Olympiad problems

1997 All-Russian Olympiad Regional Round, 8.5

Segments $AB$, $BC$ and $CA$ are, respectively, diagonals of squares $K_1$, $K_2$, $K3$. Prove that if triangle $ABC$ is acute, then it completely covered by squares $K_1$, $K_2$ and $K_3$.

1997 Iran MO (3rd Round), 2

Tags: inequalities , trigonometry , geometry , circumcircle , geometry proposed

Show that for any arbitrary triangle $ABC$, we have \[\sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \leq \frac{abc}{(a+b)(b+c)(c+a)}.\]

1989 Mexico National Olympiad, 2

Tags: number theory , Divisors

Find two positive integers $a,b$ such that $a | b^2, b^2 | a^3, a^3 | b^4, b^4 | a^5$, but $a^5$ does not divide $b^6$

2002 Mexico National Olympiad, 3

Tags: modular arithmetic , induction , strong induction , number theory unsolved , number theory , bijection

Let $n$ be a positive integer. Does $n^2$ has more positive divisors of the form $4k+1$ or of the form $4k-1$?

2001 Moldova National Olympiad, Problem 1

Tags: number theory , combinatorics

Consider the set $M=\{1,2,...,n\},n\in\mathbb N$. Find the smallest positive integer $k$ with the following property: In every $k$-element subset $S$ of $M$ there exist two elements, one of which divides the other one.

Geometry Mathley 2011-12, 5.3

Tags: Euler Line , concurrent , geometry

Let $ABC$ be an acute triangle, not being isoceles. Let $\ell_a$ be the line passing through the points of tangency of the escribed circles in the angle $A$ with the lines $AB, AC$ produced. Let $d_a$ be the line through $A$ parallel to the line that joins the incenter $I$ of the triangle $ABC$ and the midpoint of $BC$. Lines $\ell_b, d_b, \ell_c, d_c$ are defined in the same manner. Three lines $\ell_a, \ell_b, \ell_c$ intersect each other and these intersections make a triangle called $MNP$. Prove that the lines $d_a, d_b$ and $d_c$ are concurrent and their point of concurrency lies on the Euler line of the triangle $MNP$. Lê Phúc Lữ

2012 USAMO, 3

Tags: function , modular arithmetic , floor function , AMC , USA(J)MO , USAMO

Determine which integers $n > 1$ have the property that there exists an infinite sequence $a_1, a_2, a_3, \ldots$ of nonzero integers such that the equality \[a_k+2a_{2k}+\ldots+na_{nk}=0\]holds for every positive integer $k$.

2020 LMT Fall, A13

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Find the number of integers $n$ from $1$ to $2020$ inclusive such that there exists a multiple of $n$ that consists of only $5$'s. [i]Proposed by Ephram Chun and Taiki Aiba[/i]

1984 Tournament Of Towns, (056) O4

Tags: number theory , sum of digits , product of digits , Digits

The product of the digits of the natural number $N$ is denoted by $P(N)$ whereas the sum of these digits is denoted by $S(N)$. How many solutions does the equation $P(P(N)) + P(S(N)) + S(P(N)) + S(S(N)) = 1984$ have?

2023 AMC 12/AHSME, 16

Tags: AMC , AMC 12 , number theory , Chicken McNugget Theorem , 2023 AMC , 2023 AMC 12B

In Coinland, there are three types of coins, each worth $6,$ $10,$ and $15.$ What is the sum of the digits of the maximum amount of money that is impossible to have? $\textbf{(A) }11\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10$ (I forgot the order)

2008 Princeton University Math Competition, A8/B9

Tags: number theory

Find all sets of three primes $p, q$, and $r$ such that $p + q = r$ and $(r -p)(q - p) - 27p$ is a perfect square.

1994 Romania TST for IMO, 1:

Tags: quadratics , modular arithmetic , number theory proposed , number theory

Find the smallest nomial of this sequence that $a_1=1993^{1994^{1995}}$ and \[ a_{n+1}=\begin{cases}\frac{a_n}{2}&\text{if $n$ is even}\\a_n+7 &\text{if $n$ is odd.} \end{cases} \]

1997 AMC 8, 7

Tags: geometry , AMC

The area of the smallest square that will contain a circle of radius 4 is $\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 64 \qquad \textbf{(E)}\ 128$

2007 Czech-Polish-Slovak Match, 3

Tags: geometry , circumcircle , geometry proposed

A convex quadrilateral $ABCD$ inscribed in a circle $k$ has the property that the rays $DA$ and $CB$ meet at a point $E$ for which $CD^2=AD\cdot ED.$ The perpendicular to $ED$ at $A$ intersects $k$ again at point $F.$ Prove that the segments $AD$ and $CF$ are congruent if and only if the circumcenter of $\triangle ABE$ lies on $ED.$

2000 India Regional Mathematical Olympiad, 2

Tags:

Solve the equation $y^3 = x^3 + 8x^2 - 6x +8$, for positive integers $x$ and $y$.

2000 Moldova National Olympiad, Problem 8

Tags: geometry , Triangle

Points $D$ and $N$ on the sides $AB$ and $BC$ and points $E,M$ on the side $AC$ of an equilateral triangle $ABC$, respectively, with $E$ between $A$ and $M$, satisfy $AD+AE=CN+CM=BD+BN+EM$. Determine the angle between the lines $DM$ and $EN$.

MOAA Team Rounds, 2023.10

Tags: MOAA 2023

Let $S$ be the set of lattice points $(a,b)$ in the coordinate plane such that $1\le a\le 30$ and $1\le b\le 30$. What is the maximum number of lattice points in $S$ such that no four points form a square of side length 2? [i]Proposed by Harry Kim[/i]

2014 PUMaC Combinatorics B, 4

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Let there be $320$ points arranged on a circle, labeled $1$, $2$, $3$, $\dots$, $8$, $1$, $2$, $3$, $\dots$, $8$, $\dots$ in order. Line segments may only be drawn to connect points labelled with the same number. What the largest number of non-intersecting line segments one can draw? (Two segments sharing the same endpoint are considered to be intersecting).

2020 Macedonia Additional BMO TST, 4

Tags: number theory , Sequence

Prove that for all $n\in \mathbb{N}$ there exist natural numbers $a_1,a_2,...,a_n$ such that: $(i)a_1>a_2>...>a_n$ $(ii)a_i|a^2_{i+1},\forall i\in\{1,2,...,n-1\}$ $(iii)a_i\nmid a_j,\forall i,j\in \{1,2,...,n\},i\neq j$

2011 Today's Calculation Of Integral, 682

Tags: calculus , integration , trigonometry , geometry , function , calculus computations

On the $x$-$y$ plane, 3 half-lines $y=0,\ (x\geq 0),\ y=x\tan \theta \ (x\geq 0),\ y=-\sqrt{3}x\ (x\leq 0)$ intersect with the circle with the center the origin $O$, radius $r\geq 1$ at $A,\ B,\ C$ respectively. Note that $\frac{\pi}{6}\leq \theta \leq \frac{\pi}{3}$. If the area of quadrilateral $OABC$ is one third of the area of the regular hexagon which inscribed in a circle with radius 1, then evaluate $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} r^2d\theta .$ [i]2011 Waseda University of Education entrance exam/Science[/i]

2018 AMC 10, 2

Tags: AMC , AMC 10 , 2018 AMC 10B , 2018 AMC 12B , AMC 12 , rate problems

Sam drove $96$ miles in $90$ minutes. His average speed during the first $30$ minutes was $60$ mph (miles per hour), and his average speed during the second $30$ minutes was $65$ mph. What was his average speed, in mph, during the last $30$ minutes? $\textbf{(A) } 64 \qquad \textbf{(B) } 65 \qquad \textbf{(C) } 66 \qquad \textbf{(D) } 67 \qquad \textbf{(E) } 68$

1997 Moscow Mathematical Olympiad, 1

Tags: Grade 9 , 1997

In a triangle one side is $3$ times shorter than the sum of the other two. Prove that the angle opposite said side is the smallest of the triangle’s angles.

1949-56 Chisinau City MO, 1

Tags: algebra

The numbers $1, 2, ..., 1000$ are written out in a row along a circle. Starting from the first, every fifteenth number in the circle is crossed out $(1, 16, 31, ...)$, in this case, the crossed out numbers are still taken into account at each new round of the circle. How many numbers are left uncrossed?

2002 Pan African, 5

Tags: geometry , circumcircle , projective geometry

Let $\triangle{ABC}$ be an acute angled triangle. The circle with diameter AB intersects the sides AC and BC at points E and F respectively. The tangents drawn to the circle through E and F intersect at P. Show that P lies on the altitude through the vertex C.

2020 Regional Olympiad of Mexico Center Zone, 6

Tags: combinatorics , graphing lines , Coloring

Let $n,k$ be integers such that $n\geq k\geq3$. Consider $n+1$ points in a plane (there is no three collinear points) and $k$ different colors, then, we color all the segments that connect every two points. We say that an angle is good if its vertex is one of the initial set, and its two sides aren't the same color. Show that there exist a coloration such that the \\ total number of good angles is greater than $n \binom{k}{2} \lfloor(\frac{n}{k})\rfloor^2$