Olympiad problems

2004 239 Open Mathematical Olympiad, 8

Tags: geometry , circumcircle , parallelogram , parabola , ratio , geometric transformation , conic

Given a triangle $ABC$. A point $X$ is chosen on a side $AC$. Some circle passes through $X$, touches the side $AC$ and intersects the circumcircle of triangle $ABC$ in points $M$ and $N$ such that the segment $MN$ bisects $BX$ and intersects sides $AB$ and $BC$ in points $P$ and $Q$. Prove that the circumcircle of triangle $PBQ$ passes through a fixed point different from $B$. [b]proposed by Sergej Berlov[/b]

2016 Grand Duchy of Lithuania, 4

Tags: divide , divisible , number theory

Determine all positive integers $n$ such that $7^n -1$ is divisible by $6^n -1$.

2021 Science ON grade VIII, 2

Tags: combinatorics , counting

Let $n\ge 3$ be an integer. Let $s(n)$ be the number of (ordered) pairs $(a;b)$ consisting of positive integers $a,b$ from the set $\{1,2,\dots ,n\}$ which satisfy $\gcd (a,b,n)=1$. Prove that $s(n)$ is divisible by $4$ for all $n\ge 3$. [i] (Vlad Robu) [/i]

1982 National High School Mathematics League, 8

Tags: inequalities

$a,b$ are two different positive real numbers, then which one is the largest? $$A=(a+\frac{1}{a})(b+\frac{1}{b}), B=(\sqrt{ab}+\frac{1}{\sqrt{ab}})^2, C=(\frac{a+b}{2}+\frac{2}{a+b})^2.$$ $\text{(A)}A\qquad\text{(B)}B\qquad\text{(C)}C\qquad\text{(D)}$Not sure.

2021 Iranian Geometry Olympiad, 3

Tags: geometry , perimeter , semicircle , angle

As shown in the following figure, a heart is a shape consist of three semicircles with diameters $AB$, $BC$ and $AC$ such that $B$ is midpoint of the segment $AC$. A heart $\omega$ is given. Call a pair $(P, P')$ bisector if $P$ and $P'$ lie on $\omega$ and bisect its perimeter. Let $(P, P')$ and $(Q,Q')$ be bisector pairs. Tangents at points $P, P', Q$, and $Q'$ to $\omega$ construct a convex quadrilateral $XYZT$. If the quadrilateral $XYZT$ is inscribed in a circle, find the angle between lines $PP'$ and $QQ'$. [img]https://cdn.artofproblemsolving.com/attachments/3/c/8216889594bbb504372d8cddfac73b9f56e74c.png[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

1983 All Soviet Union Mathematical Olympiad, 354

Tags: inequalities , algebra , digit

Natural number $k$ has $n$ digits in its decimal notation. It was rounded up to tens, then the obtained number was rounded up to hundreds, and so on $(n-1)$ times. Prove that the obtained number $m$ satisfies inequality $m < \frac{18k}{13}$. (Examples of rounding: $191\to190\to 200, 135\to140\to 100$.)

2014 Contests, 2

Tags: number theory , diophantine equation

$p$ is a prime. Find the all $(m,n,p)$ positive integer triples satisfy $m^3+7p^2=2^n$.

2016 IFYM, Sozopol, 7

Tags: plane , combinatorics

A grasshopper hops from an integer point to another integer point in the plane, where every even jump has a length $\sqrt{98}$ and every odd one - $\sqrt{149}$. What’s the least number of jumps the grasshopper has to make in order to return to its starting point after odd number of jumps?

2020 Ukrainian Geometry Olympiad - April, 2

Tags: geometry , equal angles , circles , tangent , symmetry

Let $\Gamma$ be a circle and $P$ be a point outside, $PA$ and $PB$ be tangents to $\Gamma$ , $A, B \in \Gamma$ . Point $K$ is an arbitrary point on the segment $AB$. The circumscirbed circle of $\vartriangle PKB$ intersects $\Gamma$ for the second time at point $T$, point $P'$ is symmetric to point $P$ wrt point $A$. Prove that $\angle PBT = \angle P'KA$.

2016 AIME Problems, 2

Tags:

There is a $40\%$ chance of rain on Saturday and a $30\%$ of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$.

Russian TST 2015, P1

Tags: square , geometry

The points $A', B', C', D'$ are selected respectively on the sides $AB, BC, CD, DA$ of the cyclic quadrilateral $ABCD$. It is known that $AA' = BB' = CC' = DD'$ and \[\angle AA'D' =\angle BB'A' =\angle CC'B' =\angle DD'C'.\]Prove that $ABCD$ is a square.

2020 Tuymaada Olympiad, 1

Tags: number theory

For each positive integer $m$ let $t_m$ be the smallest positive integer not dividing $m$. Prove that there are infinitely many positive integers which can not be represented in the form $m + t_m$. [i](A. Golovanov)[/i]

1992 China National Olympiad, 3

Tags: number theory

Let sequence $\{a_1,a_2,\dots \}$ with integer terms satisfy the following conditions: 1) $a_{n+1}=3a_n-3a_{n-1}+a_{n-2}, n=2,3,\dots$ ; 2) $2a_1=a_0+a_2-2$ ; 3) for arbitrary natural number $m$, there exist $m$ consecutive terms $a_k, a_{k-1}, \dots ,a_{k+m-1}$ among the sequence such that all such $m$ terms are perfect squares. Prove that all terms of the sequence $\{a_1,a_2,\dots \}$ are perfect squares.

2010 Contests, 3

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How many ordered triples of integers $(x, y, z)$ are there such that \[ x^2 + y^2 + z^2 = 34 \, ? \]

2009 VTRMC, Problem 4

Tags: geometry , circles

Two circles $\alpha,\beta$ touch externally at the point $X$. Let $A,P$ be two distinct points on $\alpha$ different from $X$, and let $AX$ and $PX$ meet $\beta$ again in the points $B$ and $Q$ respectively. Prove that $AP$ is parallel to $QB$.

1915 Eotvos Mathematical Competition, 2

Tags: geometry , perimeter , geometric inequalities

Triangle $ABC$ lies entirely inside a polygon. Prove that the perimeter of triangle $ABC$ is not greater than that of the polygon.

2010 LMT, 34

Tags:

A [i]prime power[/i] is an integer of the form $p^k,$ where $p$ is a prime and $k$ is a nonnegative integer. How many prime powers are there less than or equal to $10^6?$ Your score will be $16-80|\frac{\textbf{Your Answer}}{\textbf{Actual Answer}}-1|$ rounded to the nearest integer or $0,$ whichever is higher.

2019 Dutch IMO TST, 2

Tags: algebra , system of equations , max and min , max , min

Determine all $4$-tuples $(a,b, c, d)$ of positive real numbers satisfying $a + b +c + d = 1$ and $\max (\frac{a^2}{b},\frac{b^2}{a}) \cdot \max (\frac{c^2}{d},\frac{d^2}{c}) = (\min (a + b, c + d))^4$

Kyiv City MO Juniors Round2 2010+ geometry, 2019.9.31

Tags: geometry , inradius , incircle

A circle $k$ of radius $r$ is inscribed in $\vartriangle ABC$, tangent to the circle $k$, which are parallel respectively to the sides $AB, BC$ and $CA$ intersect the other sides of $\vartriangle ABC$ at points $M, N; P, Q$ and $L, T$ ($P, T \in AB$, $L, N \in BC$ and $M, Q\in AC$). Denote by $r_1,r_2,r_3$ the radii of inscribed circles in triangles $MNC, PQA$ and $LTB$. Prove that $r_1+r_2+r_3=r$.

2010 Poland - Second Round, 3

Tags: ceiling function , graph theory , combinatorics

The $n$-element set of real numbers is given, where $n \geq 6$. Prove that there exist at least $n-1$ two-element subsets of this set, in which the arithmetic mean of elements is not less than the arithmetic mean of elements in the whole set.

1988 China Team Selection Test, 1

Tags: quadratic , inequalities

Suppose real numbers $A,B,C$ such that for all real numbers $x,y,z$ the following inequality holds: \[A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) \geq 0.\] Find the necessary and sufficient condition $A,B,C$ must satisfy (expressed by means of an equality or an inequality).

2021 China Team Selection Test, 6

Tags: partial order , graph theory , combinatorics

Let $n(\ge 2)$ be an integer. $2n^2$ contestants participate in a Chinese chess competition, where any two contestant play exactly once. There may be draws. It is known that (1)If A wins B and B wins C, then A wins C. (2)there are at most $\frac{n^3}{16}$ draws. Proof that it is possible to choose $n^2$ contestants and label them $P_{ij}(1\le i,j\le n)$, so that for any $i,j,i',j'\in \{1,2,...,n\}$, if $i<i'$, then $P_{ij}$ wins $P_{i'j'}$.

2007 Princeton University Math Competition, 6

Tags: function

Find the number of ordered triplets of nonnegative integers $(m, n, p)$ such that $m+3n+5p \le 600$.

1962 AMC 12/AHSME, 35

Tags:

A man on his way to dinner short after $ 6: 00$ p.m. observes that the hands of his watch form an angle of $ 110^{\circ}.$ Returning before $ 7: 00$ p.m. he notices that again the hands of his watch form an angle of $ 110^{\circ}.$ The number of minutes that he has been away is: $ \textbf{(A)}\ 36 \frac23 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 42 \qquad \textbf{(D)}\ 42.4 \qquad \textbf{(E)}\ 45$

2018-2019 SDML (High School), 11

Tags: system of equations , algebra

For the system of equations $x^2 + x^2y^2 + x^2y^4 = 525$ and $x + xy + xy^2 = 35$, the sum of the real $y$ values that satisfy the equations is $ \mathrm{(A) \ } 2 \qquad \mathrm{(B) \ } \frac{5}{2} \qquad \mathrm {(C) \ } 5 \qquad \mathrm{(D) \ } 20 \qquad \mathrm{(E) \ } \frac{55}{2}$