA$. Prove that the sum of the radii of the circles $ k_1$ and $ k_2$ equals the length of $ [FA]$.

2010 LMT, 13

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Let $ABC$ be a non-degenerate triangle inscribed in a circle, such that $AB$ is the diameter of the circle. Let the angle bisectors of the angles at $A$ and $B$ meet at $P.$ Determine the maximum possible value of $\angle APB,$ in degrees.

1976 AMC 12/AHSME, 7

Tags: algebra , function , polynomial

If $x$ is a real number, then the quantity $(1-|x|)(1+x)$ is positive if and only if $\textbf{(A) }|x|<1\qquad\textbf{(B) }|x|>1\qquad\textbf{(C) }x<-1\text{ or }-1<x<1\qquad$ $\textbf{(D) }x<1\qquad \textbf{(E) }x<-1$

2024 Junior Balkan Team Selection Tests - Moldova, 4

Tags: geometry

Let $ABC$ be a triangle with $AB<AC$ and $\omega$ be its circumcircle. The tangent line to $\omega$ at $A$ intersects line $BC$ at $D$ and let $E$ be a point on $\omega$ such that $BE$ is parallel to $AD$. $DE$ intersects segment $AB$ and $\omega$ at $F$ and $G$, respectively. The circumcircle of $BGF$ intersects $BE$ at $N$. The line $NF$ intersects lines $AD$ and $EA$ at $S$ and $T$, respectively. Prove that $DGST$ is cyclic.

2009 AMC 12/AHSME, 4

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Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could [i]not[/i] be the total value of the four coins, in cents? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 45 \qquad \textbf{(E)}\ 55$

1982 All Soviet Union Mathematical Olympiad, 331

Tags: number theory , algebra

Once upon a time, three boys visited a library for the first time. The first decided to visit the library every second day. The second decided to visit the library every third day. The third decided to visit the library every fourth day. The librarian noticed, that the library doesn't work on Wednesdays. The boys decided to visit library on Thursdays, if they have to do it on Wednesdays, but to restart the day counting in these cases. They strictly obeyed these rules. Some Monday later I met them all in that library. What day of week was when they visited a library for the first time?

2004 AMC 10, 14

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The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is $ 20$ cents. If she had one more quarter, the average value would be $ 21$ cents. How many dimes does she have in her purse? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 3\qquad \textbf{(E)}\ 4$

2002 Federal Math Competition of S&M, Problem 2

Tags: geometry , fibonacci

The (Fibonacci) sequence $f_n$ is defined by $f_1=f_2=1$ and $f_{n+2}=f_{n+1}+f_n$ for $n\ge1$. Prove that the area of the triangle with the sides $\sqrt{f_{2n+1}},\sqrt{f_{2n+2}},$ and $\sqrt{f_{2n+3}}$ is equal to $\frac12$.

2018 Costa Rica - Final Round, 5

Tags: number theory , divisible , divide

Let $a$ and $ b$ be even numbers, such that $M = (a + b)^2-ab$ is a multiple of $5$. Consider the following statements: I) The unit digits of $a^3$ and $b^3$ are different. II) $M$ is divisible by $100$. Please indicate which of the above statements are true with certainty.

2002 India IMO Training Camp, 1

Tags: geometric transformation , trapezoid , homothety , power of a point , radical axis , geometry

Let $A,B$ and $C$ be three points on a line with $B$ between $A$ and $C$. Let $\Gamma_1,\Gamma_2, \Gamma_3$ be semicircles, all on the same side of $AC$ and with $AC,AB,BC$ as diameters, respectively. Let $l$ be the line perpendicular to $AC$ through $B$. Let $\Gamma$ be the circle which is tangent to the line $l$, tangent to $\Gamma_1$ internally, and tangent to $\Gamma_3$ externally. Let $D$ be the point of contact of $\Gamma$ and $\Gamma_3$. The diameter of $\Gamma$ through $D$ meets $l$ in $E$. Show that $AB=DE$.

1999 South africa National Olympiad, 1

Tags: inequalities , perimeter , ceiling function , triangle inequality , geometry

How many non-congruent triangles with integer sides and perimeter 1999 can be constructed?

2014 Baltic Way, 19

Tags: greatest common divisor , number theory

Let $m$ and $n$ be relatively prime positive integers. Determine all possible values of \[\gcd(2^m - 2^n, 2^{m^2+mn+n^2}- 1).\]

2004 China Western Mathematical Olympiad, 4

Tags: logarithm , number theory , search , function , euler , totient function , prime number

Let $\mathbb{N}$ be the set of positive integers. Let $n\in \mathbb{N}$ and let $d(n)$ be the number of divisors of $n$. Let $\varphi(n)$ be the Euler-totient function (the number of co-prime positive integers with $n$, smaller than $n$). Find all non-negative integers $c$ such that there exists $n\in\mathbb{N}$ such that \[ d(n) + \varphi(n) = n+c , \] and for such $c$ find all values of $n$ satisfying the above relationship.

2015 Stars Of Mathematics, 3

Tags: geometry , cyclic quadrilateral , circumcircle

Let $ABCD$ be cyclic quadrilateral,let $\gamma$ be it's circumscribed circle and let $M$ be the midpoint of arc $AB$ of $\gamma$,which does not contain points $C,D$.The line that passes through $M$ and the intersection point of diagonals $AC,BD$,intersects $\gamma$ in $N\neq M$. Let $P,Q$ be two points situated on $CD$,such that $\angle{AQD}=\angle{DAP}$ and $\angle{BPC}=\angle{CBQ}$.Prove that circles $\odot(NPQ)$ and $\gamma$ are tangent.

1995 Vietnam Team Selection Test, 2

Tags: number theory , modular arithmetic

For any nonnegative integer $ n$, let $ f(n)$ be the greatest integer such that $ 2^{f(n)} | n \plus{} 1$. A pair $ (n, p)$ of nonnegative integers is called nice if $ 2^{f(n)} > p$. Find all triples $ (n, p, q)$ of nonnegative integers such that the pairs $ (n, p)$, $ (p, q)$ and $ (n \plus{} p \plus{} q, n)$ are all nice.

2011 IMAR Test, 1

Tags: geometry , locus , locus problems , concurrency

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 $P$.

2021 HMNT, 4

Tags: number theory

The sum of the digits of the time $19$ minutes ago is two less than the sum of the digits of the time right now. Find the sum of the digits of the time in $19$ minutes. (Here, we use a standard $12$-hour clock of the form $hh:mm$.)

2003 AMC 12-AHSME, 19

Tags: probability

Let $ S$ be the set of permutations of the sequence $ 1, 2, 3, 4, 5$ for which the first term is not $ 1$. A permutation is chosen randomly from $ S$. The probability that the second term is $ 2$, in lowest terms, is $ a/b$. What is $ a \plus{} b$? $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 19$

1990 IMO Shortlist, 3

Tags: graph theory , combinatorics , extremal combinatorics , coloring

Let $ n \geq 3$ and consider a set $ E$ of $ 2n \minus{} 1$ distinct points on a circle. Suppose that exactly $ k$ of these points are to be colored black. Such a coloring is [b]good[/b] if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly $ n$ points from $ E$. Find the smallest value of $ k$ so that every such coloring of $ k$ points of $ E$ is good.

MBMT Team Rounds, 2015 F5 E2

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An unfair $6$-sided die has faces labeled $1$, $2$, $3$, $4$, $5$, and $6$. The probability that a die lands with a certain face up is proportional to the number on the face. What is the probability that at least one of the first three rolls is a $1$ or a $2$?