Olympiad problems

2017 India IMO Training Camp, 2

Tags: number theory

Let $a,b,c,d$ be pairwise distinct positive integers such that $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}$$ is an integer. Prove that $a+b+c+d$ is [b]not[/b] a prime number.

2008 Turkey MO (2nd round), 3

Tags: function , combinatorics unsolved , combinatorics

There is a connected network with $ 2008$ computers, in which any of the two cycles don't have any common vertex. A hacker and a administrator are playing a game in this network. On the $ 1st$ move hacker selects one computer and hacks it, on the $ 2nd$ move administrator selects another computer and protects it. Then on every $ 2k\plus{}1th$ move hacker hacks one more computer(if he can) which wasn't protected by the administrator and is directly connected (with an edge) to a computer which was hacked by the hacker before and on every $ 2k\plus{}2th$ move administrator protects one more computer(if he can) which wasn't hacked by the hacker and is directly connected (with an edge) to a computer which was protected by the administrator before for every $ k>0$. If both of them can't make move, the game ends. Determine the maximum number of computers which the hacker can guarantee to hack at the end of the game.

2018 Online Math Open Problems, 21

Tags:

Let $\bigoplus$ and $\bigotimes$ be two binary boolean operators, i.e. functions that send $\{\text{True}, \text{False}\}\times \{\text{True}, \text{False}\}$ to $\{\text{True}, \text{False}\}$. Find the number of such pairs $(\bigoplus, \bigotimes)$ such that $\bigoplus$ and $\bigotimes$ distribute over each other, that is, for any three boolean values $a, b, c$, the following four equations hold: 1) $c \bigotimes (a \bigoplus b) = (c \bigotimes a) \bigoplus (c \bigotimes b);$ 2) $(a \bigoplus b) \bigotimes c = (a \bigotimes c) \bigoplus (b \bigotimes c);$ 3) $c \bigoplus (a \bigotimes b) = (c \bigoplus a) \bigotimes (c \bigoplus b);$ 4) $(a \bigotimes b) \bigoplus c = (a \bigoplus c) \bigotimes (b \bigoplus c).$ [i]Proposed by Yannick Yao

1993 AMC 12/AHSME, 28

Tags: analytic geometry , geometry , AMC

How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1 \le x \le 4$ and $1 \le y \le 4$? $ \textbf{(A)}\ 496 \qquad\textbf{(B)}\ 500 \qquad\textbf{(C)}\ 512 \qquad\textbf{(D)}\ 516 \qquad\textbf{(E)}\ 560 $

2010 Dutch IMO TST, 2

Tags: functional equation , maximum , algebra

Find all functions $f : R \to R$ which satisfy $f(x) = max_{y\in R} (2xy - f(y))$ for all $x \in R$.

2024 ELMO Shortlist, G5

Tags: geometry

Let $ABC$ be a triangle with circumcenter $O$ and circumcircle $\omega$. Let $D$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $P$ and $Q$ be points on the circumcircles of triangles $AOB$ and $AOC$, respectively, such that $A$, $P$, and $Q$ are collinear. Prove that if the circumcircle of triangle $OPQ$ is tangent to $\omega$ at $T$, then $\angle BTD=\angle CAP$. [i]Tiger Zhang[/i]

2014 Cezar Ivănescu, 2

Tags: inequalities , algebra

[b]a)[/b] Let be two nonegative integers $ n\ge 1,k, $ and $ n $ real numbers $ a,b,\ldots ,c. $ Prove that $$ (1/a+1/b+\cdots 1/c)\left( a^{1+k} +b^{1+k}+\cdots c^{1+k} \right)\ge n\left(a^k+b^k+\cdots +c^k\right) . $$ [b]b)[/b] If $ 1\le d\le e\le f\le g\le h\le i\le 1000 $ are six real numbers, determine the minimum value the expression $$ d/e+f/g+h/i $$ can take.

2013 Dutch IMO TST, 1

Tags: algebra , system of equations

Determine all 4-tuples ($a, b,c, d$) of real numbers satisfying the following four equations: $\begin{cases} ab + c + d = 3 \\ bc + d + a = 5 \\ cd + a + b = 2 \\ da + b + c = 6 \end{cases}$

Durer Math Competition CD Finals - geometry, 2017.C+1

Tags: geometry , circles , parallel

Given a plane with two circles, one with points $A$ and $B$, and the other with points $C$ and $D$ are shown in the figure. The line $AB$ passes through the center of the first circle and touches the second circle while the line $CD$ passes through the center of the second circle and touches the first circle. Prove that the lines $AD$ and $BC$ are parallel. [img]https://cdn.artofproblemsolving.com/attachments/e/e/92f7b57751e7828a6487a052d4869e27e658b2.png[/img]

1993 Baltic Way, 16

Tags: geometry unsolved , geometry

Two circles, both with the same radius $r$, are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points $A,B$ and the other at points $C,D$, so that $|AB|=|BC|=|CD|=14\text{cm}$. Another line intersects the circles at $E,F$, respectively $G,H$ so that $|EF|=|FG|=|GH|=6\text{cm}$. Find the radius $r$.

1997 Slovenia National Olympiad, Problem 1

Tags: number theory

Let $k$ be a positive integer. Prove that: (a) If $k=m+2mn+n$ for some positive integers $m,n$, then $2k+1$ is composite. (b) If $2k+1$ is composite, then there exist positive integers $m,n$ such that $k=m+2mn+n$.

2023 Polish Junior Math Olympiad First Round, 7.

Tags: 3D geometry

Let $ABCDEF$ be a regular hexagon with side length $2$. Point $M$ is the midpoint of diagonal $AE$. The pentagon $ABCDE$ is folded along segments $BD$, $BM$, and $DM$ in such a way that points $A$, $C$, and $E$ coincide. As a result of this operation, a tetrahedron is obtained. Determine its volume.

1940 Moscow Mathematical Olympiad, 057

Tags: tangent circles , tangent , geometry , circle construction , geometric construction

Draw a circle that has a given radius $R$ and is tangent to a given line and a given circle. How many solutions does this problem have?

2017 Ukraine Team Selection Test, 11

Tags: IMO Shortlist , geometry , mixtilinear incircle , projective geometry , median , geometry solved

Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$. [i]Proposed by Evan Chen, Taiwan[/i]

2021 Brazil Team Selection Test, 1

Tags: combinatorics , game

Players $A$ and $B$ play a game on a blackboard that initially contains 2020 copies of the number 1 . In every round, player $A$ erases two numbers $x$ and $y$ from the blackboard, and then player $B$ writes one of the numbers $x+y$ and $|x-y|$ on the blackboard. The game terminates as soon as, at the end of some round, one of the following holds: $(1)$ one of the numbers on the blackboard is larger than the sum of all other numbers; $(2)$ there are only zeros on the blackboard. Player $B$ has to pay to player $A$ an amount in reais equivalent to the quantity of numbers left on the blackboard after the game ends. Show that player $A$ can earn at least 8 reais regardless of the moves taken by $B$ Ps.: Easier version of [url = https://artofproblemsolving.com/community/c6h2625868p22698110]ISL 2020 C8[/url]

2012 National Olympiad First Round, 7

Tags: function

How many $f:\mathbb{R} \rightarrow \mathbb{R}$ are there satisfying $f(x)f(y)f(z)=12f(xyz)-16xyz$ for every real $x,y,z$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 0 \qquad \textbf{(E)}\ \text{None}$

2021 Peru EGMO TST, 4

Tags: combinatorics

There are $300$ apples in a table and the heaviest apple is [b]not[/b] heavier than three times the weight of the lightest apple. Prove that the apples can be splitted in sets of $4$ elements such that [b]no[/b] set is heavier than $\frac{3}{2}$ times the weight of any other set.

2006 China Team Selection Test, 1

Tags: geometry , circumcircle , geometric transformation , reflection , projective geometry , power of a point , radical axis

The centre of the circumcircle of quadrilateral $ABCD$ is $O$ and $O$ is not on any of the sides of $ABCD$. $P=AC \cap BD$. The circumecentres of $\triangle{OAB}$, $\triangle{OBC}$, $\triangle{OCD}$ and $\triangle{ODA}$ are $O_1$, $O_2$, $O_3$ and $O_4$ respectively. Prove that $O_1O_3$, $O_2O_4$ and $OP$ are concurrent.

2020 Taiwan TST Round 3, 2

Tags: combinatorics , Taiwan

There are $N$ monsters, each with a positive weight. On each step, two of the monsters are merged into one, whose weight is the sum of weights for the two original monsters. At the end, all monsters will be merged into one giant monster. During this process, if at any mergence, one of the two monsters has a weight greater than $2.020$ times the other monster's weight, we will call this mergence [b]dangerous[/b]. The dangerous level of a sequence of mergences is the number of dangerous mergence throughout its process. Prove that, no matter how the weights being distributed among the monsters, "for every step, merge the lightest two monsters" is always one of the merging sequences that obtain the minimum possible dangerous level. [i]Proposed by houkai[/i]

2020 Putnam, B1

Tags: Putnam , abstract algebra , Putnam 2020

For a positive integer $n$, define $d(n)$ to be the sum of the digits of $n$ when written in binary (for example, $d(13)=1+1+0+1=3$). Let \[ S=\sum_{k=1}^{2020}(-1)^{d(k)}k^3. \] Determine $S$ modulo $2020$.

2017 CCA Math Bonanza, T9

Tags:

Aida made three cubes with positive integer side lengths $a,b,c$. They were too small for her, so she divided them into unit cubes and attempted to construct a cube of side $a+b+c$. Unfortunately, she was $648$ blocks off. How many possibilities of the ordered triple $\left(a,b,c\right)$ are there? [i]2017 CCA Math Bonanza Team Round #9[/i]

2020 Princeton University Math Competition, A7

Tags: geometry

Let $ABC$ be a triangle with sides $AB = 34$, $BC = 15$, $AC = 35$ and let $\Gamma$ be the circle of smallest possible radius passing through $A$ tangent to $BC$. Let the second intersections of $\Gamma$ and sides $AB$, $AC$ be the points $X, Y$ . Let the ray $XY$ intersect the circumcircle of the triangle $ABC$ at $Z$. If $AZ =\frac{p}{q}$ for relatively prime integers $p$ and $q$, find $p + q$.

2000 Slovenia National Olympiad, Problem 1

Tags: combinatorics

Let $n$ be the number of ordered $5$-tuples $(a_1,a_2,\ldots,a_5)$ of positive integers such that $\frac1{a_1}+\frac1{a_2}+\ldots+\frac1{a_5}=1$. Is $n$ an even number?

2014 ASDAN Math Tournament, 20

Tags: 2014 , General Test

$ABCD$ is a parallelogram, and circle $S$ (with radius $2$) is inscribed insider $ABCD$ such that $S$ is tangent to all four line segments $AB$, $BC$, $CD$, and $DA$. One of the internal angles of the parallelogram is $60^\circ$. What is the maximum possible area of $ABCD$?

2016 Iran Team Selection Test, 5

Tags: geometry , perpendicular bisector

Let $AD,BF,CE$ be altitudes of triangle $ABC$.$Q$ is a point on $EF$ such that $QF=DE$ and $F$ is between $E,Q$.$P$ is a point on $EF$ such that $EP=DF$ and $E$ is between $P,F$.Perpendicular bisector of $DQ$ intersect with $AB$ at $X$ and perpendicular bisector of $DP$ intersect with $AC$ at $Y$.Prove that midpoint of $BC$ lies on $XY$.