Olympiad problems

2004 Estonia National Olympiad, 3

On the sides $AB , BC$ of the convex quadrilateral $ABCD$ lie points $M$ and $N$ such that $AN$ and $CM$ each divide the quadrilateral $ABCD$ into two equal area parts. Prove that the line $MN$ and $AC$ are parallel.

1994 Korea National Olympiad, Problem 1

Tags: number theory

Consider the equation $ y^2\minus{}k\equal{}x^3$, where $ k$ is an integer. Prove that the equation cannot have five integer solutions of the form $ (x_1,y_1),(x_2,y_1\minus{}1),(x_3,y_1\minus{}2),(x_4,y_1\minus{}3),(x_5,y_1\minus{}4)$. Also show that if it has the first four of these pairs as solutions, then $ 63|k\minus{}17$.

1968 Poland - Second Round, 2

Tags: construction , geometry , geometric construction , orthocenter

Given a circle $ k $ and a point inside it $ H $. Inscribe a triangle in the circle such that this point $ H $ is the point of intersection of the triangle's altitudes.

2007 Junior Balkan Team Selection Tests - Moldova, 1

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

The numbers $d_1, d_2,..., d_6$ are distinct digits of the decimal number system other than $6$. Prove that $d_1+d_2+...+d_6= 36$ if and only if $(d_1-6) (d_2-6) ... (d_6 -6) = -36$.

2024 Mathematical Talent Reward Programme, 10

Tags: combinatorics

In MTRP district there are $10$ cities. Bob the builder wants to make roads between cities in such a way so that one can go from one city to the other through exactly one unique path. The government has allotted him a budget of Rs. $20$ and each road requires a positive integer amount (in Rs.) to build. In how many ways he can build such a network of roads? It is known that in the MTRP district, any positive integer amount of rupees can be used to construct a road, and that the full budget is used by Bob in the construction. Write the last two digits of your answer.

2008 Mexico National Olympiad, 2

Tags: number theory , 3d geometry , greatest common divisor , geometry

We place $8$ distinct integers in the vertices of a cube and then write the greatest common divisor of each pair of adjacent vertices on the edge connecting them. Let $E$ be the sum of the numbers on the edges and $V$ the sum of the numbers on the vertices. a) Prove that $\frac23E\le V$. b) Can $E=V$?

2023 Grand Duchy of Lithuania, 4

Tags: number theory , divide , factorial

Note that $k\ge 1$ for an odd natural number $$k! ! = k \cdot (k - 2) \cdot ... \cdot 1.$$ Prove that $2^n$ divides $(2^n -1)!! -1$ for all $n \ge 3$.

2010 LMT, 19

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Let $f(x)=x^2-2x+1.$ For some constant $k, f(x+k) = x^2+2x+1$ for all real numbers $x.$ Determine the value of $k.$

2020 CCA Math Bonanza, I3

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Compute the remainder when $\left(\frac{2^5}{2}\right)^5$ is divided by $5$. [i]2020 CCA Math Bonanza Individual Round #3[/i]

2000 Vietnam Team Selection Test, 2

Tags: algebra , function

Let $a > 1$ and $r > 1$ be real numbers. (a) Prove that if $f : \mathbb{R}^{+}\to\mathbb{ R}^{+}$ is a function satisfying the conditions (i) $f (x)^{2}\leq ax^{r}f (\frac{x}{a})$ for all $x > 0$, (ii) $f (x) < 2^{2000}$ for all $x < \frac{1}{2^{2000}}$, then $f (x) \leq x^{r}a^{1-r}$ for all $x > 0$. (b) Construct a function $f : \mathbb{R}^{+}\to\mathbb{ R}^{+}$ satisfying condition (i) such that for all $x > 0, f (x) > x^{r}a^{1-r}$ .

2022 Taiwan Mathematics Olympiad, 5

Tags: geometry

Let $J$ be the $A$-excenter of an acute triangle $ABC$. Let $X$, $Y$ be two points on the circumcircle of the triangle $ACJ$ such that $\overline{JX} = \overline{JY} < \overline{JC}$. Let $P$ be a point lies on $XY$ such that $PB$ is tangent to the circumcircle of the triangle $ABC$. Let $Q$ be a point lies on the circumcircle of the triangle $BXY$ such that $BQ$ is parallel to $AC$. Prove that $\angle BAP = \angle QAC$. [i]Proposed by Li4.[/i]

1976 IMO Longlists, 44

Tags: floor function , geometry

A circle of radius $1$ rolls around a circle of radius $\sqrt{2}$. Initially, the tangent point is colored red. Afterwards, the red points map from one circle to another by contact. How many red points will be on the bigger circle when the center of the smaller one has made $n$ circuits around the bigger one?

2007 Austria Beginners' Competition, 3

Tags: inequalities , algebra

For real numbers $x \ge 0$ and $y \ge 0$, write $A= \frac{x+y}{2}$ for the arithmetic mean and $G=\sqrt{xy}$ for the geometric mean of $x$ and $y$. Furthermore, let $W= \frac{\sqrt{x}+\sqrt{y}}{2}$ be the arithmetic mean of $\sqrt{x}$ and $\sqrt{y}$. Prove that $$G\le W^2 \le A.$$ Determine all $x$ and $y$ such that $G= W^2 = A$

2017 Peru Iberoamerican Team Selection Test, P6

Tags: number theory

For each positive integer $k$, let $S(k)$ be the sum of the digits of $k$ in the decimal system. Prove that there exists a positive integer $k$, which does not have the digit $9$ in its decimal representation, such that: $$S(2^{24^{2017}}k)=S(k)$$

2023 India National Olympiad, 2

Tags: inequalities , polynomial , algebra

Suppose $a_0,\ldots, a_{100}$ are positive reals. Consider the following polynomial for each $k$ in $\{0,1,\ldots, 100\}$: $$a_{100+k}x^{100}+100a_{99+k}x^{99}+a_{98+k}x^{98}+a_{97+k}x^{97}+\dots+a_{2+k}x^2+a_{1+k}x+a_k,$$where indices are taken modulo $101$, [i]i.e.[/i], $a_{100+i}=a_{i-1}$ for any $i$ in $\{1,2,\dots, 100\}$. Show that it is impossible that each of these $101$ polynomials has all its roots real. [i]Proposed by Prithwijit De[/i]

1959 AMC 12/AHSME, 29

Tags: function , algebra

On a examination of $n$ questions a student answers correctly $15$ of the first $20$. Of the remaining questions he answers one third correctly. All the questions have the same credit. If the student's mark is $50\%$, how many different values of $n$ can there be? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ \text{the problem cannot be solved} $

2023 All-Russian Olympiad, 4

Tags: geometry

Let $\omega$ be the circumcircle of triangle $ABC$ with $AB<AC$. Let $I$ be its incenter and let $M$ be the midpoint of $BC$. The foot of the perpendicular from $M$ to $AI$ is $H$. The lines $MH, BI, AB$ form a triangle $T_b$ and the lines $MH, CI, AC$ form a triangle $T_c$. The circumcircle of $T_b$ meets $\omega$ at $B'$ and the circumcircle of $T_c$ meets $\omega$ at $C'$. Prove that $B', H, C'$ are collinear.

2016 Peru IMO TST, 12

Tags: geometry

Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.

2016 Iran Team Selection Test, 4

Tags: geometry

Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$. [i]Proposed by El Salvador[/i]

2018 District Olympiad, 3

Tags: 3d geometry , cube , parallelepiped , collinear , geometry

Let $ABCDA'B'C'D'$ be the rectangular parallelepiped. Let $M, N, P$ be midpoints of the edges $[AB], [BC],[BB']$ respectively . Let $\{O\} = A'N \cap C'M$. a) Prove that the points $D, O, P$ are collinear. b) Prove that $MC' \perp (A'PN)$ if and only if $ABCDA'B'C'D'$ is a cube.

2020 CHMMC Winter (2020-21), 9

Tags: nt

For a positive integer $m$, let $\varphi(m)$ be the number of positive integers $k \le m$ such that $k$ and $m$ are relatively prime, and let $\sigma(m)$ be the sum of the positive divisors of $m$. Find the sum of all even positive integers $n$ such that \[ \frac{n^5\sigma(n) - 2}{\varphi(n)} \] is an integer.

1999 National Olympiad First Round, 18

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Let $ t_{k} \left(n\right)$ show the sum of $ k^{th}$ power of digits of positive number $ n$. For which $ k$, the condition that $ t_{k} \left(n\right)$ is a multiple of 3 does not imply the condition that $ n$ is a multiple of 3? $\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ \text{None}$

2021 Winter Stars of Mathematics, 4

Tags: algebra , number theory

Let $a_0 = 1, \ a_1 = 2,$ and $a_2 = 10,$ and define $a_{k+2} = a_{k+1}^3+a_k^2+a_{k-1}$ for all positive integers $k.$ Is it possible for some $a_x$ to be divisible by $2021^{2021}?$ [i]Flavian Georgescu[/i]

2007 China Team Selection Test, 3

Tags: inequalities , 3-variable inequality , cyclic inequality , square root inequality

Find the smallest constant $ k$ such that $ \frac {x}{\sqrt {x \plus{} y}} \plus{} \frac {y}{\sqrt {y \plus{} z}} \plus{} \frac {z}{\sqrt {z \plus{} x}}\leq k\sqrt {x \plus{} y \plus{} z}$ for all positive $ x$, $ y$, $ z$.

2024 ELMO Shortlist, C5

Tags: combinatorics

Let $\mathcal{S}$ be a set of $10$ points in a plane that lie within a disk of radius $1$ billion. Define a $move$ as picking a point $P \in \mathcal{S}$ and reflecting it across $\mathcal{S}$'s centroid. Does there always exist a sequence of at most $1500$ moves after which all points of $\mathcal{S}$ are contained in a disk of radius $10$? [i]Advaith Avadhanam[/i]