Olympiad problems

2019 Online Math Open Problems, 11

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Jay is given $99$ stacks of blocks, such that the $i$th stack has $i^2$ blocks. Jay must choose a positive integer $N$ such that from each stack, he may take either $0$ blocks or exactly $N$ blocks. Compute the value Jay should choose for $N$ in order to maximize the number of blocks he may take from the $99$ stacks. [i]Proposed by James Lin[/i]

2012 Kosovo National Mathematical Olympiad, 2

Tags: inequality , geometry

Let $a,b,c$ be the lengths of the sides of a triangle. Prove that, $\left|\frac {a}{b}+\frac {b}{c}+\frac {c}{a}-\frac {b}{a}-\frac {c}{b}-\frac {a}{c}\right|<1$

2000 IMO, 6

Tags: homothety , triangle , geometry , incircle , circumcircle

Let $ AH_1, BH_2, CH_3$ be the altitudes of an acute angled triangle $ ABC$. Its incircle touches the sides $ BC, AC$ and $ AB$ at $ T_1, T_2$ and $ T_3$ respectively. Consider the symmetric images of the lines $ H_1H_2, H_2H_3$ and $ H_3H_1$ with respect to the lines $ T_1T_2, T_2T_3$ and $ T_3T_1$. Prove that these images form a triangle whose vertices lie on the incircle of $ ABC$.

2015 Federal Competition For Advanced Students, 1

Tags: algebra , inequalities

Let $a$, $b$, $c$, $d$ be positive numbers. Prove that $$(a^2 + b^2 + c^2 + d^2)^2 \ge (a+b)(b+c)(c+d)(d+a)$$ When does equality hold? (Georg Anegg)

2010 IFYM, Sozopol, 3

Tags: algebra , polynomial

Let $n\ge 2$ be an even integer and $a,b$ real numbers such that $b^n=3a+1$. Show that the polynomial $P(X)=(X^2+X+1)^n-X^n-a$ is divisible by $Q(X)=X^3+X^2+X+b$ if and only if $b=1$.

1997 Pre-Preparation Course Examination, 4

Tags: number theory

Let $n$ and $k$ be two positive integers. Prove that there exist infinitely many perfect squares of the form $n \cdot 2^k - 7$.

2017 Moscow Mathematical Olympiad, 5

Tags: 3d geometry , geometry

$8$ points lie on the faces of unit cube and form another cube. What can be length of edge of this cube?

2019 BMT Spring, 14

Tags: hexagon , reflection , geometric inequality , geometry

A regular hexagon has positive integer side length. A laser is emitted from one of the hexagon’s corners, and is reflected off the edges of the hexagon until it hits another corner. Let $a$ be the distance that the laser travels. What is the smallest possible value of $a^2$ such that $a > 2019$? You need not simplify/compute exponents.

2025 Israel TST, P2

Tags: configuration , incenter , geometry , circumcircle

Triangle $\triangle ABC$ is inscribed in circle $\Omega$. Let $I$ denote its incenter and $I_A$ its $A$-excenter. Let $N$ denote the midpoint of arc $BAC$. Line $NI_A$ meets $\Omega$ a second time at $T$. The perpendicular to $AI$ at $I$ meets sides $AC$ and $AB$ at $E$ and $F$ respectively. The circumcircle of $\triangle BFT$ meets $BI_A$ a second time at $P$, and the circumcircle of $\triangle CET$ meets $CI_A$ a second time at $Q$. Prove that $PQ$ passes through the antipodal to $A$ on $\Omega$.

2018 Harvard-MIT Mathematics Tournament, 6

Tags: hmmt , combinatorics , geometry

Call a polygon [i]normal[/i] if it can be inscribed in a unit circle. How many non-congruent normal polygons are there such that the square of each side length is a positive integer?

Kyiv City MO Seniors 2003+ geometry, 2018.10.4

Tags: altitude , angle , geometry

In the acute-angled triangle $ABC$, the altitudes $BP$ and $CQ$ were drawn, and the point $T$ is the intersection point of the altitudes of $\Delta PAQ$. It turned out that $\angle CTB = 90 {} ^ \circ$. Find the measure of $\angle BAC$. (Mikhail Plotnikov)

1993 National High School Mathematics League, 10

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The last two digits of number of $\left[\frac{10^{93}}{10^{31}+1}\right]$ is________.

2016 239 Open Mathematical Olympiad, 8

Tags: combinatorial geometry , combinatorics

There are $n$ triangles inscribed in a circle and all $3n$ of their vertices are different. Prove that it is possible to put a boy in one of the vertices in each triangle, and a girl in the other, so that boys and girls alternate on a circle.

2022 USA TSTST, 3

Tags: algebra

Determine all positive integers $N$ for which there exists a strictly increasing sequence of positive integers $s_0<s_1<s_2<\cdots$ satisfying the following properties: [list=disc] [*]the sequence $s_1-s_0$, $s_2-s_1$, $s_3-s_2$, $\ldots$ is periodic; and [*]$s_{s_n}-s_{s_{n-1}}\leq N<s_{1+s_n}-s_{s_{n-1}}$ for all positive integers $n$ [/list]

2015 NIMO Summer Contest, 15

Tags: funky , system of equations , algebra

Suppose $x$ and $y$ are real numbers such that \[x^2+xy+y^2=2\qquad\text{and}\qquad x^2-y^2=\sqrt5.\] The sum of all possible distinct values of $|x|$ can be written in the form $\textstyle\sum_{i=1}^n\sqrt{a_i}$, where each of the $a_i$ is a rational number. If $\textstyle\sum_{i=1}^na_i=\frac mn$ where $m$ and $n$ are positive realtively prime integers, what is $100m+n$? [i] Proposed by David Altizio [/i]

1998 All-Russian Olympiad Regional Round, 8.2

Tags: parallelogram , trigonometry , geometry

Given a parallelogram ABCD, let M and N be the midpoints of the sides BC and CD. Can the lines AM, AN divide the angle BAD into three equal angles?

2025 Macedonian Mathematical Olympiad, Problem 4

Tags: number theory

Let $P(x)=a x^{75}+b$ be a polynomial where $a$ and $b$ are coprime integers in the set $\{1,2,\dots,151\}$, and suppose it satisfies the following condition: there exists at most one prime $p$ such that for every positive integer $k$, $p\mid P(k)$. Prove that for every prime $q \neq p$ there exists a positive integer $k$ for which $q^2 \mid P(k).$

2006 All-Russian Olympiad Regional Round, 11.2

Tags: polynomial , trinomial , algebra

Product of square trinomials $x^2 - a_1x + b_1$, $x^2 - a_2x + b_2$, $...$, $x^2-a_nx + b_n$ is equal to the polynomial $P(x) = x^{2n} +c_1x^{2n-1} +c_2x^{2n-2} +...+ c_{2n-1}x + c_{2n}$, where the coefficients are $c_1$, $c_2$, $...$ , $c_{2n}$ are positive. Show that for some $k$ ($1\le k \le n$) the coefficients $a_k$ and $b_k$ are positive.

2023 MOAA, 8

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In the coordinate plane, Yifan the Yak starts at $(0,0)$ and makes $11$ moves. In a move, Yifan can either do nothing or move from an arbitrary point $(i,j)$ to $(i+1,j)$, $(i,j+1)$ or $(i+1,j+1)$. How many points $(x,y)$ with integer coordinates exist such that the number of ways Yifan can end on $(x,y)$ is odd? [i]Proposed by Yifan Kang[/i]

2023 Belarusian National Olympiad, 8.3

Tags: geometry

In the triangle $ABC$ points $M$ and $N$ are the midpoints of sides $AC$ and $AB$ respectively. $I$ is the incenter of the triangle. It is known that the angle $MIC$ is a right angle. Find the angle $NIB$.

2017 Morocco TST-, 2

Tags: binary , string , combinatorics

The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?

Denmark (Mohr) - geometry, 2006.1

Tags: geometry , area

The star shown is symmetric with respect to each of the six diagonals shown. All segments connecting the points $A_1, A_2, . . . , A_6$ with the centre of the star have the length $1$, and all the angles at $B_1, B_2, . . . , B_6$ indicated in the figure are right angles. Calculate the area of the star. [img]https://1.bp.blogspot.com/-Rso2aWGUq_k/XzcAm4BkAvI/AAAAAAAAMW0/277afcqTfCgZOHshf_6ce2XpinWWR4SZACLcBGAsYHQ/s0/2006%2BMohr%2Bp1.png[/img]

MOAA Team Rounds, 2022.10

Tags: number theory , combinatorics

Three integers $A, B, C$ are written on a whiteboard. Every move, Mr. Doba can either subtract $1$ from all numbers on the board, or choose two numbers on the board and subtract $1$ from both of them whilst leaving the third untouched. For how many ordered triples $(A, B, C)$ with $1 \le A < B < C\le 20$ is it possible for Mr. Doba to turn all three of the numbers on the board to $0$?

1988 China Team Selection Test, 3

Tags: vector , lattice , geometric transformation , geometry , analytic geometry , combinatorics

A polygon $\prod$ is given in the $OXY$ plane and its area exceeds $n.$ Prove that there exist $n+1$ points $P_{1}(x_1, y_1), P_{2}(x_2, y_2), \ldots, P_{n+1}(x_{n+1}, y_{n+1})$ in $\prod$ such that $\forall i,j \in \{1, 2, \ldots, n+1\}$, $x_j - x_i$ and $y_j - y_i$ are all integers.

2010 Korea - Final Round, 2

Tags: circumcircle , geometry , incenter

Let $ I$ be the incentre and $ O$ the circumcentre of a given acute triangle $ ABC$. The incircle is tangent to $ BC$ at $ D$. Assume that $ \angle B < \angle C$ and the segments $ AO$ and $ HD$ are parallel, where $H$ is the orthocentre of triangle $ABC$. Let the intersection of the line $ OD$ and $ AH$ be $ E$. If the midpoint of $ CI$ is $ F$, prove that $ E,F,I,O$ are concyclic.