Olympiad problems

2012 Turkey Team Selection Test, 3

Tags: combinatorics

Two players $A$ and $B$ play a game on a $1\times m$ board, using $2012$ pieces numbered from $1$ to $2012.$ At each turn, $A$ chooses a piece and $B$ places it to an empty place. After $k$ turns, if all pieces are placed on the board increasingly, then $B$ wins, otherwise $A$ wins. For which values of $(m,k)$ pairs can $B$ guarantee to win?

2003 China Western Mathematical Olympiad, 4

Tags: geometry

Given that the sum of the distances from point $ P$ in the interior of a convex quadrilateral $ ABCD$ to the sides $ AB, BC, CD, DA$ is a constant, prove that $ ABCD$ is a parallelogram.

2012 Iran MO (3rd Round), 1

Tags: algorithm , combinatorics

We've colored edges of $K_n$ with $n-1$ colors. We call a vertex rainbow if it's connected to all of the colors. At most how many rainbows can exist? [i]Proposed by Morteza Saghafian[/i]

2019 Thailand Mathematical Olympiad, 8

Tags: geometry , incenter , circumcircle , collinear , collinearity

Let $ABC$ be a triangle such that $AB\ne AC$ and $\omega$ be the circumcircle of this triangle. Let $I$ be the center of the inscribed circle of $ABC$ which touches $BC$ at $D$. Let the circle with diameter $AI$ meets $\omega$ again at $K$. If the line $AI$ intersects $\omega$ again at $M$, show that $K, D, M$ are collinear.

1977 Putnam, A5

Tags:

Prove that $$\binom{pa}{pb}=\binom{a}{b} (\text{mod } p)$$ for all integers $p,a,$ and $b$ with $p$ a prime, $p>0,$ and $a>b>0.$

Kvant 2022, M2688

Tags: geometry

Let $T_a, T_b$ and $T_c$ be the tangent points of the incircle $\Omega$ of the triangle $ABC$ with the sides $BC, CA$ and $AB{}$ respectively. Let $X, Y$ and $Z{}$ be points on the circle $\Omega$ such that $A{}$ lies on the ray $YX$, $B{}$ lies on the ray $ZY$ and $C{}$ lies on the ray $XZ$. Let $P{}$ be the intersection point of the segments $ZX$ and $T_bT_c$, and similarly $Q=XY \cap T_cT_a$ and $R=YZ\cap T_aT_b$. Prove that the points $A, B$ and $C{}$ lie on the lines $RP, PQ$ and $QR{}$, respectively. [i]Proposed by L. Shatunov (11th grade student)[/i]

2015 Vietnam Team selection test, Problem 6

Tags: algebra

Find the smallest positive interger number $n$ such that there exists $n$ real numbers $a_1,a_2,\ldots,a_n$ satisfied three conditions as follow: a. $a_1+a_2+\cdots+a_n>0$; b. $a_1^3+a_2^3+\cdots+a_n^3<0$; c. $a_1^5+a_2^5+\cdots+a_n^5>0$.

2003 All-Russian Olympiad Regional Round, 9.5

Tags: combinatorics

$100$ people came to the party. Then those who had no acquaintances among those who came left. Among those who remained, then those who had exactly $1$ friend , also left. Then those who had exactly $2$, $3$, $4$,$ . .$ , $99$ acquaintances among those remaining at the time of their departure did the same..What is the largest number of people left at the end?

2009 International Zhautykov Olympiad, 2

Tags: inequalities , function , algebra

Find all real $ a$, such that there exist a function $ f: \mathbb{R}\rightarrow\mathbb{R}$ satisfying the following inequality: \[ x\plus{}af(y)\leq y\plus{}f(f(x)) \] for all $ x,y\in\mathbb{R}$

2020 BMT Fall, 7

Tags: geometry

Circle $\Gamma$ has radius $10$, center $O$, and diameter $\overline{AB}$. Point $C$ lies on $\Gamma$ such that $AC = 12$. Let $P$ be the circumcenter of $\vartriangle AOC$. Line $AP$ intersects $\Gamma$ at $Q$, where $Q$ is different from $A$. Then the value of $\frac{AP}{AQ}$ can be expressed in the form $\frac{m}{n}$, where m and $n$ are relatively prime positive integers. Compute $m + n$.

2008 Czech and Slovak Olympiad III A, 3

Tags: greatest common divisor , number theory

Find all pairs of integers $(a,b)$ such that $a^2+ab+1\mid b^2+ab+a+b-1$.

2015 Online Math Open Problems, 8

Tags:

Determine the number of sequences of positive integers $1 = x_0 < x_1 < \dots < x_{10} = 10^{5}$ with the property that for each $m=0,\dots,9$ the number $\tfrac{x_{m+1}}{x_m}$ is a prime number. [i]Proposed by Evan Chen[/i]

2000 Italy TST, 4

Tags: combinatorics

On a mathematical competition $ n$ problems were given. The final results showed that: (i) on each problem, exactly three contestants scored $ 7$ points; (ii) for each pair of problems, exactly one contestant scored $ 7$ points on both problems. Prove that if $ n \geq 8$, then there is a contestant who got $ 7$ points on each problem. Is this statement necessarily true if $ n \equal{} 7$?

2024 Olimphíada, 2

Tags: combinatorics

Philipe has two congruent regular $4046$-gons. He invites Philomena to a trick he's planning: he'll give her one of the $4046$-gons and she will paint in red $2023$ vertices of her polygon. Without knowing which vertices she chose, he'll paint $k$ vertices of the remaining polygon. After this, he wants to rotate the polygons so that each painted vertices from Philomena's polygon is corresponding to some painted vertice from Philipe's polygon. What is the minimal value of $k$ for which he can choose his vertices so that, no matter how she paints her polygon, the trick is always possible.

2016 AIME Problems, 10

Tags: sequence

A strictly increasing sequence of positive integers $a_1, a_2, a_3, \ldots$ has the property that for every positive integer $k$, the subsequence $a_{2k-1}, a_{2k}, a_{2k+1}$ is geometric and the subsequence $a_{2k}, a_{2k+1}, a_{2k+2}$ is arithmetic. Suppose that $a_{13} = 2016$. Find $a_1$.

2024 CCA Math Bonanza, L1.2

Tags:

Find the number of odd three digit positive integers $x$ satisfying $x^2 \equiv 1 \pmod{8}$ and the product of the digits of $x$ is odd. [i]Lightning 1.2[/i]

2002 APMO, 5

Tags: function , induction , algebra

Let ${\bf R}$ denote the set of all real numbers. Find all functions $f$ from ${\bf R}$ to ${\bf R}$ satisfying: (i) there are only finitely many $s$ in ${\bf R}$ such that $f(s)=0$, and (ii) $f(x^4+y)=x^3f(x)+f(f(y))$ for all $x,y$ in ${\bf R}$.

2022 Taiwan TST Round 3, 5

Tags: circumcircle , concyclic , geometry

Let $ABC$ be an acute triangle with circumcenter $O$ and circumcircle $\Omega$. Choose points $D, E$ from sides $AB, AC$, respectively, and let $\ell$ be the line passing through $A$ and perpendicular to $DE$. Let $\ell$ intersect the circumcircle of triangle $ADE$ and $\Omega$ again at points $P, Q$, respectively. Let $N$ be the intersection of $OQ$ and $BC$, $S$ be the intersection of $OP$ and $DE$, and $W$ be the orthocenter of triangle $SAO$. Prove that the points $S$, $N$, $O$, $W$ are concyclic. [i]Proposed by Li4 and me.[/i]

2016 India PRMO, 10

Tags: algebra , floor function , maximum

Let $M$ be the maximum value of $(6x-3y-8z)$, subject to $2x^2+3y^2+4z^2 = 1$. Find $[M]$.

2019 IMO Shortlist, C6

Tags: combinatorics , combinatorial geometry , angle

Let $n>1$ be an integer. Suppose we are given $2n$ points in the plane such that no three of them are collinear. The points are to be labelled $A_1, A_2, \dots , A_{2n}$ in some order. We then consider the $2n$ angles $\angle A_1A_2A_3, \angle A_2A_3A_4, \dots , \angle A_{2n-2}A_{2n-1}A_{2n}, \angle A_{2n-1}A_{2n}A_1, \angle A_{2n}A_1A_2$. We measure each angle in the way that gives the smallest positive value (i.e. between $0^{\circ}$ and $180^{\circ}$). Prove that there exists an ordering of the given points such that the resulting $2n$ angles can be separated into two groups with the sum of one group of angles equal to the sum of the other group.

2004 National Olympiad First Round, 17

Tags: geometry , trigonometry

Let $R$ and $T$ be points respectively on sides $[BC]$ and $[CD]$ of a square $ABCD$ with side length $6$ such that $|CR|+|RT|+|TC|=12$. What is $\tan (\widehat{RAT})$ $ \textbf{(A)}\ 2\sqrt 3 \qquad\textbf{(B)}\ \sqrt 3 \qquad\textbf{(C)}\ \dfrac 13 \qquad\textbf{(D)}\ \dfrac 12 \qquad\textbf{(E)}\ 1 $

1987 IMO Longlists, 71

Tags: number theory , prime factorization , algebra

To every natural number $k, k \geq 2$, there corresponds a sequence $a_n(k)$ according to the following rule: \[a_0 = k, \qquad a_n = \tau(a_{n-1}) \quad \forall n \geq 1,\] in which $\tau(a)$ is the number of different divisors of $a$. Find all $k$ for which the sequence $a_n(k)$ does not contain the square of an integer.

1980 AMC 12/AHSME, 14

Tags: function , quadratic , algebra , system of equations

If the function $f$ is defined by \[ f(x)=\frac{cx}{2x+3} , ~~~x\neq -\frac 32 , \] satisfies $x=f(f(x))$ for all real numbers $x$ except $-\frac 32$, then $c$ is $\text{(A)} \ -3 \qquad \text{(B)} \ - \frac{3}{2} \qquad \text{(C)} \ \frac{3}{2} \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \text{not uniquely determined}$

2016 Hong Kong TST, 2

Tags: geometry , incenter , circumcircle

Suppose that $I$ is the incenter of triangle $ABC$. The perpendicular to line $AI$ from point $I$ intersects sides $AC$ and $AB$ at points $B'$ and $C'$ respectively. Points $B_1$ and $C_1$ are placed on half lines $BC$ and $CB$ respectively, in such a way that $AB=BB_1$ and $AC=CC_1$. If $T$ is the second intersection point of the circumcircles of triangles $AB_1C'$ and $AC_1B'$, prove that the circumcenter of triangle $ATI$ lies on the line $BC$

2021 Sharygin Geometry Olympiad, 9.6

Tags: geometry , trapezoid , equal angles

The diagonals of trapezoid $ABCD$ ($BC\parallel AD$) meet at point $O$. Points $M$ and $N$ lie on the segments $BC$ and $AD$ respectively. The tangent to the circle $AMC$ at $C$ meets the ray $NB$ at point $P$; the tangent to the circle $BND$ at $D$ meets the ray $MA$ at point $R$. Prove that $\angle BOP =\angle AOR$.