Olympiad problems

2023 Brazil Cono Sur TST, 1

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A quadrilateral $ABCD$ is inscribed in a circle and the lenght of side $AD$ equals the sum of the lenghts of the sides $AB$ and $CD$. Prove that the angle bisectors of $\angle ABC$ and $\angle BCD$ meet on the side $AD$.

2019 Oral Moscow Geometry Olympiad, 3

Restore the acute triangle $ABC$ given the vertex $A$, the foot of the altitude drawn from the vertex $B$ and the center of the circle circumscribed around triangle $BHC$ (point $H$ is the orthocenter of triangle $ABC$).

2017 Ecuador Juniors, 6

Tags: number theory , perfect cube

Find all primes $p$ such that $p^2- p + 1$ is a perfect cube.

2010 National Chemistry Olympiad, 11

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Enzymes convert glucose $(M=180.2)$ to ethanol $(M=46.1)$ according to the equation \[ \text{C}_6\text{H}_{12}\text{O}_6 \rightarrow 2\text{C}_2\text{H}_2\text{OH} + 2\text{CO}_2 \] What is the maximum mass of ethanol that can be made from $15.5$ kg of glucose? $ \textbf{(A)}\hspace{.05in}0.256 \text{kg}\qquad\textbf{(B)}\hspace{.05in}0.512 \text{kg} \qquad\textbf{(C)}\hspace{.05in}3.96 \text{kg}\qquad\textbf{(D)}\hspace{.05in}7.93 \text{kg}\qquad$

2005 Federal Math Competition of S&M, Problem 4

Tags: game , combinatorics

On each cell of a $2005\times2005$ chessboard, there is a marker. In each move, we are allowed to remove a marker that is a neighbor to an even number of markers (but at least one). Two markers are considered neighboring if their cells share a vertex. (a) Find the least number $n$ of markers that we can end up with on the chessboard. (b) If we end up with this minimum number $n$ of markers, prove that no two of them will be neighboring.

2021 Israel TST, 2

Tags: combinatorial geometry , combinatorics , distance

Let $n>1$ be an integer. Hippo chooses a list of $n$ points in the plane $P_1, \dots, P_n$; some of these points may coincide, but not all of them can be identical. After this, Wombat picks a point from the list $X$ and measures the distances from it to the other $n-1$ points in the list. The average of the resulting $n-1$ numbers will be denoted $m(X)$. Find all values of $n$ for which Hippo can prepare the list in such a way, that for any point $X$ Wombat may pick, he can point to a point $Y$ from the list such that $XY=m(X)$.

1998 National Olympiad First Round, 15

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Twelve couples are seated around a circular table such that all of men are seated side by side, and every women are seated to opposite of her husband. In every step, a woman and a man next to her are swapping. What is the least possible number of swapping until all couples are seated side by side? $\textbf{(A)}\ 36 \qquad\textbf{(B)}\ 55 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\ 66 \qquad\textbf{(E)}\ \text{None}$

2004 AMC 8, 14

Tags: geometry

What is the area enclosed by the geoboard quadrilateral below? [asy] int i,j; for(i=0; i<11; i=i+1) { for(j=0; j<11; j=j+1) { dot((i,j)); } } draw((0,5)--(4,0)--(10,10)--(3,4)--cycle, linewidth(0.7)); [/asy] $\textbf{(A)} 15\qquad \textbf{(B)} 18\tfrac12\qquad \textbf{(C)} 22\tfrac12\qquad \textbf{(D)} 27\qquad \textbf{(E)} 41\qquad$

2021 AMC 12/AHSME Fall, 14

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Suppose that $P(z), Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $2$, $3$, and $6$, respectively, and constant terms $1$, $2$, and $3$, respectively. Let $N$ be the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z)=R(z)$. What is the minimum possible value of $N$? $\textbf{(A)}\: 0\qquad\textbf{(B)} \: 1\qquad\textbf{(C)} \: 2\qquad\textbf{(D)} \: 3\qquad\textbf{(E)} \: 5$

1979 Dutch Mathematical Olympiad, 2

Tags: number theory , system of equations , system , diophantine

Solve in $N$: $$\begin{cases} a^3=b^3+c^3+12a \\ a^2=5(b+c) \end{cases}$$

2017 Math Prize for Girls Problems, 1

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A bag contains 4 red marbles, 5 yellow marbles, and 6 blue marbles. Three marbles are to be picked out randomly (without replacement). What is the probability that exactly two of them have the same color?

1946 Putnam, A4

Tags: function , derivative , calculus

Let $g(x)$ be a function that has a continuous first derivative $g'(x)$. Suppose that $g(0)=0$ and $|g'(x)| \leq |g(x)|$ for all values of $x.$ Prove that $g(x)$ vanishes identically.

1999 Italy TST, 2

Tags: inequalities , geometry

Let $D$ and $E$ be points on sides $AB$ and $AC$ respectively of a triangle $ABC$ such that $DE$ is parallel to $BC$ and tangent to the incircle of $ABC$. Prove that \[DE\le\frac{1}{8}(AB+BC+CA) \]

2005 Sharygin Geometry Olympiad, 6

Tags: area of a triangle , area , geometry

Side $AB$ of triangle $ABC$ was divided into $n$ equal parts (dividing points $B_0 = A, B_1, B_2, ..., B_n = B$), and side $AC$ of this triangle was divided into $(n + 1)$ equal parts (dividing points $C_0 = A, C_1, C_2, ..., C_{n+1} = C$). Colored are the triangles $C_iB_iC_{i+1}$ (where $i = 1,2, ..., n$). What part of the area of the triangle is painted over?

2016 Turkey Team Selection Test, 7

Tags: combinatorics , arithmetic sequence

$A_1, A_2,\dots A_k$ are different subsets of the set $\{1,2,\dots ,2016\}$. If $A_i\cap A_j$ forms an arithmetic sequence for all $1\le i <j\le k$, what is the maximum value of $k$?

2023 239 Open Mathematical Olympiad, 5

Tags: combinatorics , permutation

On a table, cards numbered $1, 2, \ldots , 200$ are laid out in a row in some order, and a line is drawn on the table between some two of them. It is allowed to swap two adjacent cards if the number on the left is greater than the number on the right. After a few such moves, the cards were arranged in ascending order. Prove we have swapped pairs of cards separated by the line no more than 1884 times.

2012 Sharygin Geometry Olympiad, 19

Tags: projective geometry , incenter , geometry , 3d geometry , prism , dilation , geometric transformation

Two circles with radii 1 meet in points $X, Y$, and the distance between these points also is equal to $1$. Point $C$ lies on the first circle, and lines $CA, CB$ are tangents to the second one. These tangents meet the first circle for the second time in points $B', A'$. Lines $AA'$ and $BB'$ meet in point $Z$. Find angle $XZY$.

2017 AMC 12/AHSME, 4

Tags: percent

Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip? $\textbf{(A)}\ 30 \%\qquad\textbf{(B)}\ 40 \%\qquad\textbf{(C)}\ 50 \%\qquad\textbf{(D)}\ 60 \%\qquad\textbf{(E)}\ 70 \%$

1996 Putnam, 5

Tags: floor function

Given a finite binary string $S$ of symbols $X,O$ we define $\Delta(S)=n(X)-n(O)$ where $n(X),n(O)$ respectively denote number of $X$'s and $O$'s in a string. For example $\Delta(XOOXOOX)=3-4=-1$. We call a string $S$ $\emph{balanced}$ if every substring $T$ of $S$ has $-2\le \Delta(T)\le 2$. Find number of balanced strings of length $n$.

2009 Indonesia MO, 4

Tags: combinatorics , floor function

In an island, there exist 7 towns and a railway system which connected some of the towns. Every railway segment connects 2 towns, and in every town there exists at least 3 railway segments that connects the town to another towns. Prove that there exists a route that visits 4 different towns once and go back to the original town. (Example: $ A\minus{}B\minus{}C\minus{}D\minus{}A$)

2016 Sharygin Geometry Olympiad, P5

Tags: geometry

In quadrilateral $ABCD$, $AB = CD$, $M$ and $K$ are the midpoints of $BC$ and $AD$.Prove that the angle between $MK$ and $AC$ is equal to the half-sum of angles $BAC$ and $DCA$ [i](Proposed by M.Volchkevich)[/i]

2007 Stanford Mathematics Tournament, 10

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Evaluate \[\sum_{k=1}^{2007}(-1)^{k}k^{2}\]

2012 Grigore Moisil Intercounty, 4

Tags: circumcircle , complex number

[b]a)[/b] Let $ A $ denote the complex numbers of modulus $ 1/3, $ and $ B $ denote the complex numbers of modulus at least $ 1/2. $ Show that $ A+B=AB\neq\mathbb{C} . $ [b]b)[/b] Prove that there is no family $ Y $ of complex numbers that satisfies $ X+Y=XY\neq\mathbb{C} , $ where $ X $ denotes the complex numbers of modulus $ 1. $

2003 Baltic Way, 9

Tags: induction , combinatorics

It is known that $n$ is a positive integer and $n \le 144$. Ten questions of the type “Is $n$ smaller than $a$?” are allowed. Answers are given with a delay: for $i = 1, \ldots , 9$, the $i$-th question is answered only after the $(i + 1)$-th question is asked. The answer to the tenth question is given immediately. Find a strategy for identifying $n$.

2006 Princeton University Math Competition, 1

Tags: combinatorics , graph theory

What is the greatest possible number of edges in a planar graph with $12$ vertices? A planar graph is one that can be drawn in a plane with none of the edges crossing (they intersect only at vertices).